Content-Type: multipart/mixed; boundary="-------------0110200752528" This is a multi-part message in MIME format. ---------------0110200752528 Content-Type: text/plain; name="01-390.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-390.comments" 393 pages (including the Mathematica notebook of appendixA not reported here): my PHD-thesis ---------------0110200752528 Content-Type: text/plain; name="01-390.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-390.keywords" information, entropy ---------------0110200752528 Content-Type: application/x-tex; name="tesi.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="tesi.tex" \documentclass{report} \title{Algorithmic Information Theoretic Issues in Quantum Mechanics} \author{Gavriel Segre - PHD thesis} \usepackage{amsmath,amssymb,graphicx} \numberwithin{equation}{section} \newtheorem{definition}{DEFINITION}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{conjecture}{Conjecture}[section] \newtheorem{remark}{Remark}[section] \newtheorem{axiom}{AXIOM}[section] \newtheorem{constraint}{CONSTRAINT}[section] \newtheorem{diagram}{DIAGRAM}[section] \newenvironment{hypothesis}{HP: \begin{center}} {\end{center}} \newenvironment{thesis}{TH: \begin{center}} {\end{center}} \newenvironment{proof}{\begin{center}PROOF: \end{center}} {$ \blacksquare $} \newtheorem{example}{Example}[section] \begin{document} \maketitle \tableofcontents \part{Preliminaries} \section{Warning} The project of this work, going beyond the possibility of realization (at least mine) during doctoral studies, has not been completed and has to be considered as \textbf{open}. To underline the intellectual path I tried to pursue, I have conserved the title and mentioning to the unrealized sections instead of eliminating them, thinking that they add anyway some cbit of classical information. \section{Acknowledgments} First of all I would like to thank my PHD-tutor, prof. Rimini, for having believed I was worth to be allowed to follow my own way and for many encouragements and skills. \smallskip Then I would like to thank doct. Benatti for having read, analyzed and often constructively criticized many parts of this work, and for having taught me so many things that it is difficult to me to mention all of them. In particular I am grateful to him for clarifying me the key point why some diagonalization-proofs don't generalize noncommutatively. \smallskip I then want strongly to thank prof. Jona-Lasinio for many invaluable remarks and teachings and, in particular, for having suggested me the idea of considering sequences of free coin tosses in view of Voiculescu's Central Limit Theorem. \smallskip Then I want to thank prof. Rasetti who introduced me to Word Problems, as well as to the Turing Barrier Problem of Quantum Mechanics and all the related issues. \smallskip I am very grateful to prof. Van Lambalgen for having given me a copy of his wonderful dissertation. \smallskip Then I would like to thank doct. Winter for having suggested me to partecipate to the Second Bielefeld Workshop on Quantum Information and Computation, giving me the motivations to pursue walking on my way. \smallskip Last but not least, I would like to thank prof. Odifreddi for many useful suggestions. \newpage \section{Notation} \bigskip \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $ \forall $ & for all (universal quantificator) \\ $ \exists $ & exist (existential quantificator) \\ $ \exists \; ! $ & exist and is unique \\ $ x \; = \; y $ & x is equal to y \\ $ x \; := \; y $ & x is defined as y \\ $ 2^{S} $ & power-set of the set S \\ $ S^{0} $ & interior of the topological space S \\ $ \bar{S} $ & closure of the topological space S \\ $ {\mathcal{D}} ( A_{1} \, , \, A_{2} ) $ & description methods of $ A_{2} $ through $ A_{1} $ \\ $ {\mathcal{R}} $ & universe of description \\ $ \Sigma $ & binary alphabet $ \{ 0 , 1 \} $ \\ $ \Sigma_{n} $ & n-letters' alphabet \\ $ \Sigma^{\star} $ & strings on the alphabet $ \Sigma $ \\ $ \Sigma^{\infty} $ & sequences on the alphabet $ \Sigma $ \\ $ \vec{x} $ & string \\ $ \lambda $ & empty string \\ $ | \vec{x} | $ & length of the string $ \vec{x} $ \\ $ string(n) $ & $ n^{th} $ string in quasilexicographic order \\ $ |n | $ & length of the $ n^{th} $ string in quasilexicographic order \\ $ \bar{x} $ & sequence \\ $ <_{p} $ & prefix order relation \\ $ \cdot $ & concatenation operator \\ $ x_{n} $ & $ n^{th} $ digit of the string $\vec{x} $ or of the sequence $ \bar{x} $ \\ $ \vec{x}(n) $ & prefix of length n of the string $ \vec{x} $ or of the sequence $ \bar{x} $ \\ $ \vec{x}(n,m) $ & substring of the sequence $ \bar{x} $ obtained taking the digits from the $n^{th}$ to the $m^{th}$ \\ $ \vec{x}^{n} $ & string made of n repetitions of the string $ \vec{x} $ \\ $ \vec{x} ^{\infty} $ & sequence made of infinite repetitions of the string $ \vec{x} $ \\ $ S \, \Sigma^{ \infty } $ & sequences having the strings of S as prefixes \\ $ \vec{x} \Sigma^{\infty} $ & sequences having the string $ \vec{x} $ as prefix \\ $ N_{i}( \vec{y}) $ & number of occurence of the letter i in the string $ \vec{y} $ \\ $ N_{i}^{n} ( \bar{x} ) $ & number of successive letters i ending in position n of the sequence $ \bar{x} $ \\ $ {\mathcal{I}} ( a , n , \vec{b} ) $ & string obtained inserting the letter a at the $ n^{th} $ place of the string $ \vec{b} $ \\ $ L_{D,P} $ & average code-word length w.r.t. the code D and the distribution P \\ $ H(P) $ & Shannon's entropy of the distribution P \\ $ K( \vec{x} ) $ & simple algorithmic entropy of the string $ \vec{x} $ \\ $ I( \vec{x} ) $ & prefix algorithmic entropy of the string $ \vec{x} $ \\ $ \Sigma(n) $ & busy-beaver function \\ $ P_{U} ( \vec{x} ) $ & universal probability of the string $ \vec{x}$ w.r.t. the universal Chaitin computer U \\ $ \Omega_{U} $ & halting probability of the universal Chaitin computer U \\ \hline \end{tabular} \newpage \begin{tabular}{|c|c|} $ CHAITIN-m-RANDOM ( \Sigma^{\star}) $ & Chaitin-m-random strings \\ $ CHAITIN-RANDOM ( \Sigma^{\star}) $ & Chaitin-random strings \\ $ {\mathcal{N}} ( \bar{x} ) $ & numeric representation of the sequence $ \bar{x} $ \\ $ CHAITIN-RANDOM(\Sigma^{\infty} )$ & Martin L\"{o}f- Solovay -Chaitin-random sequences \\ $ BRUDNO-RANDOM(\Sigma^{\infty} )$ & Brudno-random sequences \\ $ q-PSEUDORANDOM ( \Sigma^{\star} , V ) $ & pseudorandom strings of level q w.r.t. V \\ $ MARTINL\ddot{O}F-q-RANDOM( \Sigma^{\star} ) $ & Martin L\"{o}f pseudorandom strings of level q \\ $ \mu-RANDOM( \Sigma^{\infty} \, , \, \delta ) $ & Martin L\"{o}f $\mu$-random sequences w.r.t. $ \delta $ \\ $ \mu-RANDOM( \Sigma^{\infty}) $ & Martin L\"{o}f $\mu$-random sequences \\ $ {\mathcal{P}} (M) $ & unary predicates over the set M \\ $ {\mathcal{P}}_{TYPICAL} (CPS) $ & typical properties of CPS \\ $ {\mathcal{L}}_{RANDOMNESS} (CPS) $ & laws of randomness of CPS \\ $ P-CONF-RANDOM( \Sigma^{\infty} ) $ & conformistically-random sequences w.r.t. P \\ $ CONF-RANDOM( \Sigma^{\infty} ) $ & conformistically-random sequences \\ $ EXT[S] $ & subsequence extraction function w.r.t. S \\ $ CHURCH-RANDOM(\Sigma^{\infty} )$ & Church-random sequences \\ $ A \, \bigvee \, B $ & coarsest refinement of the partitions A and B \\ $ h_{CDS} $ & Kolmogorov-Sinai entropy of CDS \\ $ {\mathbb{N}} $ & natural numbers \\ $ {\mathbb{Z}} $ & integers numbers \\ $ {\mathbb{A}}_{n} $ & algebraic numbers of order n \\ $ {\mathbb{A}} $ & algebraic numbers \\ $ {\mathbb{Q}} $ & rational numbers \\ $ {\mathbb{R}} $ & real numbers \\ $ {\mathbb{C}} $ & complex numbers \\ $ {\mathbb{C}}P^{n} $ & complex projective space of order n \\ $ G_{k,n} ({\mathbb{C}}) $ & complex Grassmann manifold of order (k,n) \\ $ \aleph_{n} $ & $ (n+1)^{th} $ infinite cardinal \\ $ \Re(z) $ & real part of the complex number z \\ $ \Im(z) $ & imaginary part of the complex number z \\ $ f_{1} \, \stackrel{ + }{\leq} \, f_{2} $ & $ f_{1} $ is additively less or equal to $ f_{2} $ \\ $ f_{1} \, \stackrel{ + }{=} \, f_{2} $ & $ f_{1} $ is additively equal to $ f_{2} $ \\ $ f_{1} \, \stackrel{ \times }{\leq} \, f_{2} $ & $ f_{1} $ is multiplicatively less or equal to $ f_{2} $ \\ $ f_{1} \, \stackrel { \times }{=} \, f_{2} $ & $ f_{1} $ is multiplicatively equal to $ f_{2} $ \\ $ a \, | \, b $ & a divides b \\ $ a \, \nmid \, b $ & a does not divide b \\ gcd(a,b) & greatest common divisor of (a,b) \\ lcm(a,b) & least common multiple of (a,b) \\ $ \lfloor x \rfloor $ & floor of x: the greater integer less than or equal to x \\ $ \lceil x \rceil $ & ceiling of x: the least integer greater than or equal to x \\ $ x \; mod \, n $ & remainder: $ x - n \lfloor \frac{x}{n} \rfloor $ \\ $ MAP(A,B) $ & maps from A to B \\ $ f : A \mapsto B $ & map from A to B \\ $ \stackrel{ \circ } {MAP}(A,B) $ & partial maps from A to B \\ \hline \end{tabular} \newpage \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $ f : A \stackrel{ \circ } { \mapsto } B $ & partial map from A to B \\ $ C_{M} \, ( NC_{M} ) $ & mathematically-classical (mathematically-nonclassical) \\ $ C_{\Phi} \, ( NC_{\Phi} ) $ & physically-classical (physically-nonclassical) \\ $ REC $ & recursive \\ $ \Delta_{0}^{0} $ & computable \\ $ \varphi_{e}^{(n)} $ & n-ary partial recursive function with G\"{o}del number e \\ $ {\mathcal{W}}_{e}^{(n)} $ & halting set of $ \varphi_{e}^{(n)} $ \\ $ {\mathcal{L}}({\mathcal{H}}) $ & lattice of all the closed linear subspaces of the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{O}} ({\mathcal{H}} ) $ & linear operators on the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{B}}({\mathcal{H}}) $ & bounded linear operators on the Hilbert space $ {\mathcal{H}} $ \\ $ \| \cdot \|_{n} $ & $ n^{th} $ operatorial norm on $ {\mathcal{B}}({\mathcal{H}}) $ \\ $ | a | $ & modulus of the bounded operator a \\ $ {\mathcal{C}}_{n} ({\mathcal{H}}) $ & n-class bounded operators on the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{C}}_{1} ({\mathcal{H}}) $ & trace-class bounded operators on the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{C}}_{2} ({\mathcal{H}}) $ & Hilbert-Schmidt bounded operators on the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{C}} ({\mathcal{H}}) $ & noncommutative infinitesimals on the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{I}}_{\alpha} ({\mathcal{H}}) $ & noncommutative infinitesimals of order $ \alpha $ on the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{D}}(M) $ & classical probability distributions over M \\ $ {\mathcal{D}} ({\mathcal{H}}) $ & density operators over the Hilbert space $ {\mathcal{H}} $ \\ $ D_{T} ( \vec{p}^{(A)} , \vec{p}^{(B)} ) $ & classical trace distance among $ \vec{p}^{(A)} $ and $ \vec{p}^{(B)} $ \\ $ D_{T} ( \rho^{(A)} , \rho^{(B)} ) $ & quantum trace distance among $ \rho^{(A)} $ and $ \rho^{(B)} $ \\ $ F ( \vec{p}^{(A)} , \vec{p}^{(B)} ) $ & classical fidelity among $ \vec{p}^{(A)} $ and $ \vec{p}^{(B)} $ \\ $ F ( \rho^{(A)} , \rho^{(B)} ) $ & quantum fidelity among $ \rho^{(A)} $ and $ \rho^{(B)} $ \\ $ D_{A} ( \vec{p}^{(A)} , \vec{p}^{(B)} ) $ & classical angle distance among $ \vec{p}^{(A)} $ and $ \vec{p}^{(B)} $ \\ $ D_{A} ( \rho^{(A)} , \rho^{(B)} ) $ & quantum angle distance among $ \rho^{(A)} $ and $ \rho^{(B)} $ \\ $ A_{+} $ & positive part of the $ \star $-algebra A \\ $ A_{p.s.d} $ & part with discrete spectrum of the $ C^{\star} $-algebra A \\ $ A_{sa} $ & self-adjoint part of the $ \star $-algebra A \\ $ PUR( \rho \, , \, {\mathcal{H}} ) $ & purifications of the density operator $ \rho $ w.r.t. the Hilbert space $ {\mathcal{H}} $ \\ $ {\mathcal{U}}(A) $ & unitary group of the $ \star $-algebra A \\ $ {\mathcal{P}}(A) $ & projections of the $ \star $-algebra A \\ $ QL(A) $ & quantum logic of the $W^{\star} $-algebra A \\ $ S(A) $ & states on the $ W^{\star} $-algebra A \\ $ S(A)_{n} $ & normal states on the $ W^{\star} $-algebra A \\ $ \rho_{\omega} $ & density operator of the normal state $ \omega $ \\ $ \omega_{\mu} $ & state of the classical probability measure $ \mu $ \\ $ \Xi(A) $ & pure states on the $ W^{\star} $-algebra A \\ $ POINTS(A) $ & points on the $ W^{\star} $-algebra A \\ $ Sp(a) $ & spectrum of a \\ $ S_{\tau} ( A ) $ & $ \tau $ - invariant states on the $ W^{\star} $-algebra A \\ $ \Xi_{\tau} ( A ) $ & $ \tau $ - invariant pure states on the $ W^{\star} $-algebra A \\ $ \Delta_{\omega}(a) $ & dispersion of the state $ \omega $ on the element a of the $ W^{\star} $-algebra A \\ $ ( \mathcal{H}_{\omega} \, , \, \pi_{\omega} \, , \, | \Omega_{\omega} > ) $ & GNS-representation of the $C^{\star}$-algebra A w.r.t. the state $ \omega $ \\ $ AUT(A) $ & automorphisms of the $ W^{\star} $-algebra A \\ $ INN(A) $ & inner automorphisms of the $ W^{\star} $-algebra A \\ $ OUT(A) $ & outer automorphisms of the $ W^{\star} $-algebra A \\ \hline \end{tabular} \newpage \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $ GR-AUT( G \, , \, A) $ & automorphisms' groups of A representing G \\ $ GR-INN( G \, , \, A) $ & inner automorphisms' groups of A representing G \\ $ GR-OUT( G \, , \, A) $ & outer automorphisms' groups of A representing G \\ $ CPU(A,B) $ & channels from the $ W^{\star} $-algebra A to the $ W^{\star} $-algebra B \\ $ CPU(A) $ & channels on the $ W^{\star} $-algebra A \\ $ \alpha_{\star} $ & dual of the channel $ \alpha $ \\ $ OPU(A) $ & operational partition of unity over the $ W^{\star} $-algebra A \\ $ \{ \alpha_{i}( {\mathcal{V}}) \} $ & channels' set of the operational partition of unity $ {\mathcal{V}} $ \\ $ R ( {\mathcal{V}} ) $ & reduction channel of the operational partition of unity $ {\mathcal{V}} $ \\ $ A' $ & commutant of the Von Neumann algebra A \\ $ {\mathcal{Z}}(A) $ & centre of the Von Neumann algebra A \\ R & hyperfinite $ II_{1} $-type factor \\ $ R_{\lambda} $ & hyperfinite $ III_{\lambda} $-type factor $ ( 0 \, < \, \lambda \, < \, 1 ) $ \\ $ cardinality_{NC} (A) $ & noncommutative cardinality of the noncommutative set A \\ $ \Sigma_{NC} $ & noncommutative binary alphabet \\ $ \Sigma_{NC}^{\star} $ & noncommutative space of qubits' strings \\ $ \Sigma_{NC}^{\infty} $ & noncommutative space of qubits' sequences \\ $ M_{n} (a) $ & $ n^{th} $ moment of the algebraic random variable a \\ $ E (a) $ & expectation value of the algebraic random variable a \\ $ Var(a) $ & variance of the algebraic random variable a \\ $ Z_{a} $ & characteristic function of the algebraic random variable a \\ $ \mu_{a} $ & classical probability measure of the self-adjoint algebraic random variable a \\ $ v_{a} $ & result of the measurement of the self-adjoint algebraic random variable a \\ $ ZQ_{a}(t) $ & characteristic function of the noncommutative random variable a \\ $ ZC_{Ap}(t) $ & characteristic function of the classical approximation Ap \\ $ END( \, A \, , \, \omega \, ) $ & endomorphisms of the algebraic probability space $ ( \, A \, , \, \omega \, ) $ \\ $ AUT( \, A \, , \, \omega \, ) $ & automorphism of the algebraic probability space $ ( \, A \, , \, \omega \, ) $ \\ $ tr_{\omega} $ & Dixmier trace \\ $ O(n) $ & $ n^{th} $ orthogonal group \\ $ Spin(n) $ & $ n^{th} $ spin group \\ $ \nabla $ & Levi-Civita connection of the (pseudo)riemannian manifold $ ( M \, , \, g ) $ \\ $ \triangle_{g} $ & Laplace-Beltrami operator on the (pseudo)riemannian manifold $ ( M \, , \, g ) $ \\ $ d \mu (g) $ & metric measure of the (pseudo)riemannian manifold $ ( M \, , \, g ) $ \\ $ Is[ ( M \, , \, g )] $ & isometries' group of the (pseudo)riemannian manifold $ ( M \, , \, g ) $ \\ $ \Gamma ( M , E ) $ & sections of the fibre bundle $ E \, \stackrel{\pi}{\rightarrow} \, M $ \\ $ O(M) \, \stackrel{\pi}{\rightarrow} \, M $ & orthonormal frame bundle of the riemannian manifold $ ( M \, , \, g ) $ \\ $ S(M) \, \stackrel{\pi}{\rightarrow} \, M $ & spin bundle on the spin manifold $ ( M \, , \, g ) $ \\ $ C(M) \, \stackrel{\pi}{\rightarrow} \, M $ & Clifford bundle on the spin manifold $ ( M \, , \, g ) $ \\ $ \int_{NC} $ & noncommutative integral on the spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ \\ $ d_{NC} $ & noncommutative differential on the spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ \\ $ J[ p(z) ] $ & Julia set of the polynomial p(z) on the complex field \\ $ d \Lambda_{D} $ & Hausdorff measure on a set with Hausdorff dimension D \\ $ {\mathcal{M}} $ & Mandelbrot's set \\ $ d ( \omega_{1} , \omega_{2} ) $ & noncommutative geodesic distance between two states \\ $ ( \, D \omega_{1} \, : \, D \omega_{1} \, )_{t} $ & noncommutative Radon-Nikodym derivative between two states \\ $ \sigma^{\omega}_{t} $ & modular group of the state $ \omega $ \\ $ < \chi \, | \, {\mathcal{R}} > $ & presentation with generating system $ \chi $ and defining relators $ {\mathcal{R}} $ \\ \hline \end{tabular} \newpage \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $ F_{n} $ & free group of rank n \\ $ l^{2}(G) $ & Hilbert space of the discrete group G \\ $ {\mathcal{L}} (G) $ & left group Von Neumann algebra of the discrete group G \\ $ {\mathcal{R}} (G) $ & right group Von Neumann algebra of the discrete group G \\ $ C_{n} $ & $ n^{th} $ Catalan number \\ $ CSM ( M , N ) $ & classical statistical model w.r.t M and N \\ $ g_{CSM ( M , N )} $ & Fisher-Rao riemannian metric w.r.t. CSM ( M , N ) \\ $ \hat{\Pi}_{random} $ & Coleman-Lesniewski's operator \\ $ I_{Q} ( | \psi > ) $ & Svozil's quantum algorithmic information of $ | \psi > $ w.r.t. Q \\ $ {\mathcal{P}}_{C} (A) $ & commutative predicates over the $ W^{\star}$-algebra A \\ $ {\mathcal{P}}_{NC} (A) $ & noncommutative predicates over the $ W^{\star}$-algebra A \\ $ {\mathcal{P}}_{C}^{TYPICAL} (APS) $ & typical commutative properties of APS \\ $ {\mathcal{P}}_{NC}^{TYPICAL} (APS) $ & typical noncommutative properties of APS \\ $ KOLMOGOROV_{C}(APS) $ & Kolmogorov commutatively-random elements of APS \\ $ KOLMOGOROV_{NC}(APS) $ & Kolmogorov noncommutatively-random elements of APS \\ $ {\mathcal{L}}_{RANDOMNESS}^{C} (APS) $ & commutative laws of randomness of APS \\ $ {\mathcal{L}}_{RANDOMNESS}^{C} (APS) $ & noncommutative laws of randomness of APS \\ $ RANDOM( \Sigma_{NC}^{\infty} ) $ & random sequences of qubits \\ $ [ G \, , \, G ] $ & commutator subgroup of the group G \\ $ G_{1} \, \star \, G_{2} $ & free product of the groups $ G_{1} $ and $ G_{2} $ \\ $ A_{1} \, \star \, A_{2} $ & free product of the algebraic spaces $ A_{1} $ and $ A_{2} $ \\ $ ( A_{1} , \omega_{1} ) \, \star \, ( A_{2} , \omega_{2} ) $ & free product of $ ( A_{1} , \omega_{1} ) $ and $ ( A_{2} , \omega_{2} ) $ \\ ENSEMBLE[ CPS \, , \, n ] & ensemble of random matrices of order n w.r.t. CPS \\ $ \mu_{emp} (a) $ & empirical eigenvalue distribution of the random matrix a \\ $ \mu_{mean} (a) $ & mean eigenvalue distribution of the random matrix a \\ $ S_{Araki} ( \omega_{1} \, , \, \omega_{2} ) $ & Araki's relative entropy of $ \omega_{1} $ w.r.t. $ \omega_{2} $ \\ $ S( \omega ) $ & entropy of the state $ \omega $ \\ $ H_{\omega}(A) $ & entropy of the sub-$W^{\star}$-algebra A w.r.t. the state $ \omega $ \\ $ H_{\omega}(\alpha) $ & entropy of the the channel $ \alpha $ w.r.t. the state $ \omega $ \\ $ I( \omega \, ; \, \alpha ) $ & mutual entropy of the state $ \omega $ and the channel $ \alpha $ \\ $ DEC( \omega ) $ & decompositions of the state $ \omega $ \\ $ DEC_{EXT} ( \omega ) $ & extremal decompositions of the state $ \omega $ \\ $ DEC_{\bot} ( \omega ) $ & orthogonal decompositions of the state $ \omega $ \\ $ DEC_{Schatten} ( \omega ) $ & Schatten's decompositions of the normal state $ \omega $ \\ $ I_{acc} ( {\mathcal{E}} ) $ & classical accessible information information of the decomposition $ {\mathcal{E}} $ \\ $ I_{Holevo} ( {\mathcal{E}} ) $ & Holevo's information of the decomposition $ {\mathcal{E}} $ \\ n-GOE & gaussian orthogonal ensemble of order n \\ n-GUE & gaussian unitary ensemble of order n \\ $ gauss(D , \vec{m} , \hat{C} ; \vec{x} ) $ & D-dimensional gaussian measure of mean $ \vec{m} $ and covariance $ \hat{C} $ \\ $ gauss_{STANDARD} $ & standard gaussian measure \\ $ sc(m ,r ; x ) $ & semi-circle measure of mean m and variance $ \frac{r^{2}}{4} $ \\ $ sc_{STANDARD} $ & standard semi-circle measure \\ \hline \end{tabular} \newpage \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $ S_{Bennett} (P) $ & Bennett's entropy of the distribution P \\ $ S_{Zurek} (\rho) $ & Zurek's entropy of the density matrix $ \rho $ \\ $ S_{therm} ( \omega ) $ & thermodynamical entropy of the state $ \omega $ \\ $ S_{\text{double approach}} ( \omega ) $ & double approach entropy of the state $ \omega $ \\ $ I_{skew} ( \rho , a ) $ & skew information of the density matrix $ \rho $ w.r.t. the operator a \\ $ REC_{O} $ & recursivity in the oracle O \\ $ f \, \leq_{T} \, g $ & f is Turing reducible to g \\ $ f \, \sim_{T} \, g $ & f is Turing equivalent to g \\ $ ( {\mathcal{D}}_{T} \, , \, \leq_{T} ) $ & Turing degrees \\ $ I^{-}(S) $ & chronological past of the space-time's region's S \\ $ I^{+}(S) $ & chronological future of the space-time's region's S \\ $ J^{-}(S) $ & causal past of the space-time's region's S \\ $ J^{+}(S) $ & causal future of the space-time's region's S \\ $ D^{-}(S) $ & past domain of dependence of the space-time's region's S \\ $ D^{+}(S) $ & future domain of dependence of the space-time's region's S \\ $ D(S) $ & domain of dependence of the space-time's region's S \\ $ \approx_{Dirac} $ & weak equality \\ $ [ a \, , \, b ]_{EFF} $ & effective commutator of a and b \\ $ I^{(EFF)}_{skew} ( \rho \, , \, a ) $ & effective skew-information of $ \rho $ w.r.t. a \\ $ r-\lim_{n \rightarrow \infty} $ & recursive limit for $ n \, \rightarrow \, \infty $ \\ $ REC_{Nielsen} (A) $ & Nielsen-computable part of the $ W^{\star}$-algebra A \\ $ COMP-ST(B) $ & computability structures on the Banach space B \\ $ REC_{Pour \; El} (B , {\mathcal{S}}) $ & Pour-El computable vectors of B w.r.t. $ {\mathcal{S}} $ \\ $ REC_{Pour \; El}-{\mathcal{O}}( {\mathcal{H}}) $ & effectively determined linear operators on $ {\mathcal{H}} $ w.r.t. $ {\mathcal{S}} $ \\ $ Bloch ( \vec{r} ) $ & one qubit density operator w.r.t. the Bloch-sphere's vector $ \vec{r}$ \\ L(G) & language generated by the Chomsky's grammar G \\ FIN & finite languages \\ REG & regular languages \\ LIN & linear languages \\ CF & context-free languages \\ CS & context-sensitive languages \\ RE & recursively enumerable languages \\ IGUS & information gathering and using system \\ PRG & pseudorandom number generator \\ r.e. & recursively enumerable \\ w.r.t. & with respect to \\ l.h.s. & left-hand side \\ r.h.s. & right-hand side \\ iff & if and only if \\ i.e. & id est \\ e.g. & exempli gratia \\ \hline \end{tabular} \newpage \section{Introduction} \label{sec:Introduction} The new exciting research field of Quantum Computation has opened a cross-fertilization area among Theoretical Physics and Theoretical Computer Science that, beside the intrinsic technological difficulties in the physical implementation essentially owed to a not-sufficient technological ability in contrasting decoherence, is expected to be a strategic point for developments in both fields \cite{Preskill-98}, \cite{Nielsen-Chuang-00}, \cite{Barndorff-Nielsen-Gill-Jupp-01}, \cite{Gruska-01}. From the other side Quantum Computation Theory, concerning the algorithmic evolution of quantum information, may be seen as a sub-discipline of Quantum Information Theory, a research field that, in spite of its recent exciting developments, is a very older object of investigation in Mathematical Physics \cite{Ohya-Petz-93}, \cite{Ingarden-Kossakowski-Ohya-97}. From a foundational perspective the first natural question is: \begin{center} \textbf{What is quantum information?} \end{center} \smallskip Such an innocent question is, surprisingly, still open. \smallskip The more reasonable way of proceeding to answer this question could consist in following the same footsteps Classical Information Theory undertook to become a well-established mathematical theory \cite{Khinchin-57}, \cite{Billingsley-65}, \cite{Ihara-93},\cite{Kakihara-99} of common engeneering application \cite{Cover-Thomas-91}. Though the invention of Information Theory must be tributed with no doubt to Claude E. Shannon's 1948 paper \textit{"The Mathematical Theory of Communication"} \cite{Shannon-Weaver-71} the mathematical foundation of the concept of \textbf{classical information} was given, among the fifthies and the sixties, by the great mathematician Andrei Nicolaevich Kolmogorov \cite{Shiryayev-94}, \cite{AMS-LMS-00}. \medskip He observed that there exist three conceptually different ways of approaching the problem of defining the notion of \textbf{amount of information}: \begin{enumerate} \item the \textbf{combinatorial approach} \item the \textbf{probabilistic approach} \item the \textbf{algorithmic approach} \end{enumerate} \medskip The \textbf{combinatorial approach} furnishes a definition of the information content of an object that is \textbf{contextual}, i.e. depends from the particular context (collection of objects) in which such an object is considered, but is \textbf{weight independent}, i.e. it doesn't depend on the specification of a way of weighting the contribution of different elements of such context. So, given an object x belonging to a set X of N elements (the context) the \textbf{combinatorial approach}, invented by R. Hartley in 1928, defines the amount of information of x simply as: \begin{equation} I_{combinatorial} (x) \; := \; \log_{2} N \end{equation} Let us suppose, for example that x is a n-letter word in an alphabet of s letters containing $ m_{i} $ occurences of the $ i^{th} $ letter ($ m_{1} + \cdots + m_{s} \, = \, n $). Since there are: \begin{equation} C( m_{1}, \cdots , m_{s}) \; := \; \frac{n !}{ m_{1} ! \cdots m_{s} ! } \end{equation} words of this kind, one has that: \begin{equation} I_{combinatorial} (x) \; = \; \log_{2} C( m_{1}, \cdots , m_{s}) \end{equation} As $ n, m_{1} , \cdots m_{s} $ tend to infinity, Stirling's asymptotic formula implies that: \begin{equation} \label{eq:asymptotic combinatorial information} I_{combinatorial} (x) \; \sim \; \sum_{i=1}^{s} \frac{ m_{i} }{n} \log_{2} \frac{ m_{i} }{n} \end{equation} \medskip The \textbf{probabilistic approach} furnishes a definition of the information content of an object that is both \textbf{contextual} and \textbf{weight dependent}. So given an object x belonging to a set X of N elements (the context) such the the $ i^{th} $ element is considered with weight (probability) $ p_{i} $ , the \textbf{probabilistic approach}, that invented by Shannon in 1948 and often considered as Classical Information Theory tout court, defines the amount of information of x simply as: \begin{equation} \label{eq:probabilistic information} I_{probabilistic} (x) \; := \; - \sum_{i=1}^{N} p_{i} \log_{2} p_{i} \end{equation} It must be noted, at this point, that the asymptotic formula eq.\ref{eq:asymptotic combinatorial information} can be obtained, in the probabilistic approach by eq.\ref{eq:probabilistic information} applying the Law of Large Numbers. Anyway, at this point, Kolmogorov underlines the importance that such a result can be obtained getting rid of the \textbf{weight dependence}, observing that it is precisely what he guaranteed for other two important notions he introduced time before, namely the \textbf{$\epsilon$-entropy} $H_{\epsilon}(K) $ and the \textbf{$\epsilon$-capacity} of compact classes of functions describing, respectively, the amount of information necessary for distinguishing some individual function in the class of functions K and the amount of information that can be coded by elements of K under the condition that elements of K no closer than $ \epsilon $ to each other can be reliably distinguished. In the same way Kolmogorov stressed the importance of getting-rid of the \textbf{context dependence}, i.e. to find an \textbf{intrinsic} notion of the amount of information of an object. This led him to introduce the \textbf{algorithmic approach} that is, indeed, both \textbf{weight independent} and \textbf{context independent}: in the \textbf{algorithmic approach} the amount of information of an object x with respect to a given computer C is defined as the length of the shortest program for C computing (i.e. algorithmically-describing ) x: \begin{equation} \label{eq:naife algorithmic information} I_{algorithmic} ( C ; x) \; := \; \begin{cases} \min \{ length(p) , C(p) = x \} & \text{if x is computable by the computer C}, \\ + \infty & \text{otherwise}. \end{cases} \end{equation} The independence of this notion from the particular computer C is then established by Kolmogorov through the proof of the so called \textbf{Invariance Theorem} certifying the existence of \textbf{optimal computers}, i.e. of computers that, up to an object-independent constant, give algorithmic-descriptions always shorter of those given by any other computer. \smallskip Kolmogorov, the father of the usual, standard, measure-theoretic axiomatization, stressed from the beginning the conceptual importance of such an \textbf{intrinsic} definition of the informational-amount of an object for the same Foundation of Probability Theory, i.e. for the explanation why Probability Theory applies to reality. The key point is that the \textbf{intrinsic nature} of the algorithmic definition of information allows to address the issue of giving an \textbf{intrinsic} characterization of randomness: an algorithmically-random object x is, informally speaking, an \textbf{algorithmically-incompressible} object, i.e. an object whose more concise algorithmic-description is its same assignation. So the grown up theory concerning the algorithmic approach to information, Algorithmic Information Theory from here and beyond, appeared from the beginning as the corner-stone for an alternative Algorithmic Foundation of Classical Probability Theory \cite{Chaitin-87}, \cite{Van-Lambalgen-87}, \cite{Calude-94}, \cite{Li-Vitanyi-97}. \smallskip Later, especially by the work of Cristian Calude, Gregory Chaitin and the Auckland's Center of Discrete Mathematics and Theoretical Computer Science, Algorithmic Information Theory revealed soon an even more fundamental rule in the Foundations of Mathematics, furnishing an extraordinarily clear information-theoretic explanation of the mathematical phenomenon of Incompleteness \cite{Odifreddi-89} (Chaitin's First Undecidability Theorem states that a formal system can't decide statements involving an algorithmic-information's amount higher than its own algorithmic-informational content for more than a fixed constant, implying the recursive undecidability of algorithmic randomness), defining the notion of Halting Probability codifying in optimal way all the undecidabilities of Mathematics (the knowledge of the first n cbits in the binary expansion of Chaitin's $ \Omega $ number would allow to decide all the n-cbit mathematical statements), stating precise bounds on its determination (Chaitin's Second Undecidability Theorem states that a formal system can't decide more than a finite number of digits in the binary expansion of $ \Omega $, such a result having been recentely streghtened by R.M. Solovay through the proof that, by a proper choice of the fixed Chaitin Universal Computer and considering as formal system the Zermelo-Fraenkel axiomatic system endowed with the Axiom of Choice, this finite number of digits reduces even to zero) and, last but not least, showing that Randomness is a pervasive phaenomenon in Pure Mathematics through a paradigmatical example, i.e. shelding new light on Jones and Matjasevic's proof that Hilbert's Tenth Problem (i.e. the problem of finding an algorithm deciding whether an arbitrary Diophantine equation has integer solutions) is undecidable by the proof of the existence of a one integer parameter, let's call it k, exponential diophantine equation such that to decide if it has a finite number of integer solutions is equivalent to decide the $k^{th}$ cbit of the Halting Probability \cite{Bennett-88} \cite{Chaitin-87}, \cite{Chaitin-90}, \cite{Chaitin-98}, \cite{Chaitin-99}, \cite{Chaitin-01}, \cite{Solovay-00}. \bigskip Furthermore Algorithmic Information Theory appeared soon to play a key rule in the Theory of Chaotic Dynamical Systems: in 1958 Kolmogorov introduced (only for K-systems, the generalization for arbitrary dynamical systems having being furnished later by Ya. Sinai) a notion that would have played a key rule for the solution of the problem of giving a metric classification of dynamical systems (that is the problem of finding a complete set of invariants that imply a metric isomorphism between dynamical systems): the metric entropy of a dynamical system, characterizing the maximal asymptotic rate of information obtained through a coarse-grained observation of dynamics; a dynamical system is called chaotic if it has strictly positive metric entropy (or Kolmogorov-Sinai entropy as such notion is more often called). For the link existing between probabilistic information and probabilistically expected algorithmic information it appeared, then, intuitive that the characterization of chaoticity in terms of the probabilistic approach to information should have a counterpart in terms of the algorithmic approach: a first formalization of such a link was established by A.A. Brudno \cite{Brudno-78}, \cite{Brudno-83}, \cite{Alekseev-Yakobson-1981} by the proof of a theorem (usually called Brudno's Theorem) stating that the Kolmogorov-Sinai entropy of a dynamical system is equal to the asympotic rate of \textbf{simple algorithmic information} of almost all its trajectories. The importance of such a link was later stressed with particular emphasis by Joseph Ford who advocated strongly what he called an Algorithmic Approach to Chaos Theory \cite{Ford-92}. It must be said, anyway, that since the version of classical algorithmic information giving rise to the correct characterization of classical algorithmic randomness is not \textbf{simple algorithmic entropy} but \textbf{prefix algorithmic entropy}, the Algorithmic Approach to Chaos Theory is equivalent to the usual one only in a weak sense as we will extensively discuss. \smallskip The most important reason why Algorithmic Information Theory is of physical relevance lies, anyway, in Thermodynamics. Many generations of physicists has been educated that the correct exorcism of Maxwell's demon \cite{Maxwell-71} was the Leon's Brilloiun one \cite{Brillouin-90}: the \textbf{acquisition of information} on the velocity of the molecules by the demon is responsible of the fact that the Second Law of Thermodynamics is not violated. The recent developments of the Thermodynamics of Computation \cite{Feynman-96}, \cite{Bennett-90b}, has shown, anyway, that Brillouin's exorcism doesn't work: by Landauer's Principle \cite{Landauer-90a} such an information's acquisition process may be realized in a thermodynamically reversible way. The correct exorcism was, instead proposed by Charles Bennett \cite{Bennett-90a}: it is the \textbf{erasure of information} by the demon that cannot e accomplished in a thermodynamically reversible way and is responsible of the preservation of the Second Law. This has, anyway, dramatic conseguence concerning the same Foundations of Statistical Mechanics: introducing the issue concerning the compatibility between the time-reversibility of motions'-equation and the phenomenological time-irreversibility of thermodynamics with Ludwig Boltzmann's own words: \begin{center} \textit{"If therefore we conceive of the world as an enormously large mechanical system composed of an enormously large number of atoms, which starts from a completelly ordered initial state, and even at present is still in a substantially ordered state, then we obtain consequences which actually agree with the observed facts; although this conception involves, from a purely theoretical - I might say philosophical - standpoint, certain new aspects with contradicts general thermodynamics based on a purely phenomenological viewpoint. General thermodynamics proceeds from the fact that, as far as we can tell from our experience up to now, all natural processes are irreversible. Hence according to the principles of phenomenology, the general thermodynamics of the second law is formulated in such a way that the unconditional irreversibility of all natural process is asserted as a so-called axiom, just as general physics based on a purely phenomenological standpoint asserts the unconditional divisibility of matter without limits as an axiom". From the section 89 of \cite{Boltzmann-95}} \end{center} let us observe that most of the answers it has received, such as the one authoritatively supported by Giovanni Gallavotti seeing in Lanford's Theorem a mathematical formalization and confirmation of Boltzmann's point of view that no inconsistency exists owing to the not observability of the time-scale on which reversibility manifests \cite{Gallavotti-99}, agree in a thing: the link between the thermodynamical entropy and the state of a dynamical system is given by the \textbf{probabilistic information} of that state. As it has been strongly supported by Wojciech Zurek \cite{Zurek-89}, \cite{Zurek-90a}, \cite{Zurek-90b}, \cite{Zurek-99} Bennett's exorcism ultimatively implies that in presence of a particular kind of information gathering and using systems (IGUS), also the \textbf{algorithmic information} of the state contribute to the thermodynamical entropy and has, conseguentially, to be taken into account. \bigskip As far as Quantum Information Theory is concerned, the whole Kolmogorovian analysis concerning the three possible approaches to Information Theory may be rephrased with no variation. Anyway, nowadays, while the probabilistic approach has received a massive attention, resulting in a theory developed almost as much as the classical Shannon's theory, the situation is radically different for the combinatorial as well as for the algorithmic approach where all is available is not so much more than a plethora of attempts. The algorithmic approach to Quantum Information Theory, for its context-independence as well as for its weight-independence, is of particular importance for the mathematical foundations of such a subject. \medskip The organization of this thesis is the following: \begin{itemize} \item In part\ref{part:Equivalent characterizations of classical algorithmic randomness} we review the various equivalent characterizations of classical-algorithmic randomness \item In part\ref{part:The road for quantum algorithmic randomness} the issue of formalizing the notion of quantum algorithmic randomness in the framework of Quantum Algorithmic Information Theory is analyzed \item In part\ref{part:Classical Algorithmic Information Theory of the results of quantum measurements} the complementary issue of analyzing the classical algorithmic information status of quantum-measurements' results is discussed \end{itemize} \part{Equivalent characterizations of classical algorithmic randomness} \label{part:Equivalent characterizations of classical algorithmic randomness} \newpage \chapter{Classical algorithmic randomness as classical algorithmic incompressibility} \label{chap:Classical algorithmic randomness as classical algorithmic incompressibility} \section{The distinction between mathematical-classicality and physical-classicality} \label{sec:The distinction between mathematical-classicality and physical-classicality} The attribute \textbf{classicality} is used by two different scientific communities with different meanings: \begin{itemize} \item it is usually used by Theoretical Physicists to express that some physical system obeys the laws of Classical Mechanics; this is, for example the acception of the adjective \textbf{classical} intended in the title of the first two volumes \textit{"Classical Dynamical Systems"} and \textit{"Classical Field Theory"} \cite{Thirring-97} of Walter Thirring's monography \textit{"A Course in Mathematical Physics"} \item it is usually used by logico-mathematicians to express the part of a theory concerning only mathematical objects with cardinality less or equal to $ \aleph_{0} $; this is, for example, the acception of the adjective \textbf{classical} intended in the title \textit{"Classical Recursion Theory"} of Piergiorgio Odifreddi's monography \cite{Odifreddi-89}, \cite{Odifreddi-99a} \end{itemize} Unfortunately such a double acception of the term \textbf{classical} have generated many confusions in the literature belonging to the intersection of the two disciplines. At a foundational level the generated confusion may be seen as a confusion between the \textbf{subject} and the \textbf{object} of a computational process, i.e. between the \textbf{attributes of the computational device} and the \textbf{attributes of the computed mathematical objects}. Hence some property (classicality/quantisticality i.e. commutativity/noncommutativity) is used in two undistinguished (and often interchanged) acceptions according to it refers: \begin{itemize} \item to the \textbf{subject of the computation}, i.e. to the computational device \item to the \textbf{object of the computation}, i.e. to the computed mathematical objects \end{itemize} \smallskip An elegant way of avoiding this kind of mistakes is to pursue the following prescription: any issue of Computability Theory must analyze separetely each cell of the following: \medskip \begin{diagram}\label{di:diagram of computation} \end{diagram} DIAGRAM OF COMPUTATION: \begin{tabular}{|c|c|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $ \frac{OBJECT}{SUBJECT} $ & $C_{M}$ & $NC_{M}$ \\ $C_{\Phi}$ & $\cdot_{11}$ & $\cdot_{12}$ \\ $NC_{\Phi}$ & $\cdot_{21}$ & $\cdot_{22}$ \\ \hline \end{tabular} \medskip with: \begin{description} \item[$C_{M}$ :] MATHEMATICALLY CLASSICAL \item[$NC_{M}$:] MATHEMATICALLY NONCLASSICAL \item[$C_{\Phi}$:] PHYSICALLY CLASSICAL \item[$NC_{\Phi}$:] PHYSICALLY NONCLASSICAL \end{description} \smallskip Let us consider, first of all, the following issue: \textmd{$ 1^{th}$ ISSUE: WHAT IS COMPUTABLE ?} \begin{itemize} \item $cell_{11} \; : \; C_{M} \, \cap \, C_{\Phi} $ There is complete agreement in the scientific community that, as to the computation by \textbf{physically classical computers} of the following set of functions: \begin{definition} \end{definition} MATHEMATICALLY CLASSICAL FUNCTIONS: (partial) functions on sets $ S \, : \, card(S) \, \leq \, \aleph_{0}$ \textbf{Church-Turing's Thesis} holds leading to the identification of the computable (partial) functions with the (partial) recursive functions \cite{Odifreddi-89}, \cite{Odifreddi-96} that we will now define. Introducing a notation we will adopt from here and beyond, we will denote the computability attribute relative to the cell $ cell_{i j} $ of the diagram\ref{di:diagram of computation} by the symbol $cell_{i j} - \Delta_{0}^{0} $. For example the above statement may be rephrased saying that the set $ C_{M} - C_{\Phi} - \Delta_{0}^{0} -\stackrel{ \circ } {MAP}(S,S) $ is the set of all the partial recursive functions over S. Let, clarify, first of all, what we mean by a partial function: a \textbf{total} (i.e. ordinary) $ f \, : \, A \, \rightarrow \, B $ is a rule associating to every element x of the set A an element f(x) of the set B: \begin{equation*} x \in A \; \stackrel{f}{\rightarrow} \; f(x) \in B \end{equation*} We will indicate the set of all the total functions from a set A to a set B by MAP(A,B). A \textbf{partial function} $ f \, : \, A \stackrel{\circ}{\rightarrow} \, B $ is a rule associating to each element x of a certain subset $ HALTING(f) \subseteq A $ of A, said the \textbf{halting set of f}, an element f(x) of the set B: \begin{equation*} x \in HALTING(f) \; \stackrel{f}{\rightarrow} \; f(x) \in B \end{equation*} We will say that: \begin{definition} \end{definition} f HALTS ON $ x\ \in A \; ( f(x) \downarrow ) $: \begin{equation} x \; \in \; HALTING(f) \end{equation} \begin{definition} \end{definition} f DOESN'T HALT ON $ x\ \in A \; ( f(x) \uparrow ) $: \begin{equation} x \; \notin \; HALTING(f) \end{equation} We will indicate the set of all the partial functions from a set A to a set B by $ \stackrel{\circ}{MAP}(A,B) $. Given two partial functions $ f_{1} , f_{2} \, \in \, \stackrel{\circ}{MAP}(A,B) $: \begin{definition} \end{definition} $ f_{1} $ IS EQUAL TO $ f_{2} \; ( f_{1} \, = \, f_{2} ) $: \begin{equation} ( HALTING( f_{1} ) \, = \, HALTING( f_{2} ) ) \; and \; (f_{1} (x) \, = \, f_{2}(x) \; \; \forall x \in HALTING( f_{1} )) \end{equation} The language of \textbf{partial functions} is of common use in Mathematical-Logic; we adopt it, anyway, also in unusual environments: from Classical (i.e. commutative) Measure Theory (in which measures' halting sets will be suitable $ \sigma $-algebras) to Operator Theory on Hilbert spaces (in which unbounded operators' halting sets will be dense subspaces) Denoted by $ {\mathbb{N}}^{\star} \; := \; \bigcup_{n \in {\mathbb{N}}} {\mathbb{N}}^{n} $ the set of all the n-ples of natural numbers: \begin{definition} \label{def:partial recursive functions on numbers} \end{definition} CLASS OF PARTIAL RECURSIVE FUNCTIONS (ON NUMBERS) $(REC-\stackrel{\circ} {MAP} ( {\mathbb{N}}^{n} \, , \, {\mathbb{N}} ))$ the smallest class of partial functions: \begin{enumerate} \item containing the initial functions: \begin{align} {\mathcal{O}} & (x) \; := \; 0 \\ {\mathcal{S}} & (x) \; := \; x+1 \\ {\mathcal{I}}_{i}^{n} & (x_{1}, \cdots , x_{n}) \; := \; x_{i} \; \; i=1 , \cdots , n \, , \, n \in {\mathbb{N}} \end{align} \item closed under \textbf{composition}, i.e. the schema that given $ \gamma_{1} , \cdots , \gamma_{m} , \psi $ produces: \begin{equation} \varphi ( \vec{x} ) \; := \; \psi ( \gamma_{1} ( \vec{x} ) , \cdots , \gamma_{m} ( \vec{x} ) ) \end{equation} \item closed under \textbf{primitive recursion}, i.e. the schema that given $ \psi \, , \, \gamma $ produces: \begin{align} \varphi( \vec{x} , 0 ) \; & := \; \psi ( \vec{x} ) \\ \varphi( \vec{x} , y+1 ) \; & := \; \gamma ( \vec{x} , y , \phi( \vec{x}, y )) \end{align} \item closed under \textbf{unrestricted $ \mu $-recursion}, i.e. the schema that given $ \psi $ produces: \begin{equation} \varphi( \vec{x} ) \; := \; \min \{ y \, : \, ( \psi( \vec{x} , z ) \downarrow \; \forall z \leq y ) \: and \: ( \psi( \vec{x} , y ) \, = \, 0 ) \} \end{equation} where $ \varphi( \vec{x} ) \, := \, \uparrow $ if there is no such function \end{enumerate} \smallskip The more fundamental properties of partial recursive functions may be collected in the following: \begin{theorem} \label{th:Goedel's numbering of partial recursive functions} \end{theorem} G\"{O}DEL'S NUMBERING OF PARTIAL RECURSIVE FUNCTIONS: It is possible to enumerate all partial recursive functions: \begin{equation*} \varphi_{e}^{(n)} \: : \: {\mathbb{N}}^{n} \, \stackrel{\circ}{\rightarrow} \, {\mathbb{N}} \end{equation*} (where the natural number \textbf{e} is called the \textbf{G\"{o}del's number} of the $ e^{th}$ n-ary partial recursive function) in such a way that the following conditions are satisfied: \begin{itemize} \item \textbf{Universality:} there is a partial recursive function of two variables $ \varphi_{z}^{(2)}(e,x) $ such that: \begin{equation} \varphi_{z}^{(2)}(e,x) \; = \; \varphi_{e}^{(1)}(e,x) \; \; \forall x \in {\mathbb{N}} \end{equation} \item \textbf{Uniform Composition:} there is a (total) recursive function of two variables \emph{comp} such that: \begin{equation} \varphi_{comp(x,y)}^{(1)}(z) \; = \; \varphi_{x}^{(1)}( \varphi_{y}^{(1)} (z)) \; \; \forall x,y,z \in {\mathbb{N}} \end{equation} \item \textbf{Fixed Point:} For every $ m \in {\mathbb{N}}_{+} $ and every recursive function f there effectively exists an x (called the fixed point of f) such that: \begin{equation} \varphi_{x}^{(m)} \; = \; \varphi_{f(x)}^{(m)} \end{equation} \end{itemize} \smallskip It is useful to introduce the following notation concerning the domains of partial recursive functions: \begin{definition} \label{def:domains of partial recursive functions} \end{definition} \begin{equation} {\mathcal{W}}_{e}^{n} \; := \; HALTING( \varphi_{e}^{(n)}) \; \; e,n \in {\mathbb{N}} \end{equation} Given an n-ary relation $ R( x_{1} , \cdots , x_{n} ) $ on $ {\mathbb{N}} $: \begin{definition} \label{def:r.e. relations} \end{definition} R IS RECURSIVELY ENUMERABLE (R.E.): \begin{equation} \exists e \in {\mathbb{N}} \; : \; R \, = \, {\mathcal{W}}_{e}^{n} \end{equation} We will identify, from here and beyond, \textbf{sets} and \textbf{unary relations} by posing: \begin{equation} {\mathcal{W}}_{e} \; := \; {\mathcal{W}}_{e}^{1} \end{equation} Given a set $ S \subset {\mathbb{N}}^{\star} $: \begin{definition} \label{def:recursive set} \end{definition} S IS RECURSIVE: the characteristic function $ \chi_{S} $: \begin{equation}\label{eq:characteristic function of a set} \chi_{S} (x) \; := \; \begin{cases} 1 & \text{if $ x \in S $}, \\ 0 & \text{otherwise}. \end{cases} \end{equation} is a total recursive function \smallskip Clearly one has that: \begin{theorem} \label{th:recursivity is stronger than recursive enumerability} \end{theorem} RECURSIVITY IS STRONGER THAN RECURSIVE ENUMERABILITY: \begin{align*} recursivity & \; \Rightarrow \; \text{ recursive enumerability} \\ \text{ recursive enumerability} & \; \nRightarrow \; recursivity \end{align*} \smallskip \begin{remark} \label{Godel numbering and self-reference} \end{remark} G\"{O}DEL NUMBERING AND SELF-REFERENCE G\"{o}del's numbering, introduced by Kurt G\"{o}del in his his famous 1931's paper \emph{"On Formally Undecidable Propositions of the Principia Mathematica and Related Systems"} \cite{Davis-65}, is a deep concept since it creates that link between \textbf{language} and \textbf{meta-language} giving rise to self-reference and all the consequences it generates through Cantor's Diagonalization. Since recursivity is equivalent to representability in an arbitary consistent formal system extending Tarski-Montowski-Robinson Arithmetics G\"{o}del's numbering may be equivalentely seen as a way of enumerating all the logical propositions concerning natural numbers. So one has a hierarchy of levels: \begin{enumerate} \item the \textbf{objects} of investigation, i.e. natural numbers \item the \textbf{language} by which properties of the objects are described, i.e. the logical propositions concerning the objects \item the \textbf{meta-language} by which properties of the \textbf{meta-objects}, i.e the logical propositions of the language (by which properties of the objects are described) are described; we will denote by \textbf{meta-proposition} a proposition of the \textbf{meta-language} \end{enumerate} Owing to G\"{o}del numbering, a \textbf{number} plays a double rule: \begin{itemize} \item as an \textbf{object} \item as the G\"{o}del number identifying a proposition of the language, i.e. as a \textbf{meta-object} \end{itemize} This can be used to pass from \textbf{meta-language} to the \textbf{language}, simply associating to the \textbf{meta-proposition} $ \varphi_{e} (x) $ concerning the \textbf{meta-object} x, i.e. the proposition of the language with G\"{o}del number x, the \textbf{proposition} $ \varphi_{e} (x) $ concerning the \textbf{object}, i.e. the number, x and, viceversa, to pass from \textbf{language} to \textbf{meta-language}, associating to the \textbf{proposition} $ \varphi_{e}(x) $ concerning the \textbf{object}, i.e. the number, x the \textbf{meta-proposition} $ \varphi_{e} (x) $ concerning the \textbf{meta-object} x, i.e. the proposition of the language with G\"{o}del number x. But then self-reference immediately appears since $ \varphi_{e}(e) $ happens to speak about itself. \item $cell_{21} \; : \; C_{M} \, \cap \, NC_{\Phi} $ There is no universally accepted answer in the scientific community to the question if a \textbf{physically nonclassical computer} can violate Church-Turing's Thesis, i.e. can compute non-recursive \textbf{mathematically classical functions}. In particular, as far as the computation by \textbf{physically quantistical computers} of \textbf{mathematically classical functions} is concerned, the common opinion among the researchers in Quantum Computation \cite{Feynman-99}, \cite{Deutsch-85}, \cite{Jozsa-98} is that \textbf{Nonrelativistic Quantum Mechanics} and \textbf{Special-relativistic Quantum Mechanics (Local Quantum Field Theories)} don't violate Church-Turing's Thesis. It must be cited, anyway, that the opposite thesis has been asserted by various authors (cfr. \cite{Castagnoli-Rasetti-Vincenti-92}, \cite{Mitchison-Jozsa-99}, \cite{Calude-Dinneen-Svozil-00} and the paragraphs 4.12 and 4.23 of \cite{Calude-Paun-01}) Furthermore it must be observed that in the Masanao Ozawa's final formalization of Quantum Turing Machines \cite{Ozawa-98a} ( saying according, to us, the last word on the consistence's problem of Deutsch's Halting Protocol) the satisfaction of the Church-Turing's Thesis is posed by hand restricting the range of the local transition function to recursive complex numbers. We will, anyway, extensively return on this point in section\ref{sec:The problem of characterizing mathematically the notion of a quantum algorithm} Finally, when \textbf{Generally-relativistic Quantum Mechanics} (both in the form of \textbf{Quantum Gravity} and in the form of some suggested \textbf{gravitationally-modificated Quantum Mechanics}) is considered, the whole story touches the strongly debated ideas of Roger Penrose about a non-computable alteration of the quantum unitary dynamics induced by gravity \cite{Penrose-89}, \cite{Penrose-96}, \cite{Anandan-98}, \cite{Penrose-00}. \item $cell_{12} \; : \; NC_{M} \, \cap \, C_{\Phi} $ As soon as one goes out from the boundaries of $ C_{M}$-Classical Recursion Theory the almost miracolous equivalence of all the different approaches (recursivity, finitely definability, Herbrand-G\"{o}del Computability, representability in consistent formal system extending Tarski-Montowski-Robinson Arithmetics, $ \lambda$-definability in Church's $ \lambda $- Calculus, flowchart computability, computability by Classical Turing Machines, by cellular automata \cite{Odifreddi-89}, by Shepherdson-Sturgis register machines \cite{Cutland-80}, LISP computability \cite{Mc-Carthy-60}, $ \cdots $) that in such a theory manifests the strong experimental verification of Church's Thesis, dramatically disappears. Just as to Computability Theory by \textbf{physically classical computers} of (partial) functions on sets $ S \, : \, cardinality(S) \, = \, \aleph_{1}$ while many different inequivalent candidate theories have been proposed: \begin{enumerate} \item the well extablished and almost always accepted theory named Computable Analysis, generated by the studies of Grzegorczyck - Lacombe \cite{Pour-El-Richards-89} \item the theory developed by the so called Markov School in the framework of Constructive Mathematics \cite{Odifreddi-89} \item the Blum - Shub - Smale 's Theory \cite{Smale-92}, \cite{Blum-Cucker-Shub-Smale-98} \end{enumerate} The relative popularity of the issue about the concurrence of such candidate theories is owed to Penrose's question if Mandelbrot's set is recursive \cite{Penrose-89}. We will partially analyze it in section\ref{sec:Brudno algorithmic entropy versus the Uspensky abstract approach} \item $cell_{22} \; : \; NC_{M} \, \cap \, NC_{\Phi} $ It's important to realize that, contrary to what is often claimed, Church-Turing's Thesis doesn't imply that the answer to the $ 1^{th} ISSUE $ contained in the cells $cell_{12}$ and $cell_{22}$ must be equal. For example Church-Turing's Thesis is not incompatible with an hypothetical situation in which Mandelbrot's set would be $ C_{\Phi} $ - incomputable but $ NC_{\Phi} $ - computable. Though some undecidability theorems and conjectures still exist (cfr. e.g. Lloyd's arguments concerning uncomputable diagonalizations in Quantum Computation \cite{Loyd-89} as well as his general consideration about the physical limits of Computation \cite{Lloyd-01}, or Geroch and Hartle's speculations concerning the eventuality that the recursive undecidability of the Homeomorphism-problem for four-manifolds \cite{Collins-Zieschang-98} may lead to the recursive undecidability of quantizing Gravity) no general mathematically formalization has been realized yet. Particular importance has, according to us, Karl Svozil's suggestion that in Quantum Algorirhmic Information Theory there should exist undecidability theorems analogues to the classical Chaitin's ones (cfr. the problem17 of \cite{Calude-96}). \end{itemize} \newpage \section{Uspensky's abstract definition of algorithmic information} \label{sec:Uspensky's abstract definition of algorithmic information} The last contribution Andrei Nikolaevich Kolmogorov left us before dying was his forum report \textit{Algorithms and Randomness}, made with and exposed by his student Vladimir Uspensky, at the First World Congress of the Bernoulli Society (September 8-14, 1986) \cite{AMS-LMS-00}. Later Unspensky formalized the Kolmogorovian approach to Algorithmic Information Theory in a very general and elegant way we will start from \cite{Uspensky-92}, \cite{Uspensky-Semenov-93}. \begin{definition} \label{def:aggregate} \end{definition} AGGREGATE: a couple $ ( X \, , \, R ) $ such that: \begin{itemize} \item X is a set \item R, called a \textbf{concordance relation}, is a computable binary relation on X \end{itemize} \medskip \begin{remark} \label{rem:context dependence of the computability constraint} \end{remark} CONTEXT-DEPENDENCE OF THE COMPUTABILITY CONSTRAINT Let us observe that, in the definition def.\ref{def:aggregate} we have imposed a \textbf{computability constraint} without specifying its precise mathematical meaning. This has been done in order of guaranteeing the maximal generality: in the different contextes corresponding to the different cells $ cell_{ij} $ of the diagram\ref{di:diagram of computation} such a constraint is formalized by the proper $ cell_{ij} - \Delta_{0}^{0}$ condition \bigskip Given two aggregates $ A_{1} \, := \, ( X_{1} \, , \, R_{1} ) $ and $ A_{2} \, := \, ( X_{2} \, , \, R_{2} ) $ it is natural to ask under which conditions we can think to elements of $ A_{2} $ as descriptions of elements of $ A_{1} $ with respect to a proper description mode; the answer is given by the following definition: \begin{definition} \end{definition} MODE OF DESCRIPTION (OF $ A_{2} $-ELEMENTS THROUGH $ A_{1} $-ELEMENTS): a relation R between elements of $ X_{1} $ and elements of $ X_{2} $ such that: \begin{equation}\label{eq:mode of description} R_{1}( x_{1} , y_{1}) \, and \, R_{2}( x_{2} , y_{2}) \, and \, R( x_{1} , x_{2}) \; \Rightarrow \; R( y_{1} , y_{2}) \; \; \forall x_{1} , x_{2} \in X_{1} , \forall y_{1} , y_{2} \in X_{2} \end{equation} \smallskip We will denote the set of all the mode of description of $ A_{2} $-elements through $ A_{1} $-elements by $ {\mathcal{D}}( A_{1} , A_{2} ) $. \smallskip Given a mode of description R among the aggregates $ A_{1} $ and $ A_{2} $: \begin{definition} \end{definition} $ x_{2} \in X_{2}$ IS A DESCRIPTION OF $ x_{1} \in X_{1} $ THROUGH THE MODE R: \begin{equation} R( x_{1} , x_{2} ) \end{equation} \smallskip All the ingredients introduced up to this point are of pure set-theoretic nature (with some constructibility constraint). The introduction of a notion characterizing the amount of not-redundant, i.e. algorithmically incompressible, information of an object $ x_{1} \in X_{1} $ with respect to the description mode R requires the introduction of some point measure quantifying the extension of the descriptions. Let us, then, define, the following notion: \begin{definition} \end{definition} METRIC AGGREGATE: a couple $ ( A \, , \mu ) $ such that: \begin{itemize} \item $ A \, := \, ( X \, , R ) $ is an aggregate \item $ \mu $ is a point measure on A \end{itemize} \smallskip Given a metric aggregate $ A_{1} \, := \, ( X_{1} \, , \, R_{1} \, , \, \mu ) $, an aggregate $ A_{2} \, := \, ( X_{2} \, , \, R_{2} \, ) $ and a mode of description R among the aggregates $ A_{1} $ and $ A_{2} $ we can finally introduce the following basic notion: \begin{definition} \label{def:algorithmic information} \end{definition} ALGORITHMIC INFORMATION OF $ x_{2} \in X_{2} $ W.R.T. THE DESCRIPTION MODE R: \begin{equation} I_{R} ( x_{2} ) \; := \; \begin{cases} \min \{ \mu ( x_{1} ) \, : \, R( x_{1} , x_{2} ) \} & \text{if $ \exists \, x_{1} \in X_{1} \, : R( x_{1} , x_{2} )$}, \\ + \infty & \text{otherwise}. \end{cases} \end{equation} \smallskip Clearly the definition def.\ref{def:algorithmic information} depends on the particular chosen description mode R. It is clear, anyway, that the whole consistence of Algorithmic Information Theory lies on the possibility of getting-rid of such a dependence. The formalization of this issue is given by the following notions: \begin{definition} \end{definition} UNIVERSE OF DESCRIPTION OF $ A_{2} $ THROUGH $ A_{1} $: a set $ {\mathcal{R}} $ of description modes of the aggregate $ A_{2} $ through the metric aggregate $ A_{1} $: \begin{equation} {\mathcal{R}} \; \subseteq \; {\mathcal{D}}( A_{1} , A_{2} ) \end{equation} \smallskip The intuitive idea we are going to formalize is that Algorithmic Information Theory is meaningful provided the involved universes of descriptions admit optimal mode of descriptions, i.e. mode of descriptions that are always more concise of all the others, up to an object-independent additive constant. This requires the introduction of two ordering relation we will use extensively in the whole dissertation. Given two real-valued partial function $ f_{1} : A \stackrel {\circ}{\rightarrow} {\mathbb{R}} \, , \, f_{2} : A \stackrel {\circ}{\rightarrow} {\mathbb{R}} $ we will say that: \begin{definition} \label{def:addittively less or equal} \end{definition} $ f_{1} $ IS ADDITIVELY LESS OR EQUAL TO $ f_{2} $ ( $ f_{1} \, \stackrel{ + }{\leq} \, f_{2} $ ) \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; f_{1} (x) \, \leq \, f_{2} (x) + c \; \; \forall x \in HALTING(f_{1}) \bigcap HALTING(f_{2}) \end{equation} \begin{definition} \label{def:addittively equal} \end{definition} $ f_{1} $ IS ADDITIVELY EQUAL TO $ f_{2} $ ( $ f_{1} \, \stackrel{ = }{\leq} \, f_{2} $ ) \begin{equation} f_{1} \, \stackrel{ + }{\leq} \, f_{2} \; and \; f_{2} \, \stackrel{ + }{\leq} \, f_{1} \end{equation} \begin{definition} \label{def:multiplicatively less or equal} \end{definition} $ f_{1} $ IS MULTIPLICATIVELY LESS OR EQUAL TO $ f_{2} $ ( $ f_{1} \, \stackrel { \times }{\leq} \, f_{2} $ ) \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; f_{1} (x) \, \leq \, f_{2} (x) \times c \; \; \forall x \in HALTING(f_{1}) \bigcap HALTING(f_{2}) \end{equation} \begin{definition} \label{def:multiplicatively equal} \end{definition} $ f_{1} $ IS MULTIPLICATIVELY EQUAL TO $ f_{2} $ ( $ f_{1} \, \stackrel { \times }{=} \, f_{2} $ ) \begin{equation} f_{1} \, \stackrel{ \times }{ \leq } \, f_{2} \; and \; f_{2} \, \stackrel{ \times }{\leq} \, f_{1} \end{equation} \medskip Let us now consider an aggregate $ A_{1} $, a metric aggregate $ A_{2} $, a universe of description $ {\mathcal{R}} $ of $ A_{1} $ through $ A_{2} $ and a particular mode of description belonging to such a universe $ U \in {\mathcal{R}} $. We will say that: \begin{definition} \label{def:optimal mode of description} \end{definition} U IS OPTIMAL W.R.T. $ {\mathcal{R}} $: \begin{equation} U \; \stackrel{ + }{\leq} \; f \; \; \forall f \in {\mathcal{R}} \end{equation} We can then introduce the following notion: \begin{definition} \end{definition} ALGORITHMIC INFORMATION THEORY IS MEANINGFUL W.R.T. $ {\mathcal{R}} $: \begin{equation} \exists \; U \in {\mathcal{R}} \; optimal \end{equation} Adhering to Uspensky's terminology let us introduce the following notion: \begin{definition} \end{definition} ALGORITHMIC ENTROPY: a function I equal to the algorithmic information w.r.t. a description mode that is optimal w.r.t to some universe of description modes. \medskip In order to discuss the first fundamental examples, let us introduce some basic notions. Given a set $ \Sigma $: \begin{definition} \label{def: classical strings on an alphabet} \end{definition} SET OF THE STRINGS ON $ \Sigma $ : \begin{equation} \Sigma^{\star} \; \equiv \; \{ \lambda \} \; \bigcup \; \cup_{ k \in {\mathbb{N}}} \Sigma^{k} \end{equation} \begin{definition} \label{def: classical sequences on an alphabet} \end{definition} SET OF THE SEQUENCES ON $ \Sigma $: \begin{equation} \Sigma^{\infty} \; \equiv \; \{ \lambda \} \; \bigcup \; \{ \bar{x} : {\mathbb{N}}_{+} \, \rightarrow \, \Sigma \} \end{equation} where $ \lambda $ denotes the \textit{empty string}. Given $ \vec{x} \in \Sigma^{\star} $ let us denote by $ \vec{x}^{n} \in \Sigma^{\star} $ the string made of n repetitions of $ \vec{x} $ and by $ \vec{x} ^{\infty} \in \Sigma^{\infty} $ the sequence made of infinite repetitions of $ \vec{x} $. It is important to remark that \cite{Calude-94}: \begin{theorem} \label{th:cardinalities of strings and sequences} \end{theorem} ON THE CARDINALITIES OF STRINGS AND SEQUENCES OVER A FINITE ALPHABET \begin{hypothesis} \end{hypothesis} \begin{equation*} cardinality ( \Sigma ) \; \in \; {\mathbb{N}} \end{equation*} \begin{thesis} \end{thesis} \begin{align*} cardinality(\Sigma^{\star}) \; & = \; \aleph_{0} \\ cardinality(\Sigma^{\infty}) \; & = \; \aleph_{1} \end{align*} We will assume from here and beyond that $ \Sigma \; := \; \{ 0 ,1 \}$. The total-ordering $ 0 \; < \; 1 $ induces the following: \begin{definition} \end{definition} QUASI-LEXICOGRAPHIC ORDERING ON $\Sigma^{\star}$ \begin{multline} \lambda \, < \, 0 \, < \, 1 \, < \, 00 \, < \, 01 \, < \, \\ 10 \, < \, 11 \, < \, 000 \, < \, 001 \, < \, \cdots 111 \, < \, \cdots \end{multline} We can then introduce the following bijection: \begin{definition} \end{definition} QUASI-LEXICOGRAPHIC MAP: \begin{equation} \begin{split} string : {\mathbb{N}} & \rightarrow \Sigma^{\star} \\ string(n) \, & \, = \text{the $ n^{th} $ string in quasi-lexicographic ordering} \end{split} \end{equation} \smallskip Let us now introduce the following ordering relation on $ \Sigma^{\star} $ \begin{definition} \end{definition} PREFIX-ORDER RELATION $ <_{p} $ ON $ \Sigma^{\star} $: \begin{equation} \vec{x} <_{p} \vec{y} \; := \; \exists \vec{z} \in \Sigma^{\star} \, : \; \vec{y} = \vec{x} \cdot \vec{z} \end{equation} Give a set $ S \, \subset \, \Sigma^{\star} $ we will say that: \begin{definition} \end{definition} S IS PREFIX-FREE: \begin{equation} ( \vec{x} <_{p} \vec{y} \Rightarrow \vec{x} = \vec{y} ) \, \forall \vec{x},\vec{y} \in S \end{equation} \medskip \begin{example} \label{ex:simple algorithmic entropy} \end{example} SIMPLE ALGORITHMIC ENTROPY Let us consider the case in which $ A_{1} \; = \; A_{2} \; = \; ( {\mathbb{N}} \, , \, = \, , \, | \cdot | ) $ with: \begin{equation} | n | \; := \; | string^{- 1} (n) | \; = \; \llcorner \log_{2} ( n + 1 ) \lrcorner \end{equation} Kolmogorov started considering as universe of mode of descriptions the whole $ {\mathcal{D}}( A_{1} , A_{2} ) $. But he immediately realized that: \begin{theorem} \label{th:simple algorithmic information theory w.r.t. all the partial functions is not meaningful} \end{theorem} ALGORITHMIC INFORMATION THEORY W.R.T. $ {\mathcal{D}}( A_{1} , A_{2} ) $ IS NOT MEANINGFUL \begin{proof} Following \cite{Li-Vitanyi-97} let us us suppose by abdurdum that there exist a function $ U \in \bigcap {\mathcal{D}}( A_{1} , A_{2} )$ such that: \begin{equation} U \; \stackrel{ + }{\leq} \; f \; \; \forall f \in {\mathcal{D}} ( A_{1} \, , \, A_{2} ) \end{equation} Let us then consider an infinite sequence $ X \, := \, \{ x_{n} \in {\mathbb{N}} \}_{n \in {\mathbb{N}}} $ such that: \begin{equation} i < j \; \Rightarrow \; x_{i} < x_{j} \; \, \forall i,j \in {\mathbb{N}} \end{equation} Considered a subsequence $ Y \, := \, \{ y_{n} \in {\mathbb{N}} \}_{n \in {\mathbb{N}}} $ of the sequence X such that: \begin{equation} \log y_{n} \; < \; \frac{ \log x_{n} }{2} \; \; \forall n \in {\mathbb{N}} \end{equation} let us introduce the function $ f \in {\mathcal{D}}( A_{1} , A_{2} ) $ conciding with U everywhere but for the points of the sequence X where it is defined as: \begin{equation} f( x_{n} ) \; := \; U ( y_{n} ) \; \; \forall n \in {\mathbb{N}} \end{equation} We have clearly that: \begin{equation} cardinality ( \{ n \in {\mathbb{N}} \, : I_{f} ( n ) \, \leq \, \frac{ I_{U} ( n )}{2} \} ) \; = \; \aleph_{0} \end{equation} that contradict the absurdum hypothesis \end{proof} \medskip Theorem\ref{th:simple algorithmic information theory w.r.t. all the partial functions is not meaningful} is the first of a set of theorems we will meet in this dissertation showing that certain quantities of Algorithmic Information Theory are meaningful only by effectivizing some notion. Indeed, requiring that to describe objects must be an effective property, one is led by Church-Turing's thesis, for reasons that will be clarified in the next section, to restrict the universe of modes of descriptions to the set $ C_{M}-C_{\Phi} -\Delta_{0}^{0} [ {\mathcal{D}}( A_{1} , A_{2} )] $ of the \textbf{partial recursive ones}. Kolmogorov realized that in this way the problem was overcome proving the following \cite{Calude-94}: \begin{theorem} \label{th:invariance theorem for simple algorithmic entropy} \end{theorem} INVARIANCE THEOREM FOR SIMPLE ALGORITHMIC ENTROPY: Algorithmic Information Theory w.r.t. $ C_{M}-C_{\Phi}- \Delta_{0}^{0} - [ {\mathcal{D}}( A_{1} , A_{2} )] $ is meaningful \smallskip As we have preannounced, the resulting algorithmic entropy, w.r.t. an optimal description mode that we will call from here and beyond a \textbf{simple universal computer}, is called the \textbf{simple algorithmic entropy} and is denoted by K. \medskip \begin{example} \label{ex:monotone algorithmic entropy} \end{example} MONOTONE ALGORITHMIC ENTROPY Let us consider the case in which $ A_{1} \; = \; A_{2} \; = \; ( \Sigma^{\star} \, , \, <_{p} \, , \, | \cdot | ) $. Exactly as in the example\ref{ex:simple algorithmic entropy} it may be proved that: \begin{theorem} \label{th:monotone algorithmic information theory w.r.t. all the partial functions is not meaningful} \end{theorem} ALGORITHMIC INFORMATION THEORY W.R.T. $ {\mathcal{D}}( A_{1} , A_{2} ) $ IS NOT MEANINGFUL \smallskip but: \begin{theorem} \end{theorem} INVARIANCE THEOREM FOR MONOTONE ALGORITHMIC ENTROPY: Algorithmic Information Theory w.r.t. $ C_{M}-C_{\Phi}- \Delta_{0}^{0} - [ {\mathcal{D}}( A_{1} , A_{2} )] $ is meaningful \smallskip As we have preannounced, the resulting algorithmic entropy is called the \textbf{monotone algorithmic entropy} \medskip \begin{remark} \end{remark} FROM BINARY STRINGS TO NATURAL NUMBERS AND VICEVERSA For pure simplicity reasons we have defined simple algorithmic entropy for natural numbers and monotone algorithmic information for binary strings. By the quasi-lexicographic bijection the corrispondent notions, namely simple algorithmic entropy of strings and prefix algorithmic entropy of natural numbers are immediately obtained. For the same reason from here and beyond everything stated for $ \Sigma^{\star} $ may be immediately translated in terms of $ {\mathbb{N}} $ and viceversa. \medskip Up to now we have considered the case in which the two metric aggregates coincide. Anyway one can clearly introduce also the following mixed notions: \begin{example} \label{ex:decision algorithmic entropy} \end{example} DECISION ALGORITHMIC ENTROPY: Let us assume $ A_{1} \; = ( {\mathbb{N}} \, , \, = \, , \, | \cdot | ) $ while $ A_{2} \; = \; ( \Sigma^{\star} \, , \, <_{p} \, , \, | \cdot | )$. Exactly as in the example\ref{ex:simple algorithmic entropy} it may be proved that: \begin{theorem} \label{th:decision algorithmic information theory w.r.t. all the partial functions is not meaningful} \end{theorem} ALGORITHMIC INFORMATION THEORY W.R.T. $ {\mathcal{D}}( A_{1} , A_{2} ) $ IS NOT MEANINGFUL \smallskip but: \begin{theorem} \end{theorem} INVARIANCE THEOREM FOR DECISION ALGORITHMIC ENTROPY: Algorithmic Information Theory w.r.t. $ C_{M}-C_{\Phi}- \Delta_{0}^{0} - [ {\mathcal{D}}( A_{1} , A_{2} )] $ is meaningful \smallskip As we have preannounced, the resulting algorithmic entropy is called the \textbf{monotone algorithmic entropy} \medskip \begin{example} \label{ex:prefix algorithmic entropy} \end{example} PREFIX ALGORITHMIC ENTROPY: Let us assume $ A_{1} \; = \; ( \Sigma^{\star} \, , \, <_{p} \, , \, | \cdot | )$ while $ A_{2} \; = ( {\mathbb{N}} \, , \, = \, , \, | \cdot | ) $. Exactly as in the example\ref{ex:simple algorithmic entropy} it may be proved that: \begin{theorem} \label{th:prefix algorithmic information theory w.r.t. all the partial functions is not meaningful} \end{theorem} ALGORITHMIC INFORMATION THEORY W.R.T. $ {\mathcal{D}}( A_{1} , A_{2} ) $ IS NOT MEANINGFUL \smallskip but: \begin{theorem} \label{th:invariance theorem for prefix algorithmic entropy} \end{theorem} INVARIANCE THEOREM FOR PREFIX ALGORITHMIC ENTROPY: Algorithmic Information Theory w.r.t. $ C_{M}-C_{\Phi}- \Delta_{0}^{0} - [ {\mathcal{D}}( A_{1} , A_{2} )] $ is meaningful \smallskip As we have preannounced, the resulting algorithmic entropy, w.r.t. an optimal description mode that we will call from here and beyond a \textbf{Chaitin universal computer}, is called the \textbf{prefix algorithmic entropy} and is denoted by I. \medskip While the \textbf{decision entropy} and \textbf{monotone entropy} are of scarce utility, \textbf{simple entropy} and \textbf{prefix entropy} are of fundamental importance \newpage \section{Why prefix algorithmic entropy is better than simple algorithmic entropy} \label{sec:Why prefix entropy is better than simple entropy} Let us now compare simple algorithmic entropy and prefix algorithmic entropy. Though more intuitive, simple algorithmic entropy has a list of inconveniences that, after decades of debates among different attempts, led the scientific community to realize that the correct way of formulating Classical Algorithmic Information Theory involves prefix algorithmic entropy: \begin{enumerate} \item \textbf{classical probabilistic information}, namely \textbf{Shannon entropy}, satisfies the \textbf{subadditivity property}: \begin{equation} \label{eq:subadditivity of probabilistic information} I_{probabilistic} ( X , Y ) \; \leq \; I_{probabilistic} (X) + I_{probabilistic} (Y) \end{equation} with the equality holding iff the classical random variables X and Y are independent, where the joint probabilistic information $ I_{probabilistic} ( X , Y ) $ will be defined in section\ref{sec:From the communicational-compression of the Quantum Coding Theorems to the algorithmic-compression in Quantum Computation}. As we will see therein the subadditivity property remain preserved in the noncommutative generalization, i.e. eq.\ref{eq:subadditivity of probabilistic information} holds also in Quantum Probability Theory, where $ I_{probabilistic} $ is the \textbf{quantum probabilistic information}, namely \textbf{Von Neumann entropy}, ( X , Y ) denotes a state over a tensor product Von Neumann algebra $ A_{1} \bigotimes A_{2} $ having X and Y as marginal states, the equality holding iff ( X , Y ) is not entangled. The intuitive meaning of eq.\ref{eq:subadditivity of probabilistic information} (the information of a compound system is less or equal to the information of its parts) lead to think that such a condition should hold also for \textbf{classical algorithmic information}. As to \textbf{simple algorithmic entropy}, anyway, the subadditivity condition is violated by a disturbing logarithmic addendum causing that: \begin{theorem} \label{th:not subadditivity of simple algorithmic entropy} \end{theorem} NOT SUBADDITIVITY OF SIMPLE ALGORITHMIC ENTROPY \begin{equation} K ( ( \vec{x} , \vec{y} )) \; \not \stackrel{ + }{\leq} \; K( \vec{x}) \, + \, K(\vec{y}) \end{equation} The subadditivity property is instead satisfied by \textbf{prefix algorithmic entropy}: \begin{theorem} \label{th:subadditivity of prefix algorithmic entropy} \end{theorem} SUBADDITIVITY OF PREFIX ALGORITHMIC ENTROPY \begin{equation} I ( ( \vec{x} , \vec{y} )) \; \stackrel{ + }{\leq} \; I( \vec{x}) \, + \, I(\vec{y}) \end{equation} \item intuitive reasoning suggests that $ C_{M} - C_{ \Phi } $-algorithmic information should be \textbf{monotonic over prefixes}. Anyway one has that: \begin{theorem} \label{th:not monotonicity over prefixes of simple algorithmic entropy} \end{theorem} NOT MONOTONICITY OVER PREFIXES OF SIMPLE ALGORITHMIC INFORMATION \begin{equation} \vec{x} \, <_{p} \, \vec{y} \; \nRightarrow \; K ( \vec{x} ) \, \stackrel{ + }{\leq} \, K ( \vec{y} ) \end{equation} while: \begin{theorem} \label{th:monotonicity over prefixes of prefix algorithmic entropy} \end{theorem} MONOTONICITY OVER PREFIXES OF PREFIX ALGORITHMIC INFORMATION \begin{equation} \vec{x} \, <_{p} \, \vec{y} \; \Rightarrow \; I( \vec{x} ) \, \stackrel{ + }{\leq} \, I ( \vec{y} ) \end{equation} \item since the \textbf{probabilistic approach} and the \textbf{algorithmic approach} to $ C_{M} - C_{ \Phi } $ - Information Theory are different ways of formalizing the same object of investigation, it is natural to suppose that \textbf{$ C_{M} - C_{ \Phi } $-probabilistic information} and \textbf{$ C_{M} - C_{ \Phi } $-algorithmic information} are strictly connected notions. While the link is very clear in term of \textbf{prefix algorithmic entropy}, anyway, it is much obscure in terms of \textbf{simple algorithmic entropy}. To show this it is necessary to introduce some notion of $ C_{M} $ - Coding Theory: \begin{definition} \label{def:mathematically classical code} \end{definition} $ C_{M} $ - CODE: a partial function $ D : \Sigma^{\star} \stackrel{ \circ } { \mapsto } \Sigma^{\star} $ of decoding associating to each word $ \vec{x} $ belonging to the set HALTING(D) of \textbf{code words} its \textbf{source word} $ D(\vec{x})$. \medskip Given a $ C_{M} $ - code D and a source word $ \vec{x} \in \Sigma^{\star} $: \begin{definition} \end{definition} SET OF THE D - CODE WORDS OF $\vec{x}$: the (eventually empty) set $ D^{-1} (\vec{x})$. \medskip Let us observe that the definition\ref{def:mathematically classical code} doesn't require nor the surjectivity of a code (i.e. that each source word is codificable) neither the injectivity of a code (i.e. that each source word has only one code word). \medskip Let us now introduce a particular fundamental kind of code: \begin{definition} \end{definition} PREFIX-CODE: a code $ D : \Sigma^{\star} \stackrel{ \circ } { \mapsto } \Sigma^{\star} $ such that HALTING(D) is prefix-free \medskip The more fundamental property of prefix-codes is given by the following: \begin{theorem} \label{th:Kraft inequality} \end{theorem} KRAFT'S INEQUALITY \begin{hypothesis} \end{hypothesis} \begin{align*} I \text{ index set} & : cardinality(I) \leq \aleph_{0} \\ \{ l_{i} \in & {\mathbb{N}} \}_{i \in I} \end{align*} \begin{thesis} \end{thesis} \begin{equation} \exists \, D : \Sigma^{\star} \stackrel{ \circ } { \mapsto } \Sigma^{\star} \text{ prefix code} : \{ | \vec{x} | , \vec{x} \in HALTING(D) \} \, = \, \{ l_{i} \in {\mathbb{N}} \}_{i \in I} \; \Leftrightarrow \; \sum_{ i \in I} 2^{- l_{i}} \, \leq \, 1 \end{equation} We will appreciate the importance of Kraft Inequality as soon as we will introduce the \textbf{universal algorithmic probability} and the \textbf{Halting Probability}. \medskip Let us now start the probabilistic analysis of $ C_{M} $ - Coding Theory. Let us suppose that the generic source-word $ \vec{x} $ occur with probability $ P ( \vec{x} ) $. Given an injective prefix-code $ D : \Sigma^{\star} \stackrel{ \circ } { \mapsto } \Sigma^{\star} $ we can then introduce the: \begin{definition} \end{definition} AVERAGE CODE WORD LENGTH OF THE CODE D W.R.T. THE SOURCE CODE DISTRIBUTION P: \begin{equation} L_{D,P} \; := \; \sum_{\vec{x} \in HALTING(D)} P ( \vec{x} ) | D (\vec{x}) | \end{equation} It is clear that, in a communicational situation, the objective of a transmitter is to minimize the average code word length. Clearly a coding strategy will be the more clever the more it will assign short code words to highly probable source words and viceversa, in order to minimize the average code word length. \begin{definition} \end{definition} MINIMAL AVERAGE CODE WORD LENGTH ALLOWED BY THE DISTRIBUTION P: \begin{equation} L \; := \; \min \{ L_{D,P} \, , \, D \, prefix-code \} \end{equation} \begin{definition} \end{definition} OPTIMAL PREFIX-CODE W.R.T. THE SOURCE CODE DISTRIBUTION P: a prefix-code D such that: \begin{equation} L_{D,P} \; = \; L \end{equation} \smallskip The probabilistic approach to $ C_{M} - C_{\Phi} $ Information Theory is based on the following notion: \begin{definition} \label{def:Shannon entropy of a distribution} \end{definition} SHANNON ENTROPY OF THE DISTRIBUTION P: \begin{equation} H(P) \; := \; - \sum_{\vec{x} \in \Sigma^{\star}} P ( \vec{x} ) \log_{2} P ( \vec{x} ) \end{equation} The corner stone of $ C_{M} - C_{\Phi} $ Probabilistic Information Theory is the following: \begin{theorem} \label{th:mathematically classical mathematically physical noiseless coding theorem} \end{theorem} $ C_{M} - C_{\Phi} $ NOISELESS CODING THEOREM \begin{equation} H(P) \; \leq \; L \; \leq \; H(P)+1 \end{equation} \smallskip Let us now observe that \textbf{prefix algorithmic entropy} may be used to define a particular code: by definition we have that: \begin{equation} I( \vec{x} ) \, = \, | \vec{x}^{\star} | \end{equation} where $ \vec{x}^{\star} $ is the shortest input for the fixed universal Chaitin computer giving $ \vec{x} $ as output (or the first one in quasi-lexicographic order if there are many). The map $ D_{I} : \Sigma^{\star} \stackrel{ \circ } { \mapsto } \Sigma^{\star} $ defined by: \begin{equation} D_{I}( \vec{x} ) \; := \; \vec{x}^{\star} \end{equation} is by construction a prefix-code. \smallskip Since the code $ D_{I} $ is of pure algorithmic nature, it would be very reasonable to think that it may me optimal only for some ad hoc probability distribution, i.e. that for a generic probability distribution P the average code word length of $ D_{I} $ w.r.t. P: \begin{equation} L_{ D_{I} , P } \; = \; \sum_{\vec{x} \in HALTING( D_{I} )} P( \vec{x}) I( \vec{x}) \end{equation} won't achieve the optimal bound of H(P) stated by Theorem\ref{th:mathematically classical mathematically physical noiseless coding theorem} \smallskip But here the deep link between the \textbf{probabilistic-approach} and the \textbf{algorithmic-approach} makes the miracle: under mild assumptions about the distribution P the code $ D_{I} $ is optimal as is stated by the following: \begin{theorem} \label{th:link between mathematically classical mathematically physical probabilistic information and mathematically classical mathematically physical algorithmic information} \end{theorem} LINK BETWEEN $C_{M}-C_{\Phi}$ PROBABILISTIC INFORMATION AND $C_{M}-C_{\Phi}$ ALGORITHMIC INFORMATION \begin{hypothesis} \end{hypothesis} P $ C_{M}-C_{\Phi} - \Delta_{0}^{0} $ probability distribution over $ \Sigma^{\star} $ \begin{thesis} \end{thesis} \begin{equation} \exists c_{P} \in {\mathbb{R}}_{+} \; : \; 0 \, \leq \, L_{D_{I},P} - H(P) \, \leq \, c_{P} \end{equation} \smallskip Such a result of a substantial equivalence between \textbf{Shannon entropy} and \textbf{average algorithmic prefix entropy} has a strongly weaker counterpart in terms of \textbf{algorithmic simple entropy}. Indeed the two algorithmic entropies are linked by the following: \begin{theorem} \label{th:first Solovay theorem} \end{theorem} FIRST SOLOVAY'S THEOREM: \begin{align} I( \vec{x} ) \; & \; = \; K( \vec{x} ) \, + \, K( string^{- 1} ( K (\vec{x} )) \, + \, O ( K ( string^{- 1} K( string^{- 1} ( K (\vec{x} ))))) \\ K( \vec{x} ) \; & \; = \; I( \vec{x} ) \, - \, K( string^{- 1} ( K (\vec{x} )) \, - \, O ( I ( string^{- 1} I( string^{- 1} ( I (\vec{x} ))))) \end{align} that substituted in the Theorem\ref{th:link between mathematically classical mathematically physical probabilistic information and mathematically classical mathematically physical algorithmic information} gives: \begin{equation} - c_{P} \; \leq \; L_{D_{K},P} - H(P) \; \leq \; \sum_{\vec{x}} P ( \vec{x} ) K( string ^{-1} (C ( string ^{-1} ( \vec{x})))) \end{equation} which is bounded only if $ \sum_{\vec{x}} P ( \vec{x} ) K( string ^{-1} (C ( string ^{-1} ( \vec{x})))) $ converges. \item called U the fixed Chaitin universal computer let us introduce the main actors of some of the most fascinating developes of $ C_{M} - C_{\Phi} $ Algorithmic Information Theory: \begin{definition} \label{def:universal algorithmic probability} \end{definition} UNIVERSAL ALGORITHMIC PROBABILITY OF $ \vec{x} \in \Sigma^{\star} $: \begin{equation} P_{U} (\vec{x}) \; := \; \sum_{\vec{y} \in \Sigma^{\star} : U( \vec{y} ) = \vec{x} } 2^{- |\vec{y} | } \end{equation} \begin{definition} \label{def:halting probability} \end{definition} HALTING PROBABILITY: \begin{equation} \Omega_{U} \; := \; \sum_{ \vec{x} \in \Sigma^{\star}} P_{U} (\vec{x}) \end{equation} These notion has a very intuitive meaning: \begin{itemize} \item $ P_{U} (\vec{x}) $ is the probability that the computer U gives as output the string $ \vec{x} $ under an uniformely random distributed input. \item $ \Omega_{U} $ gives the probability that the computer U halts under an uniformely random distributed input. \end{itemize} Such a probabilistic meaning,anyway, lies on the fact that U is a \textbf{Chaitin computer} so that its halting set is prefix-free and hence Theorem\ref{th:Kraft inequality} implies that: \begin{equation} \label{eq:universal algorithmic probability takes values in the unitary interval} 0 \; \leq \; P_{U} (\vec{x}) \; \leq \; 1 \; \; \forall \vec{x} \in \Sigma^{\star} \end{equation} and that: \begin{equation} \label{eq:halting probability takes values in the unitary interval} 0 \; \leq \; \Omega_{U} \; \leq \; 1 \end{equation} If we considered \textbf{simple algorithmic information} instead of \textbf{prefix algorithmic information} and hence we adopted a \textbf{non Chaitin computer}, anyway, the halting set of U wouldn't be prefix-free anymore, so that Theorem\ref{th:Kraft inequality} wouldn't imply eq.\ref{eq:universal algorithmic probability takes values in the unitary interval} and eq.\ref{eq:halting probability takes values in the unitary interval}. \item Unlike \textbf{prefix algorithmic information}, \textbf{simple algorithmic information} is affected by oscillations that exclude the possibility of using it to define the notion of algorithmic randomness for sequences in an enough robust way as we will show in the next section \end{enumerate} \newpage \section{Chaitin random strings and sequences of cbits} Let us suppose to make 100 independent tosses of a coin. If we obtained head all times we would certainly claim the the used coin is not fair. But let us observe that, assuming that the coin is fair, the string of 100 heads have the same exact probability, i.e. $ 2^{- 100} $, of any other binary string of 100 cbits. So, which foundation could we give at our claim that the coin is not fair? The first to analyze this problem was Laplace that dedicated to this issue the Fifth Principle among the \textit{"General Principles of the calculus of probabilities"} making the content of the third chapter of his pioneering work \cite{Laplace-51}; it is worth to report his own words: \begin{center} \textit{"Sixth Principle: Each of the causes to which an observed event may be attributed is indicated with just as much likelihood as there is probability that the event will take place supposing the event to be constant. The probability of the existence of any one of these causes is then a fraction whose numerator is the probability of the event resulting from this cause and whose denominator is the sum of the similar probabilities relative to all the causes; if these various causes considerated \`{a} priori, are unequally probable, it is necessary in place of the probability of the event resulting from each cause, to employ the product of this probability by the possibility of the cause itself. This is the fundamental principle of this branch of the analysis of chances which consists in passing from events to causes.} \textit{This principle gives the reason why we attribute regular events to a particular cause. Some philosophers have thought that these events are less possible than others and that at the play of heads and tails, for example, the combiantion in which heads occurs twenty successive times is less easy in its nature than those where head and tails are mixed in irregular manner. But this opinion suppose that past events have an influence on the possibility of future events which is not at all admissible. The regular combinations occur more rarely only because they are less numerous. $ \cdots $} \textit{Thus at the play of head and tail the occurence of heads a hundred successive times appears to us extraordinary because of the almost infinite number of combinations which may occur in a hundred throws; and if divide the combinations in two regular series containing an order easy to comprehend, and into irregular series, the latter are incomparably more numerous"} \end{center} Laplace catchs the following basic points: \begin{itemize} \item what the string made of one hundred heads have of particular is to possess some kind of regularity \item this string has the same probability $ 2^{- 100} $ of every other string \item the fact that if this string of results occurs we can claim the coin was unfair is founded by the observation that the fraction of the set of 100 cbit strings made by strings having some kind of regularity, i.e. the probability that a string of this kind occurs, is enormously low and , conseguentially, the probability that a string of this kind occurs is extraordinarily low. \end{itemize} The only thing Laplace wasn't able to explain, as anyone else for little less than two centuries, was the exact meaning of the locution \emph{\textbf{"to possess some regularity"}}. In this dissertation we will see how Classical Algorithmic Information Theory gives many equilavent mathematical characterization of this \textbf{absence of regularity} or, as we say it nowadays , of this \textbf{algorithmic randomness}. Among these characterization the more important one is with no doubt that as \textbf{algorithmic incompressibility}. As an \textbf{algorithmically incompressible} object we mean, informally speaking , an object whose more concise algorithmic description is substantially its own assignation. So one could be to tempted to say that the string $ \vec{x} \in \Sigma^{\star} $ is algorithmically random iff: \begin{equation} \label{eq:naife simple algorithmic random string} K( \vec{x} ) \; = \; | \vec{x} | \end{equation} or iff: \begin{equation} \label{eq:naife prefix algorithmic random string} I( \vec{x} ) \; = \; | \vec{x} | \end{equation} The meaningness of these definitions, anyway, appear evident as soon as one keeps into account , as to eq.\ref{eq:naife simple algorithmic random string}, the issue of the additive constant involved in the passage from a \textbf{universal computer} to another \textbf{universal computer} and, as to eq.\ref{eq:naife prefix algorithmic random string}, the issue of the additive constant involved in the passage from a \textbf{Chaitin universal computer} to another \textbf{Chaitin universal computer}. The notion of random string originally introduced by Kolmogorov in 1965 was the following: given a constant $ c \in {\mathbb{R}}_{+} $: \begin{definition} \label{def:Kolmogorov c-random string} \end{definition} $ \vec{x} \in \Sigma^{\star} $ IS c-KOLMOGOROV-RANDOM: \begin{equation} K( \vec{x} ) \; \geq \; | \vec{x} | \, - \, c \end{equation} \smallskip Before of analyzing the analogous notion involving \textbf{prefix algorithmic information} instead of \textbf{simple algorithmic information} let us introduce the following preliminary notion: \begin{definition} \label{def:busy beaver function} \end{definition} BUSY BEAVER FUNCTION: the function $ \Sigma : {\mathbb{N}} \, \rightarrow \, {\mathbb{N}} $: \begin{equation} \Sigma(n) \; := \; \max_{ \vec{x} \in \Sigma^{n}} I( \vec{x} ) \end{equation} It obeys the following \cite{Calude-94}: \begin{theorem} \end{theorem} \begin{equation} \Sigma(n) \; \stackrel{ + }{=} \; n \, + \, I ( string(n)) \end{equation} \smallskip Chaitin's idea was that of defining the random strings of length n to be the strings with maximal prefix entropy among the strings of length n. So, given a natural number m: \begin{definition} \label{def:Chaitin m-random string} \end{definition} $ \vec{x} \in \Sigma^{\star} $ IS CHAITIN m-RANDOM: \begin{equation} I( \vec{x} ) \; \geq \; \Sigma( | \vec{x} | ) \, - \, m \end{equation} We will denote the set of al the Chaitin m-random binary strings by $CHAITIN-m-RANDOM( \Sigma^{\star}$. a 0-Chatin random string is often called simply a \textbf{Chaitin random}. Following this terminology we will pone: \begin{equation} CHAITIN-RANDOM( \Sigma^{\star}) \; := \; CHAITIN-0-RANDOM( \Sigma^{\star}) \end{equation} \smallskip \begin{remark} \label{rem:impossibility of a sharp distinction between regularity and randomness for strings} \end{remark} IMPOSSIBILITY OF A SHARP DISTINCTION BETWEEN REGULARITY AND RANDOMNESS FOR STRINGS It is essential to observe that the introduction of additive constants in both definition\ref{def:Kolmogorov c-random string} and definition\ref{def:Chaitin m-random string} solves the problem of the inconsistence of, respectively, definition\ref{eq:naife simple algorithmic random string} and definition\ref{eq:naife prefix algorithmic random string} only in a partial way: indeed definition\ref{def:Kolmogorov c-random string} and definition\ref{def:Chaitin m-random string} continue to depend upon, respectively, the fixed \textbf{universal computer} and the fixed \textbf{universal Chaitin computer}. The improvement is that under these ansatzs one doesn't lose algorithmic randomness of strings but passes simply from algorithmic randomness relative to a certain constant to algorithmic randomness relative to a different constant. But in this way one has to look at the transition from regular to random strings as a continuous, asymptotic one: indeed the effective connotation of randomness given by the specification that a certain string $ \vec{x}\in \Sigma^{\star} $ is m-Chaitin random is the more significative the more high is the difference $ | \vec{x} | - m $. A sharp distinction is possible only for sequences. In chapter\ref{chap:Classical algorithmic randomness as stability of the relative frequences under proper classical algorithmic place selection rules} we will give a clear, intuitive explanation of this fact in terms of Classical Gambling Theory. Unfortunately, as we will show in part\ref{part:The road for quantum algorithmic randomness}, this point hasn't received the necessary consideration in most the attempts of defining quantum algorithmic randomness, that , erroneously to our opinion, concentrate the analysis to strings of qubits considering this case as simpler and only later, in a derivate mode, pass to analyze sequences of qubit. We anticipate here that our point of view is opposite: since exactly as in the classical case a sharp distinction between regularity and randomness is possible only for sequences of qubits, the analysis of quantum algorithmic randomness has to start from sequences of qubit. \bigskip Let us now observe that there exist many reasons to prefer Chaitin-randomness to Kolmogorov-randomness: \begin{enumerate} \item the adoption of Chaitin randomness allows to give a clear quantitative foundation to the observation Laplace himself realized almost two centuries ago, namely that the strings not having any kind of regularity, the patternless ones, are the overwhelming majority: \begin{theorem} \end{theorem} \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; cardinality ( CHAITIN-RANDOM( \Sigma^{n})) \, > \, 2^{n - c} \; \; \forall n \in {\mathbb{N}} \end{equation} \item Robert Solovay has proved that Chaitin randomness is stronger than Kolmogorov randomness \item Chaitin randomness may be easily extended to binary sequences defining, informally speaking, an algorithmic random sequence as one whose prefixes are all Chaitin algorithmic random. As we will now show, the \textbf{phenomenon of the oscillations of simple algorithmic entropy} avoid this possibility for Kolmogorov randomness. \end{enumerate} \medskip Let us introduce , first of all, a useful notation. Given a sequence $ \bar{x} \in \Sigma^{\infty} $ let us denote by $ x_{n} $ its $ n^{th} $ digit, by $ \vec{x}(n) $ its prefix of length n and by $ \vec{x}(n , m) $ ($ n \leq m$) the substring of $ \bar{x} $ obtained taking its digits from the $ n^{th} $ to the $ m^{th} $, namely: \begin{equation} \vec{x}(n , m) \; := \; x_{n} \cdots x_{m} \, \in \, \Sigma^{m-n} \end{equation} Let us then observe that by identifying the generic string $ \vec{x} \in \Sigma^{\infty} $ with the sequence $ \vec{x} 0^{\infty} \in $ we can look at $ \Sigma^{\star} $ as a proper subset of $ \Sigma^{\infty} $. Let us then introduce the following useful map: \begin{definition} \label{def:numeric representation} \end{definition} NUMERIC REPRESENTATION: $ {\mathcal{N}} : \Sigma^{\infty} \, \mapsto \, [ 0 ,1) $: \begin{equation} {\mathcal{N}} ( \bar{x} ) \; := \; \sum_{n = 1}^{\infty} \frac{x_{n}}{2^{n}} \end{equation} whose restriction $ {\mathcal{N}} |_{ \Sigma^{\infty} - \Sigma^{\star}} $ is a bijection and allows, conseguentially, to identify $ \Sigma^{\infty} $ with the set $ [ 0 ,1 ) $. Let us then introduce the probability measure: \begin{definition} \end{definition} UNBIASED PROBABILITY MEASURE ON $ \Sigma^{\infty} $: $ P_{unbiased} \, : \, 2^{ \Sigma^{\infty}} \; \stackrel{\circ }{\rightarrow} \; [0,1]$ : \begin{align} HALTING(P_{unbiased}) \; & = \; {\mathcal{F}}_{cylinder} \\ P_{unbiased} ( \Gamma_{\vec{x}} ) \; & \equiv \; \frac{1}{2^{| \vec{x} |}} \; \; \forall \, \vec{x} \, \in \, \Sigma^{\star} \end{align} where: \begin{definition} \end{definition} CYLINDER SET W.R.T. $ \vec{x} \, = ( x_{1} , \ldots , x_{n} ) \, \in \, \Sigma^{\star} $: \begin{equation} \label{eq:cylinder set} \Gamma_{\vec{x}} \; \equiv \; \{ \bar{y} = ( y_{1} , y_{2} , \ldots ) \in \Sigma^{\infty} \; : \; y_{1} = x_{1} , \ldots , y_{n} = x_{n} \} \end{equation} \begin{definition} \end{definition} CYLINDER - $ \sigma $ - ALGEBRA ON $ \Sigma^{\infty} $: \begin{equation} {\mathcal{F}}_{cylinder} \; \equiv \; \sigma- \text{algebra generated by} \{ \Gamma_{\vec{x}} \, : \, \vec{x} \in \Sigma^{\star} \} \end{equation} \smallskip In the numeric representation of $ \Sigma^{\infty} $ as the real interval $ [ 0 ,1) $, $P_{unbiased} $ is, clearly, nothing but Lebesgue measure \cite{Lebesgue-73}. Denoted by $ N_{i}^{n} ( \bar{x} ) $ the number of successive $ i \in \Sigma $ ending in position n of the sequence $ \bar{x} $ the First Borel-Cantelli's Lemma implies that (cfr. the fifth section of the fourth chapter of \cite{Billingsley-95}): \begin{theorem} \label{th:on the islands of regularity of almost all sequences} \end{theorem} \begin{align} P_{unbiased} & ( \{ \bar{x} \in \Sigma^{\infty} \, : \, \limsup_{ n \rightarrow \infty} N_{0}^{n} ( \bar{x} ) \, = \, 1 \} ) \; = \; 1 \\ P_{unbiased} & ( \{ \bar{x} \in \Sigma^{\infty} \, : \, \limsup_{ n \rightarrow \infty} N_{1}^{n} ( \bar{x} ) \, = \, 1 \} ) \; = \; 1 \end{align} \smallskip Theorem\ref{th:on the islands of regularity of almost all sequences} tells us that for $ P_{unbiased} $-almost all sequences $ \bar{x} \in \Sigma^{\infty} $ there exist infinitely many n for which: \begin{equation} \vec{x}(n) \; \stackrel{ + }{=} \; \vec{x} ( 1 , n - \log_{2} n) 0^{n} \end{equation} i.e.: \begin{equation} \label{eq:oscillations of simple algorithmic entropy} K( \vec{x}(n) ) \stackrel{ + }{=} \; n - \log_{2} n \end{equation} This suggest that if we adopted the following definition of algorithmic randomness for sequences: \begin{definition} \label{def:Kolmogorov random sequence} \end{definition} $ \bar{x} \in \Sigma^{\infty} $ IS KOLMOGOROV RANDOM: \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; K ( \vec{x}(n) ) \, > \, n - c \; \; \forall n \in {\mathbb{N}} \end{equation} there wouldn't exist Kolmogorov random sequences. That this is indeed the case may be rigorously proved observing that the existence of infinitely many n such that eq.\ref{eq:oscillations of simple algorithmic entropy} holds may be proved to hold for all sequences (not only with $ P_{unbiased}$- probability one). \smallskip This doesn't happen if we use prefix algorithmic entropy. \begin{definition} \label{def:Chaitin random sequence} \end{definition} $ \bar{x} \in \Sigma^{\infty} $ IS CHAITIN RANDOM: \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; I ( \vec{x}(n) ) \, > \, n - c \; \; \forall n \in {\mathbb{N}} \end{equation} We will denote the set of all the Chaitin random sequences by $ CHAITIN -RANDOM(\Sigma^{\infty})$. By the numeric representation's map the notion of Chaitin randomness may be immediately extended to reals numbers in the following way: \begin{definition} \label{def:Chaitin random real number} \end{definition} $ x \in [ 0 \, , \, 1 ) $ IS CHAITIN RANDOM: \begin{equation} ({\mathcal{N}} | _{\Sigma^{\infty} \, - \, \Sigma^{\star}})^{- 1} (\Omega) \end{equation} We will denote the set of all the random real numbers by $ CHAITIN -RANDOM( [ 0 , 1 )) $ As we will prove later, \textit{"almost all"} the numbers in the unitary interval are Chaitin-random. In particular one has the following: \begin{theorem} \label{th:Chaitin randomness of the halting probability} \end{theorem} CHAITIN RANDOMNESS OF THE HALTING PROBABILITY: \begin{equation} \Omega \; \in \; CHAITIN-RANDOM( [ 0 , 1 )) \end{equation} Supposing now to let the fixed Chaitin universal computer to vary, Theodore A. Slaman has recentely proved the followin remarkable \cite{Kucera-Slaman-01}: \begin{theorem} \label{th:Slaman's theorem} \end{theorem} SLAMAN'S THEOREM: \begin{equation} \{ \, \Omega_{U} \, , \, : \, U \text{ Chaitin's universal computer } \, \} \; = \; CHAITIN-RANDOM( [ 0 \, , \, 1 ) ) \, \bigcap \, REC-EN( {\mathbb{R}}) \end{equation} \newpage \section{Brudno random sequences of cbits} \label{sec:Brudno random sequences of cbits} As to the definition of algorithmically random binary sequences we have seen in the previous section that the phenomenon of oscillations of simple algorithmic entropy causes that, denoted by $KOLMOGOROV-RANDOM(\Sigma^{\infty})$ the set of all the Kolmogorov-random binary sequence, one has that: \begin{theorem} \label{th:not existence o Kolmogorov random sequences} \end{theorem} NOT EXISTENCE OF KOLMOGOROV RANDOM SEQUENCES: \begin{equation} KOLMOGOROV-RANDOM( \Sigma^{\infty} ) \; = \; \emptyset \end{equation} \smallskip One could , at this point, argue that the existence of infinite islands of regularity in a generic sequence resulting in the logarithmic deficit of simple algorithmic entropy showed by an infinite number of its prefixes is a false problem since what is really relevant is the rate of plain algorithmic entropy for digit of the prefixes, i.e. the ratio $ \frac{K( \vec{x}(n)) }{n} $ for whose asymptotic behaviour the logarithmic deficits are irrilevant since obviuosly: \begin{equation} \lim_{n \rightarrow \infty} \frac{ \log_{2}(n) }{n} \; = \; 0 \end{equation} This way of reasoning led A.A. Brudno to introduce the following notions: \begin{definition} \label{def:Brudno algorithmic entropy of a sequence} \end{definition} BRUDNO ALGORITHMIC ENTROPY OF $ \bar{x} \in \Sigma^{\infty} $: \begin{equation} B( \bar{x} ) \; := \; \lim_{n \rightarrow \infty} \frac{K( \vec{x}(n)) }{n} \end{equation} \smallskip At this point one could think that considering the asympotic rate of prefix entropy instead of simple entropy would result in a different definition of the algorithmic entropy of a sequence. That this is not the case is the claim of the following: \begin{theorem} \label{th:plain-prefix insensitivity of Brudno algorithmic entropy} \end{theorem} \begin{equation} B( \bar{x} ) \; = \; \lim_{n \rightarrow \infty} \frac{I( \vec{x}(n))}{n} \end{equation} \begin{proof} It immediately follows by the fact that \cite{Staiger-99}: \begin{equation} I( \vec{x}(n) ) \, - \, K( \vec{x}(n) ) \; = \; o(n) \end{equation} \end{proof} \begin{definition} \label{def:Brudno random sequence} \end{definition} $ \bar{x} \in \Sigma^{\infty} $ IS BRUDNO RANDOM: \begin{equation} B( \bar{x} ) \; = \; 1 \end{equation} We will denote the set of all the Brudno random binary sequences by $ BRUDNO( \Sigma^{\infty}) $. \smallskip Anyway one is here faced to a problem almost always misunderstood that is the main source of a sort of incomunicability between the scientific community of mathematical physicists studying Dynamical Systems Theory and the scientific community of the logico-mathematicians and Theoretical-Computer scientists studying Algorithmic Information Theory: \begin{theorem} \label{th:Brudno randomness is weaker than Chaitin randomness} \end{theorem} BRUDNO RANDOMNESS IS WEAKER THAN CHAITIN RANDOMNESS: \begin{equation} BRUDNO-RANDOM( \Sigma^{\infty}) \; \subset \; CHAITIN -RANDOM(\Sigma^{\infty}) \end{equation} \smallskip Such a theorem was proven by Brudno himself in the last section of \cite{Brudno-83} by the explicit presentation of a Brudno-random sequence doesn't passing a Martin-L\"{o}f test. We postpone the presentation of such a proof to section\ref{sec:Equivalence between passage of a Martin Lof universal statistical test and Chaitin randomness} where the involved properties of universal sequential Martin-L\"{o}f test are introduced. \newpage \section{Brudno algorithmic entropy versus the Uspensky abstract approach} \label{sec:Brudno algorithmic entropy versus the Uspensky abstract approach} In this section we will show that definition\ref{def:Brudno algorithmic entropy of a sequence} is not compatible with Uspensky's abstract approach of defining algorithmic information discussed in section\ref{sec:Uspensky's abstract definition of algorithmic information}. Uspensky's abstract approach would, indeed, require the specification of: \begin{enumerate} \item a \textbf{concordance relation} $ {\mathcal{R}} $ \item a \textbf{point measure} $ \mu $ \end{enumerate} on $ \Sigma^{\infty} $ such to constitute a \textbf{metric aggregate} $ ( \Sigma^{\infty} \, , \, {\mathcal{R}} \, , \, \mu )$. Both these points are highly not-trivial. Remembering remark\ref{rem:context dependence of the computability constraint} we have to keep attention on the meaning of the computability constraint. Indeed, by theorem\ref{th:cardinalities of strings and sequences}, we see that the definition of the concordance relation $ {\mathcal{R}} $ exit from the boundaries of $ C_{M}$-Classical Recursion Theory. As we have anticipated in section\ref{sec:The distinction between mathematical-classicality and physical-classicality}, just as to Computability Theory by \textbf{physically classical computers} of (partial) functions on sets $ S \, : \, card(S) \, = \, \aleph_{1}$ many different inequivalent candidate theories have been proposed: \begin{enumerate} \item the Orthodox Theory generated by the studies of Grzegorczyck - Lacombe \cite{Pour-El-Richards-89} \item the theory developed by the so called Markov School in the framework of Constructive Mathematics \cite{Odifreddi-89} \item the Blum - Shub - Smale 's Theory \cite{Smale-92}, \cite{Blum-Cucker-Shub-Smale-98} \end{enumerate} The basic notion of the Othodox Theory, namely the definition of a \textbf{recursive real number}, seems rather robust: starting from the $ C_{\Phi} $-Computability of the whole $ {\mathbb{Q}} $ the strategy of defining a \textbf{recursive real number} consists in effectivizing a method for constructing $ {\mathbb{R}} $ from $ {\mathbb{Z}} $; as shown by R.M. Robinson whichever of these methods one effective: \begin{enumerate} \item the construction of $ {\mathbb{R}} $ from $ {\mathbb{Z}} $ through Cauchy sequences \item the construction of $ {\mathbb{R}} $ from $ {\mathbb{Z}}$ through nested intervals \item the construction of $ {\mathbb{R}} $ from $ {\mathbb{Z}}$ through the Dedekind Cut \item the construction of $ {\mathbb{R}} $ from $ {\mathbb{Z}}$ through the expansion to base b, where b is an integer $ > 1 $. \end{enumerate} one results in the the same set $ REC ({\mathbb{R}}) $ \cite{Pour-El-Richards-89}, \cite{Pour-El-99}. Let us review the first of these strategies: given a sequence $ \{ r_{n} \}_{ n \in {\mathbb{N}}} $ of rational numbers: \begin{definition} \end{definition} $ \{ r_{n} \} $ IS RECURSIVE: \begin{equation} \exists b , c , s \in C_{M} - C_{\Phi} - \Delta_{0}^{0}-map({\mathbb{N}},{\mathbb{N}}) \; : \; r_{n} \, = \, (- 1) ^{s(n)} \frac{ b(n) }{ c(n) } \end{equation} \begin{definition} \end{definition} $ \{ r_{n} \} $ CONVERGES EFFECTIVELY TO $ x \in {\mathbb{R}} $: \begin{equation} \exists e \in C_{M} - C_{\Phi} - \Delta_{0}^{0}-map({\mathbb{N}},{\mathbb{N}}) \; : \; m \geq e(n) \, \Rightarrow \, | r_{m} - x | < \frac{1}{2^{n}} \end{equation} \smallskip Given a real number $ x \in {\mathbb{R}} $: \begin{definition} \label{def:r.e. real numbers} \end{definition} RECURSIVELY ENUMERABLE REAL NUMBERS: \begin{multline} REC-EN( {\mathbb{R}} ) \; := \; \{ x \in {\mathbb{R}} ) \, : \\ \text{ there is a increasing, computable sequence of rationals which converges to x} \} \end{multline} \begin{definition} \label{def:recursive real number} \end{definition} RECURSIVE REAL NUMBERS: \begin{multline} REC( {\mathbb{R}} ) \; := \; \{ x \in {\mathbb{R}} ) \, : \\ \text{ there is a computable sequence of rationals which converges effectively to x } \} \end{multline} \smallskip Given a complex number $ z \in {\mathbb{C}} $: \begin{definition} \label{def:recursive complex number} \end{definition} z IS RECURSIVE: \begin{equation} \Re(z) \in REC( {\mathbb{R}} ) \; and \; \Im(z) \in REC( {\mathbb{R}} ) \end{equation} \smallskip We will denote the set of all the recursive complex numbers by $ REC ( {\mathbb{R}} ) $. It may be proved that: \begin{theorem} \end{theorem} BASIC PROPERTIES OF $ REC( {\mathbb{R}} ) $ \begin{enumerate} \item $ REC ( {\mathbb{R}} ) $ is a \textbf{closed field} \item \begin{equation} cardinality( REC ( {\mathbb{R}} ) ) \; = \; \aleph_{0} \end{equation} \end{enumerate} Obviously this immediately implies that: \begin{corollary} \label{cor:enumerability of the set of all the recursive complex numbers} \end{corollary} \begin{equation} cardinality( REC ( {\mathbb{C}} )) \; = \; \aleph_{0} \end{equation} \medskip To understand in which sense Corollary\ref{cor:enumerability of the set of all the recursive complex numbers} may be seen highly unsatisfactory and even suggest the necessity of an Eterodox Theory, let us start from the analysis Roger Penrose dedicates to the issue: \begin{center} \textbf{Is Mandelbrot's set recursive?} \end{center} in the seventh section of the fourth chapter of his book \cite{Penrose-89} and its reformulation by Lenore Blum, Felipe Cucker, Michael Shub and Steve Smale in the first two chapters of their book \cite{Blum-Cucker-Shub-Smale-98} significatively reporting a picture of Mandelbrot's set on its front cover. Given a generic complex number $ c \in {\mathbb{C}} $ let us introduce the polynomial $ p_{c} (z) \; := \; z^{2} + c $ and let us denote by $ p_{c}^{(n)} (z) $ its $ n^{th} $ iterate. \begin{definition} \label{def:Mandelbrot set} \end{definition} MANDELBROT'S SET: \begin{equation} {\mathcal{M}} \; := \; {\mathbb{C}} \, - \, \{ c \in {\mathbb{C}} \, : \, \lim_{n \rightarrow \infty } p_{c}^{(n)} (0) = \infty \} \end{equation} A key property of Mandelbrot's set is stated by the following \cite{Falconer-90}: \begin{theorem} \end{theorem} $ {\mathcal{M}} $ is the halting set of the following algorithm: \begin{center} Label[start] Input c $ x^{2} + c \; \rightarrow \; x $ If $ | x | \, \leq \, 2 $ then Goto start Output 1 Halt \end{center} To answer Penrose's query one needs an algorithm that, given the input $ c \in {\mathbb{C}} $, will decide in a finite number of steps whether or not $ c \in {\mathcal{M}} $. Penrose appeals to the Orthodox Theory, but immediately refutes it: \begin{center} \textit{"One implication of this is that even with such a simple set as the unit disc $ \cdots $ there would be no algorithm for deciding for sure $ \cdots $ whether the computable number $ x^{2} + y^{2} $ is actually equal to 1 or not, this being the criterion for deciding whether or not the computable complex number $ x + i y $ lies on the unit circle $ \cdots $ Clearly that is not what we want"} \end{center} Then he tries to follow other approaches but at the end he concludes that: \begin{center} \textit{"One is left with the strong feeling that the correct viewpoint has not yet been arrived at"} \end{center} Blum, Cucker, Shub and Smale settle Penrose's question in the framework of their foundation of Computability Theory over a generic ring \cite{Smale-92} as a generalization of the Goldstine - Von Neumann axiomatization of Flowchart Theory (cfr. par.1.5 of \cite{Odifreddi-89}). Introduced the following notion: \begin{definition} \end{definition} $ S \; \subset \; {\mathbb{C}} $ IS A SEMI-ALGEBRAIC SET: it is a Boolean combination of sets defined by polynomial equalities and disequalities \smallskip Then they prove that a necessary condition for the decidability of a set $ S \; \subset \; {\mathbb{C}} $ w.r.t. Computation Theory over the ring $ {\mathbb{R}} $ is that it is the countable union of semi-algebraic sets over $ {\mathbb{R}} $. That this is not the case for the Mandelbrot's set follows by Shishikura's Theorem stating that the boundary of $ {\mathcal{M}} $ has Hausdorff dimension two, resulting in the following: \begin{theorem} \label{th:Blum Cucker Shub Smale incomputability of Mandelbrot set} \end{theorem} $ {\mathcal{M}} $ is not Blum, Cucker, Shub and Smale computable \medskip Following Uspensky abstract approach of defining algorithmic information, one would infer from Theorem\ref{th:Blum Cucker Shub Smale incomputability of Mandelbrot set} that $ {\mathcal{M}} $ has infinite algorithmic information, constrasting which what is claimed by another book reporting a picture of (part of) the Mandelbrot's set on its front cover, namely \cite{Cover-Thomas-91}, and stating that its information content is essentially zero. That the applicability of the Uspensky abstract approach to this particular case might be problematic, anyway, results by the obsevation that it doesn't seem to exist some natural \textbf{point measure} to use in the the specification of the \textbf{metric aggregate} $ ( \Sigma^{\infty} \, , \, {\mathcal{R}} \, , \, \mu )$. This throws a shadow on the same foundation of $ NC_{M} - C_{\Phi} $ - Algorithmic Information that is, as has been recentely remarked by Chaitin in the first paragraph of the fourth chapter of \cite{Chaitin-01}, mostly an unexplored field. Chaitin discusses this subject in the usual concretelly LISP programming attitude he followed in his other two Springer books \cite{Chaitin-98}, \cite{Chaitin-99}, adding to his version of LISP a new primitive function \textbf{display} that allow to get the partial outputs of a non-halting computation, and hence considering the algorithmic information of (so produced) infinite sets of S-expressions: in this way he concretely shows that the infinite version of $ C_{\Phi} $ - Algorithmic Information Theory differs for the finite version in many respects, an example being the violation of Theorem\ref{th:subadditivity of prefix algorithmic entropy}. \newpage \section{The algorithmic approach to Classical Chaos Theory and Brudno's Theorem} Let us review the basic notions of Classical Ergodic Theory: given a classical probability space $ ( X \, , \, \mu ) $: \begin{definition} \label{def:endomorphism of a classical probability space} \end{definition} ENDOMORPHISM OF $ ( X \, , \, \mu ) $: $T \, : \, HALTING(\mu) \rightarrow HALTING(\mu) $ surjective : \begin{equation} \mu ( A ) \; = \; \mu ( T^{-1} A ) \; \; \forall A \in HALTING(\mu) \end{equation} \smallskip \begin{definition} \label{def:automorphism of a classical probability space} \end{definition} AUTOMORPHISM OF $ ( X \, , \, \mu ) $: $T \, : \, HALTING(\mu) \rightarrow HALTING(\mu) $ injective endomorphism of $ ( X \, , \, \mu ) $ \smallskip \begin{definition} \label{def:classical dynamical system} \end{definition} CLASSICAL DYNAMICAL SYSTEM: a therne $ ( X \, , \, \mu \, , \, T ) $ such that: \begin{itemize} \item $ ( X \, , \, \mu ) $ is a classical probability space \item $T \, : \, HALTING(\mu) \rightarrow HALTING(\mu) $ is an endomorphism of $ ( X \, , \, \mu ) $ \end{itemize} \medskip Given a classical dynamical system $ ( X \, , \, \mu \, , \, T ) $: \begin{definition} \label{def:reversible classical dynamical system} \end{definition} $ ( X \, , \, \mu \, , \, T ) $ IS REVERSIBLE: T is an automorphism \smallskip \begin{definition} \label{def:ergodic classical dynamical system} \end{definition} $ ( X \, , \, \mu \, , \, T ) $ IS ERGODIC: \begin{equation} lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1} \, \mu ( A \cap T^{k}(B)) \; = \; \mu(A) \, \mu(B) \; \forall \, A,B \in HALTING( \mu ) \end{equation} \begin{definition} \label{def:mixing classical dynamical system} \end{definition} $ ( X \, , \, \mu \, , \, T ) $ IS MIXING: \begin{equation} lim_{n \rightarrow \infty} \, \mu ( A \cap T^{n}(B)) \; = \; \mu(A) \, \mu(B) \; \forall \, A,B \in HALTING( \mu ) \end{equation} \medskip \begin{example} \end{example} CLASSICAL SHIFTS \begin{definition} \label{def:def:classical shift} \end{definition} CLASSICAL SHIFT OVER $ \Sigma $: a classical dynamical system $ ( \, \Sigma^{\infty} \, , \, \sigma \, , \, \mu ) $ such that: \begin{equation} \begin{split} \sigma & : \Sigma^{\infty} \rightarrow \Sigma^{\infty} \\ ( \sigma & \bar{x} ) _{n} \; := \; x_{n+1} \end{split} \end{equation} and: \begin{equation} HALTING( \mu ) \; = \; {\mathcal{F}}_{cylinder} \end{equation} \begin{remark} \label{rem:classical shift is synonimous of discrete-time stationary classical stochastic process} \end{remark} CLASSICAL SHIFT IS SYNONIMOUS OF DISCRETE-TIME STATIONARY CLASSICAL STOCHASTIC PROCESS The notion of a classical shift is nothing but a way of inglobing the Theory of Classical Stationary Stochastic Processes as a sub-discipline of Classical Ergodic Theory. As we will see in section\ref{sec:From the communicational-compression of the Quantum Coding Theorems to the algorithmic-compression in Quantum Computation} an analogous inglobation is possible in a quantum case. That the notion of a classical shift over $ \Sigma $ is indeed equivalent to the notion of a classical stationary stochastic process over $ \Sigma $ follows immediately by the following two facts: \begin{enumerate} \item every classical probability measure $ \mu $ on $ \Sigma^{\infty} $ such that $ HALTING( \mu ) \; = \; {\mathcal{F}}_{cylinder} $ individuates the classical stationary stochastic process over $ \Sigma $ with \textbf{occurence probability of strings}: \begin{equation} \label{eq:occurence probability of strings} p_{k} ( i_{1} , \cdots , i_{k} ) \; \equiv \; \mu ( \Gamma_{( i_{1} , \cdots , i_{k} )}) \; \; i_{1} , \cdots \ i_{k} \, \in \Sigma , k \in { \mathbb{N}} \end{equation} satisfying the conditions: \begin{equation} \label{eq:first constraint on the occurence probability of strings} p_{k} ( i_{1} \, , \, \cdots \, i_{k} \, ) \; \geq \; 0 \end{equation} \begin{equation} \label{eq:second constraint on the occurence probability of strings} \sum_{i \in \Sigma} p_{k+1} ( i_{1} \, , \, \cdots \, i_{k} \, i ) \; = \; p_{k} ( i_{1} \, , \, \cdots \, i_{k} \, ) \end{equation} \begin{equation} \label{eq:third constraint on the occurence probability of strings} \sum_{i \in \Sigma } p_{1} ( \, i \, ) \; = \; 1 \end{equation} \item the collection of functions: \begin{equation*} p_{k} ( i_{1} \, , \, \cdots \, i_{k} \, ) \; \; i_{1} \, , \, \cdots \, i_{k} \, \in \Sigma \, , \, k \in {\mathbb{N}} \end{equation*} expressing the \textbf{occurence probability of strings} of a classical stationary stochastic process (and, hence, satisfying the constraints eq.\ref{eq:first constraint on the occurence probability of strings}, eq.\ref{eq:second constraint on the occurence probability of strings}, eq.\ref{eq:third constraint on the occurence probability of strings}) individuates the $ \sigma$-invariant classical probability measure $ \mu $ on $ \Sigma^{\infty} $ such that $ HALTING( \mu ) \; = \; {\mathcal{F}}_{cylinder} $ and: \begin{equation} p_{k} ( i_{1} , \cdots , i_{k} ) \; = \; \mu ( \Gamma_{( i_{1} , \cdots , i_{k} )}) \; \; i_{1} , \cdots \ i_{k} \, \in \Sigma , k \in { \mathbb{N}} \end{equation} \end{enumerate} \smallskip Let us introduce some useful notion: \begin{definition} \label{def:stochastic vector} \end{definition} STOCHASTIC VECTOR OVER $ \Sigma $: $ \vec{P} \; = \; \begin{pmatrix} p_{0} \\ p_{1} \end{pmatrix}$ such that: \begin{align} p_{i} & \geq 0 \; \; i=0,1 \\ \sum_{i \in \Sigma } & p_{i} \; = \; 1 \end{align} i.e. a column vector specifying a probability distribution over $ \Sigma $. \smallskip \begin{definition} \label{def:stochastic matrix} \end{definition} STOCHASTIC MATRIX OVER $ \Sigma $: $ 2 \, \times \, 2 $ matrix: $ \hat{P} = \begin{pmatrix} p_{0,0} & p_{0 , 1 } \\ p_{1,0} & p_{1,1} \end{pmatrix} $ such that: \begin{align} p_{i,j} & \geq 0 \; \; i,j=0,1 \\ \sum_{j \in \Sigma} & p_{i,j} \; = \; 1 \; \; i=0,1 \end{align} \smallskip We can now introduce some basic classical shift: \begin{definition} \label{def:classical bernoulli shift} \end{definition} CLASSICAL BERNOULLI SHIFT OVER $ \Sigma $ W.R.T. THE STOCHASTIC VECTOR $ \vec{P} \; = \; \begin{pmatrix} p_{0} \\ p_{1} \end{pmatrix}$: the classical shift over $ \Sigma $ with measure $ \mu $ : \begin{equation} p_{k} ( \, i_{1} \, , \, \cdots \, i_{k} \, ) \; = \; \prod_{l=1}^{k} p ( \, i_{l} \, ) \; \; \forall k \in {\mathbb{N} } \end{equation} \smallskip Given a stochastic vector $ \vec{e} \; := \; \begin{pmatrix} e_{0} \\ e_{1} \end{pmatrix}$ and a stochastic matrix $ \hat{P} \; := \begin{pmatrix} p_{00} & p_{01} \\ p_{10} & p_{11} \end{pmatrix}$ such that: \begin{equation} ( \vec{P} )^{T} \hat{P} \; = \; \hat{P} \end{equation} \begin{definition} \label{def:classical markov shift} \end{definition} CLASSICAL MARKOV SHIFT OVER $ \Sigma $ W.R.T. THE STOCHASTIC VECTOR $ \vec{e} $ AND THE STOCHASTIC MATRIX $ \hat{P} $ the classical shift over $ \Sigma $ with measure $ \mu $ : \begin{equation} p_{k} ( i_{1} \, , \, \cdots \, i_{k} ) \; = \; e_{i_{1}} \, p_{i_{1} , i_{2}} \, \cdots \, p_{i_{k-1} , i_{k}} \end{equation} \smallskip \begin{remark} \label{rem:classical shift is synonimous of stationary classical information source} \end{remark} CLASSICAL SHIFT IS SYNONIMOUS OF STATIONARY CLASSICAL INFORMATION SOURCE We have seen in remark\ref{rem:classical shift is synonimous of discrete-time stationary classical stochastic process} that the notion of classical shift over $ \Sigma $ is equivalent to the notion of a stationary classical stochastic process over $ \Sigma $. But a classical stochastic process $ \{ x_{n} \} $ may be equivalentely seen as a classical information source, considering the random variable $ x_{n} $ as the letter trasmitted at time n by a sender (Alice) to a receiver (Bob) through a proper communicational channel. The resulting equivalence between the notion of a classical shift and the notion of a stationary classical information source persists at the quantum level as we will see in the section\ref{sec:The algorithmic approach to Quantum Chaos Theory: quantum algorithmic information versus quantum dynamical entropies} \medskip Given a classical probability space $ ( X \, , \mu ) $: \begin{definition} \label{def:partition of a classical probability space} \end{definition} FINITE MEASURABLE PARTITION OF $ ( X \, , \, \mu ) $: \begin{equation} \begin{split} A \, & = \; \{ \, A_{1} \, , \, \cdots \, A_{n} \} \; n \in {\mathbb{N}} \, : \\ A_{i} & \in HALTING(\mu) \; \; i \, = \, 1 \, , \, \cdots \, n \\ A_{i} & \, \cap \, A_{j} \, = \, \emptyset \; \; \forall \, i \, \neq \, j \\ \mu & ( \, X \,- \, \cup_{i=1}^{n} A_{i} \, ) \; = \; 0 \end{split} \end{equation} \smallskip We will denote the set of all the finite measurable partitions of $ ( X \, , \, \mu ) $ by $ \mathcal{P} ( \, X \, , \, \mu \, ) $. \begin{definition} \end{definition} $ A \in \mathcal{P} ( X , \mu ) $ IS FINER THAN $ B \in \mathcal{P} ( X , \mu ) $: every atom of B is the union of atoms by A \smallskip \begin{definition} \end{definition} COARSEST REFINEMENT OF $ A = \{ A_{i} \}_{i=1}^{n} $ AND $ B = \{ B_{j} \}_{j=1}^{m} \in {\mathcal{P}}( \; X \, , \mu \; ) $: \begin{equation} \begin{split} A \, & \vee \, B \; \in {\mathcal{P}}( X , \mu ) \\ A \, & \vee \, B \; \equiv \; \{ \, A_{i} \, \cap \, B_{j} \, \; i =1 , \cdots , n \; j = 1 , \cdots , m \} \end{split} \end{equation} Clearly $ \mathcal{P} ( X , \mu ) $ is closed both under coarsest refinements and under endomorphisms of $ ( X , \mu ) $. \medskip \begin{remark} \end{remark} COARSE-GRAINED MEASUREMENTS ON A CLASSICAL PROBABILITY SPACE: THE OPERATIONAL MEANING OF A CLASSICAL PARTITION Beside its abstract, mathematical formalization, the definition \ref{def:partition of a classical probability space} has a precise operational meaning. Given the classical probability space $ ( X , \mu ) $ let us suppose to make an experiment on the probabilistic universe it describes using an instrument whose resolutive power is limited in that it is not able to distinguigh events belonging to the same atom of a partition $ A = \{ A_{i} \}_{i=1}^{n} \in \mathcal{P} ( X , \mu ) $. Conseguentially the outcome of such an experiment will be a number \begin{equation} r \in \{ 1 , \cdots , n \} \end{equation} specifying the observed atom $ A_{r} $ in our coarse-grained observation of $ ( X, \mu ) $. \smallskip We will call such an experiment an \textbf{operational observation of $ ( X , \mu ) $ through the partition A}. \smallskip Considered another partition $ B = \{ B_{j} \}_{i=1}^{n} \in \mathcal{P} ( X , \mu ) $ we have obviously that the operational observation of $ ( X , \mu ) $ through the partition $ A \vee B $ is the conjuction of the two experiments consisting in the operational observations of $ ( X , \mu ) $ through the partitions, respectively, A and B. Conseguentially we may consistentely call an \textbf{operational observation of $ ( X , \mu ) $ through the partition A} more simply an \textbf{A experiment}. \medskip \begin{remark} \end{remark} THE DOUBLE MEANING OF THE CLASSICAL PROBABILISTIC SHANNON ENTROPY OF AN EXPERIMENT The experimental outcome of an operational observation of $ ( X , \mu ) $ through the partition $ A = \{ A_{i} \}_{i=1}^{n} \in \mathcal{P} ( X , \mu ) $ is a classical random variable having as distribution the stochastic vector $ ( \begin{pmatrix} \mu(A _{1} ) \\ \vdots \\ \mu(A _{n}) \end{pmatrix} $ whose Shannon entropy we will call the entropy of the partition A, according to the following: \begin{definition} \end{definition} ENTROPY OF $ A = \{ A_{i} \}_{i=1}^{n} \in \mathcal{P} ( X , \mu ) $: \begin{equation} H(A) \equiv H ( \begin{pmatrix} \mu ( A _{1} ) \\ \vdots \\ \mu ( A _{n} ) \ \end{pmatrix} ) \end{equation} with the right hand side expressed in terms of the definition\ref{def:Shannon entropy of a distribution} we introduced in section\ref{sec:Why prefix entropy is better than simple entropy}. It is fundamental, at this point, to observe that, given an experiment, one has to distinguish between two conceptually different concepts: \begin{enumerate} \item the \textbf{uncertainty of the experiment}, i.e. the amount of uncertainty on the outcome of the experiment before of realizing it \item the \textbf{information of the experiment}, i.e. amount of information gained by the outcome of the experiment \end{enumerate} The fact that in Classical Probabilistic Information Theory both these concepts are quantified by the Shannon entropy of the experiment is a conseguence of the following (cfr. pag. 62 of \cite{Billingsley-65}): \begin{theorem} \label{th:the soul of Classical Information Theory} \end{theorem} THE SOUL OF CLASSICAL INFORMATION THEORY \begin{equation} \text{information gained} \; = \; \text{uncertainty removed} \end{equation} \smallskip Theorem\ref{th:the soul of Classical Information Theory} applies, in particular, as to the partition-experiments we are discussing. \medskip Let us now consider a classical dynamical system $ CDS \, := \, ( X \, , \, \mu \, , \, T ) $. The T-invariance of $ \mu $ implies the the partitions $ A = \{ A_{i} \}_{i=1}^{n} \in \mathcal{P} ( X , \mu ) $ and $ T^{-1}A $ have equal probabilistic structure. Conseguentially the \textbf{A experiment} and the \textbf{ $T^{-1}A $-experiment} are repliques , \textbf{not necessarily independent}, of the same experiment , made at succesive times. In the same way the \textbf{$ \vee_{k=0}^{n-1} \, T^{-k} A $-experiment} is the compound experiment consisting in n replications $ A \, , \, T^{-1} A \, , \, , \cdots , \, T^{-(n-1)}A $ of the experiment corresponding to $ A \in {\mathcal{P}}(X , \mu) $. The amount of classical information per replication we obtain in this compound experiment is clearly: \begin{equation*} \frac{1}{n} \, H(\vee_{k=0}^{n-1} \, T^{-k} A ) \end{equation*} It may be proved (cfr. e.g. the second paragraph of the third chapter of \cite{Kornfeld-Sinai-00}) that when n grows this amount of classical information acquired for replication converges, so that the following quantity: \begin{equation} h( A , T ) \; \equiv \; lim_{n \rightarrow \infty} \, \frac{1}{n} \, H(\vee_{k=0}^{n-1} \, T^{-k} A ) \end{equation} does exist. In different words, we can say that $ h( A , T ) $ gives the asymptotic rate of production of classical information for replication of the A-experiment. \begin{definition} \label{def:Kolmogorov-Sinai entropy} \end{definition} \begin{equation} h_{CDS} \; \equiv \; sup_{A \in {\mathcal{P}}(X , \mu)} \, h( A , T ) \end{equation} By definition we have clearly that: \begin{equation} h_{CDS} \; \geq \; 0 \end{equation} \begin{definition} \label{def:classical chaoticity} \end{definition} CDS IS CHAOTIC: \begin{equation} h_{CDS} \; > \; 0 \end{equation} \smallskip \begin{remark} \end{remark} INFORMATION-THEORETIC NATURE OF THE CONCEPT OF CLASSICAL CHAOS Definition\ref{def:classical chaoticity} shows explicitly that the concept of classical-chaos is an information-theoretic one: a classical dynamical system is chaotic if there exist at least one experiment on the system that, no matter how many times we insist on repeating it, continue to give us classical information. That such a meaning of classical chaoticity is equivalent to the more popular one as the sensible (i.e. exponential) dependence of dynamics from the initial conditions (the so called \textbf{butterfly effect} for which the little perturbation of the atmospheric flow produced here by a butterfly's flight may produce an hurricane in Alaska) is a conseguence of Pesin's Theorem stating (under mild assumptions) the equality of the Kolmogorov-Sinai entropy and the sum of the positive Lyapunov exponents. This inter-relation may be caught observing that: \begin{itemize} \item if the system is chaotic we know that there exist an experiment whose repetition definitely continues to give information: such an information may be seen as the information on the initial condition that is necessary to furnish more and more with time if one want to keep the error on the prediction of the phase-point below a certain bound \item if the system is not chaotic the repetition of every experiment is useful only a finite number of times, after which every suppletive repetition doesn't furnish further information \end{itemize} Let us now consider the issue of symbolically translating the coarse-gained dynamics: the traditional way of proceeding is that described in the second section of \cite{Alekseev-Yakobson-1981}: given a positive integer $ n \in {\mathbb{N}} $ let us introduce the: \begin{definition} \end{definition} n-LETTERS ALPHABET: \begin{equation} \Sigma_{n} \; := \; \{ 0 , \cdots , n - 1 \} \end{equation} Clearly: \begin{equation} \Sigma_{2} \; = \; \Sigma \end{equation} Considered a partition $ A \, = \, \{ A_{i} \}_{i = 1}^{n} \in \, {\mathcal{P}}(X , \mu) $ \begin{definition} \label{def:symbolic translator w.r.t. a partition} \end{definition} SYMBOLIC TRANSLATOR OF CDS W.R.T. A: $ \psi_{A} \, : \, X \rightarrow \Sigma_{n} $: \begin{equation} \psi_{A} ( x ) \; \equiv \; j \, : \, x \in A_{j} \end{equation} In this way one associate to each point of X the letter, in the alphabet having as many letters as the number of atoms of the considered partition, labelling the atom to which the point belongs. Concatenating the letters corresponding to the phase-point at different times one can then codify $ k \in {\mathbb{N}}$ steps of the dynamics: \begin{definition} \label{def:n-point symbolic translator w.r.t. a partition} \end{definition} k-POINT SYMBOLIC TRANSLATOR OF CDS W.R.T. A: $ \psi_{A}^{(k)} \, : \, X \, \rightarrow \Sigma_{n}^{k} $: \begin{equation} \psi_{A}^{(k)} ( x ) \; \equiv \; \cdot_{j = 1}^{n} \psi ( T^{j} x ) \end{equation} and whole orbits: \begin{definition} \label{def:orbit symbolic translator w.r.t. a partition} \end{definition} $ \psi_{A}^{(\infty)} \, : \, X \, \rightarrow \, \Sigma_{n}^{\infty} $: \begin{equation} \psi_{A}^{( \infty)} ( x ) \; \equiv \; \cdot_{j = 1}^{\infty} \psi ( T^{j} x ) \end{equation} \medskip The bug of this strategy of symbolic translation is the dependence of the used alphabet from the cardinality of the partition: \begin{multline} cardinality(A) \, \neq \, cardinality(B) \; \Rightarrow \\ Range[ \psi_{A} ( x) ] \, \neq \, Range[ \psi_{B} ( x ) ] \; \; \forall x \in A \, , \, \forall A , B \in {\mathcal{P}}( \, X \, , \, \mu ) \end{multline} We will therefore adopt a different strategy of symbolic-coding using only the binary alphabet $ \Sigma $ based on the following: \begin{definition} \label{def:universal symbolic translator} \end{definition} UNIVERSAL SYMBOLIC TRANSLATOR OF CDS: $ \Psi \, : \, X \, \times \, {\mathcal{P}}( \, X \, , \, \mu ) \rightarrow {\Sigma^{\star} } $: \begin{equation} \Psi ( x \, , \, \{ A_{i} \}_{i = 1}^{n} ) \; \equiv \; string( j) \; \; , \, x \, \in \, A_{j} \end{equation} that can be again used to codify $ k \in {\mathbb{N}}$ steps of the dynamics: \begin{definition} \label{def:k-point universal symbolic translator} \end{definition} k-POINT UNIVERSAL SYMBOLIC TRANSLATOR OF CDS: $ \Psi^{(k)} \, : \, X \, \times \, {\mathcal{P}}( \, X \, , \, \mu ) \rightarrow \Sigma^{\star} $: \begin{equation} \Psi^{(k)} ( x \, , \, \{ A_{i} \}_{i = 1}^{n} ) \; \equiv \; \cdot_{j = 1}^{k} \psi ( T^{j} x \, , \, \{ A_{i} \}_{i = 1}^{n} ) \end{equation} and whole orbits: \begin{definition} \label{def:orbit universal symbolic translator} \end{definition} ORBIT UNIVERSAL SYMBOLIC TRANSLATOR OF CDS: $ \Psi^{(\infty)} \, : \, X \, \times \, {\mathcal{P}}( \, X \, , \, \mu ) \rightarrow \Sigma^{\infty} $: \begin{equation} \Psi^{(\infty)} ( x \, , \, \{ A_{i} \}_{i = 1}^{n} ) \; \equiv \; \cdot_{j = 1}^{\infty} \psi ( T^{j} x \, , \, \{ A_{i} \}_{i = 1}^{n} ) \end{equation} \smallskip The asymptotic rate of of production of simple algorithmic entropy of the binary sequence obtained translating symbolically the orbit generated by $ x in X $ through the partition $ A \; \in \; {\mathcal{P}}( \, X \, , \, \mu ) $ is clearly given by: \begin{equation} B(A , x ) \; := \; B( \Psi^{(\infty)} ( x, A ) ) \end{equation} We saw in definition\ref{def:Kolmogorov-Sinai entropy} that the Kolmogorov-Sinai entropy was defined as $ K(A , x) $ calculated on the more probabilistically-informative A-experiment; in the same way the Brudno algorithmic entropy of x is defined calculating $ B(A,x) $ on the more algorithmically-informative A-experiment: \begin{definition} \label{def:Brudno algorithmic entropy of a point} \end{definition} \begin{equation} B_{CDS} (x) \; \equiv \; sup_{A \in {\mathcal{P}}(X , \mu)} \, B(A , x ) \end{equation} \medskip We can now introduce the fundamental: \begin{theorem} \label{th:Brudno theorem} \end{theorem} BRUDNO'S THEOREM \begin{equation} h_{CDS} \; = \; B_{CDS} (x) \; \; \forall - \mu - a.e. \, x \in X \end{equation} \begin{proof} Conceptually if follows from Theorem\ref{th:link between mathematically classical mathematically physical probabilistic information and mathematically classical mathematically physical algorithmic information} and Theorem\ref{th:plain-prefix insensitivity of Brudno algorithmic entropy}. Given it for granted, the complete proof given in \cite{Brudno-83} adopting the traditional strategy of symbolic translation, let us observe that the passage to our strategy of symbolic translation simply involves the application of the quasi-lexicographic map whose bijectivity guarantees the equivalence \end{proof} \medskip Let us now consider the \textbf{algorithmic approach to Classical Chaos Theory} strongly supported by Joseph Ford, whose objective is the characterization of the concept of chaoticity of a classical dynamical system as the algorithmic-randomness of its symbolically-translated trajectories. To require such a condition for all the trajectories would be too restrictive since it is reasonable to allow a chaotic dynamical system to have a numerable number of periodic orbits. Let us introduce then following two notions: \begin{definition} \label{def:strong algorithmic chaoticity} \end{definition} CDS IS STRONGLY ALGORITHMICALLY-CHAOTIC: \begin{equation} \forall-\mu-a.e. x \in X \, , \, \exists A \in {\mathcal{P}}(X , \mu) \, : \, \Psi^{(\infty)} ( x \, , \, A) \in CHAITIN-RANDOM( \Sigma^{\infty}) \end{equation} \begin{definition} \label{def:weak algorithmic chaoticity} \end{definition} CDS IS WEAK ALGORITHMICALLY-CHAOTIC: \begin{equation} \forall-\mu-a.e. x \in X \, , \, \exists A \in {\mathcal{P}}(X , \mu) \, : \, \Psi^{(\infty)} ( x \, , \, A) \in BRUDNO-RANDOM( \Sigma^{\infty}) \end{equation} The difference between definition\ref{def:strong algorithmic chaoticity} and definition\ref{def:weak algorithmic chaoticity} follows by Theorem\ref{th:Brudno randomness is weaker than Chaitin randomness}. Clearly Theorem\ref{th:Brudno theorem} implies the following: \begin{corollary} \end{corollary} \begin{align*} CHAOTICITY \; \; & = \; \; \text{WEAK ALGORITHMIC CHAOTICITY} \\ CHAOTICITY \; \; & < \; \; \text{STRONG ALGORITHMIC CHAOTICITY} \end{align*} that shows that the algorithmic approach to Classical Chaos Theory is equivalent to the usual one only in weak sense. \newpage \chapter{Classical algorithmic randomness as passage of all the classical algorithmic statistical tests} \label{chap:Classical algorithmic randomness as passage of all the classical algorithmic statistical tests} \section{Pseudorandom generators} We have seen in chapter\ref{chap:Classical algorithmic randomness as classical algorithmic incompressibility} how the notion of classical algorithmic randomness as classical algorithmic incompressibility may be properly formalized. The deepest way of introducing the characterization of classical algorithmic randomness as passage of a certain battery of statistical tests is to analyze the issue of \textbf{random number generation}. Such an expression is an oxymoron: the same fact that there exist an algorithm by which we generate an object on a classical deterministic computer means that such an object is algorithmically-compressible trough the adopted algorithm and, hence, is not algorithmically-random. This observation was condensed by John Von Neumann in his famous sentence: \begin{center} \textit{"Anyone who considers arithmetic methods of producing random digits is, of course, in a state of sin"} \end{center} What a pseudorandom number generator (a PRG from here and beyond) outputs are \textbf{pseudorandom numbers}, i.e. numbers who mimic true algorithmic randomness up to a certain degree of accuracy, i.e pass a suffiencetely extended battery of randomness tests. Before embarking in abstract mathematical definitions about what does it mean it is rather useful to make a previous breaf historical analysis of the concrete problem of pseudo-random number generation in Theoretical Computer Science. Von Neumann himself introduced an arithmetic pseudorandom-generation method known as the \textbf{middle square method}: supponing to want to generate m random integers of 10 digits starting from a certain 10' digits integer \textbf{seed} , such method is defined through the following algorithm: \begin{enumerate} \item set $ i \leftarrow 0 $ \item set $ n_{i} \, \leftarrow \, seed $ \item \label{it:begin loop} square $ n_{i} $ to get an intermediate number M with 20 or less digits \item set $ i \leftarrow i+1 $ \item set $ n_{i} \, \leftarrow \, $ the middle ten digits of M \item if $ i \, < \, m $ then goto step\ref{it:begin loop}, else halt \end{enumerate} \medskip A serious problem with the middle-square method is that the orbit generated by many seeds is periodic with a very little period \cite{Yan-00}. \medskip In 1949 D.H. Lehmer proposed to use the \textbf{Theory of congruences}, i.e. the theory of the residue classes $ { \mathbb{Z}}_{n} $ modulo n (a ring on integers being a field if and only if n is a prime number) to generate pseudorandom numbers. Fixed the following numbers: \begin{itemize} \item n : the \textbf{modulus} , $ n \, > \, 0 $ \item $x_{0}$ : the \textbf{seed}, $ 0 \, \leq \, x_{0} \, \leq n $ \item a : the \textbf{multiplier} $ 0 \, \leq \, a \, \leq n $ \item b : the \textbf{increment} $ 0 \, \leq \, b \, \leq n $ \end{itemize} the \textbf{linear congruential generator} generates the sequence of pseudorandom numbers defined recursively by: \begin{equation} x_{j} \; := \; a x_{j-1} + b \; (mod \, n) \, j >0 \end{equation} for $ 1 \, \leq \, j \, \leq \, l $ where $ l \in { \mathbb{N}} $ is the least value such that $ x_{l+1} \, \equiv \, x_{j} \; (mod \, n) $ i.e. is the \textbf{period} of the PRG. Since the period is less or equal to the modulus $ l \, \leq \, n $ to have a PRG of sufficient quality it is necessary to use high enough moduli. For fixed n one would, then like to optimize the situation choosing the other parameters so that the period is equal to the modulus. A necessary and sufficient condition for this to happen is given by the following \cite{Knuth-98}: \begin{theorem} \end{theorem} THEOREM OF GREENBERGER, HULL, DOBELL \begin{equation} l \, = \, n \; \Leftrightarrow \; ( gcd(b,n) = 1 \, and \, a \equiv 1 (mod \, p) \forall primes \, p | n , \, and \, a \equiv 1 (mod \, 4) \, if \, 4 | n ) \end{equation} \medskip The \textbf{linear congruential generator} had (and continues to have) an immense application: for example RANDU, the random number generator common on IBM mainframes computers in the sixties, was based on a linear congruential generator with parameters $ a = 65539 \, , \, b = 0 \, , n = 2^{31} $. However it was soon found that RANDU gave rise to insidious correlations: if successive triples of the numbers that RANDU generated were used as a set of coordinates in a three-dimensional space, the generated distribution of point seems random from most viewpoints but there exist a special orientation from wich one can see that they lie in a set of planes \cite{Williams-Clearwater-98}. \medskip The linear congruential method is still of great popularity: for example it is often used the \textbf{minimal standard 32-bit generator} obtained by the choice $ a = 16807 \, , \, b = 0 \, , \, n = 2^{31} - 1 $ adopted by many programming languages such as TURBO PASCAL even if its lack of randomness may be easily visualized \cite{Denker-Woyczysnki-98}. \medskip To avoid the bugs of linear congruential generators, a new kind of generators, the \textbf{shift-register-generator}, was then introduced. In these generators each successive number depends upon many preceding values. The basic operation may still be the \textbf{modular addition} or other functions such as the \textbf{exlusive or}. For example a commonly used algorithm is the following: \begin{equation} x_{j} \; := \; x_{j-p} \, XOR \, x_{j-q} \end{equation} The best choice of the pair of integers p and q happens when p and q are \textbf{Mersenne primes} such that \cite{Landau-Binder-00} : \begin{equation} p^{2} + q^{2} + 1 \text{ is prime} \end{equation} A common choice of this kind is $ p = 250 \, , \, q = 103 $. \medskip Anyway it would be an error to think that this method (or their evolutions such as the \textbf{subtract with carry generators} or the \textbf{subtract with carry Weyl generators}) is, in absolute, better than the linear congruential method: there exist Monte-carlo simulations of the bidimensional Ising model for which the minimal standard 32-bit generator is not inficiated by systematic errors of younger algorithms producing deviations from Onsager's known solution. \medskip Let us conclude this hystorical review mentioning Wolfram's suggestion of using some TYPE 3 elementary cellular automata as pseudo-random generators \cite{Wolfram-94} that he himself adopted for the integer-random number generator of Mathematica (Rule[30] while for real numbers it is used a Marsaglia-Zaman subtract with borrow \cite{Wolfram-96}). \bigskip This panoramic on concrete pseudorandom number generators, hasn't, anyway, touched the basis question: what is a \textbf{pseudorandom generator}? The answer that we have already anticipated, i.e. an algorithm producing binary strings passing an enough number of \textbf{randomness statistical tests}, may appear satisfying since anyone, eventually from his undergraduate $ \chi^{2} $ - test experiences, has an intuitive idea of what a randomness statistical test is: a well-defined procedure that allows to catch some kind of regularity of the statistical data. \medskip For all practical purposes the definition of the notion of pseudo-random generator may indeed be accomplished concretely specifying a set of randomness statistical tests it has to pass: (e.g. the Kolmogorov-Smirnov test + the Frequency test + the Serial test + the Gap test + the Partition test + the Coupon collector's test + the Permutation test + the Run test + the Maximum-of t-test + the Collision test + the Birthday spacing test + Serial correlation test \cite{Knuth-98}). The higher is the number of elements of this list of randomness statistical tests, the higher is the quality of the generator. \medskip From a conceptual point of view, anyway, such a definition is unsatisfactory since: \begin{itemize} \item it ultimately doesn't say what a \textbf{statistical randomness test} is \item it doesn't clarify the structure of the set of the \textbf{statistical randomness tests} and, conseguentially, the meaning of considering a subset of it. \end{itemize} A satisfactory answer to both these points may be given introducing Per Martin - L\"{o}f's Theory of Randomness Statistical tests. \medskip The idea behind Martin - L\"{o}f's Theory is the following: in statistical practice we are given an element x of some sample space (associated with some distribution that we will assume from here and beyond to be the unbiased one) and we want to test the hypothesis: \textbf{x is a typical outcome}. Being \textbf{typical} means \textit{"belonging to every reasonable majority"}. An element will be \textbf{random} just in the case it lies in the intersections of all such majorities. A level of a statistical test is a set of strings which are found to be relatively not-random (by the test). Each level is a subset of the previous level, containing less and less strings, considered more and more not-random. the number of strings decreases exponentially fast at each level. So a test contains at level 0 all possible strings, at level 2 at most $ \frac{1}{2} $ of the strings, at level three only $ \frac{1}{4} $ of all strings and so on; accordingly at level m the test contains at most $ 2^{n-m} $ strings of length n. \smallskip \begin{definition} \label{def:uniform-test for strings} \end{definition} A recursively enumerable (r.e.) set $ V \, \subset \, \Sigma^{\star} \, \times \, {\mathbb{N}}_{+} $ is a \textbf{Martin - L\"{o}f test} if: \begin{enumerate} \item \begin{equation} V_{m+1} \; \subset \; V_{m} \; \; \forall m \geq 1 \end{equation} where: \begin{equation} \label{eq:critical region for the uniform-test} V_{m} \; := \; \{ \vec{x} \in \Sigma^{\star} \, : \, ( \vec{x} , m ) \in V \} \end{equation} is called the \textbf{critical region of the test at level $ \frac{1}{2^{m} }$} \item \begin{equation} cardinality( \Sigma^{n} \cap V_{m} ) \; < \; 2^{n-m} \; \; \forall n \geq m \geq 1 \end{equation} \end{enumerate} \medskip Given a Martin - L\"{o}f test V and an integer number q: \begin{definition} \end{definition} SET OF THE q-PSEUDORANDOM STRINGS FOR V: \begin{equation} q-PSEUDORANDOM ( \Sigma^{\star} ; V ) \; := \; \{ \vec{x} \in \Sigma^{\star} \, : \, \vec{x} \notin V_{q} \, and \, q < | \vec{x} | \} \end{equation} \medskip Given a Martin - L\"{o}f test U: \begin{definition} \end{definition} U IS UNIVERSAL: for every Martin - L\"{o}f test V there exist a constant c (depending upon U and V) such that: \begin{equation} V_{m+c} \; \subset \; U_{m} \; \; m = 1, 2, \cdots \end{equation} \medskip A universal Martin - L\"{o}f statistical test is a Martin - L\"{o}f statistical test that is as strong as any other Martin-Martin - L\"{o}f test. So it is reasonable to fix once and for all a universal Martin - L\"{o}f test U and, given an integer number q, define: \begin{definition} \end{definition} SET OF THE MARTIN L\"{O}F - q - RANDOM STRINGS: \begin{equation} MARTIN L\ddot{O}F - q - RANDOM( \Sigma^{\star} ) \; := \; q-PSEUDORANDOM ( \Sigma^{\star} ; U ) \end{equation} \medskip We can then introduce the following basic notion: \begin{definition} \label{def:Martin Lof PRG} \end{definition} MARTIN L\"{O}F PRG OF QUALITY $ q \in {\mathbb{N}}$: a PRG whose outputs belongs to $ MARTIN L\ddot{O}F - q - RANDOM( \Sigma^{\star} ) $ \medskip It must be remarked, any way, that this is not the only possible way one can follow in order of formalizing the concept of a PRG. For example, following Oded Goldreich \cite{Goldreich-01}, one can found the Theory of Pseudorandom Generation on Structural Complexity Theory \cite{Odifreddi-99a} defining a PRG as an \textbf{efficient} algorithm that stretches short random strings into longer strings that are computationally indistinguishable from long random strings, in the sense that the difference can't be certified in an \textbf{efficient} way. Let us shortly show how this can be formalized. \smallskip Adhering to Goldreich's terminology we will speak of \textbf{Classical Turing machines} instead of \textbf{partial recursive functions} remembering that by the \textbf{Church-Turing's Thesis} that is exactly the same. Given a Turing machine M: \begin{definition} \label{def:polynomial time Turing machine} \end{definition} M IS POLYNOMIAL-TIME: there exist a polynomial p such that for every $ \vec{x} \in \Sigma^{\star} $, when invoked on input x, M halts after at most $ p ( | \vec{x} | ) $ steps \medskip We will will consider the following particular kind of sequence of probability distributions: \begin{definition} \end{definition} PROBABILITY ENSEMBLE OVER $ \Sigma^{\star} $: a sequence $ \{ P_{n} \}_{n \in {\mathbb{N}}} $ of probability distributions over $ \Sigma^{\star} $ with the property that there exist a polynomial p such that: \begin{equation} P_{n} ( \vec{x} ) \, > \, 0 \; \Rightarrow \; | \vec{x} | \, = \, p(n) \end{equation} \smallskip Given two probability ensembles $ \{ P_{n} \}_{n \in {\mathbb{N}}} $ and $ \{ Q_{n} \}_{n \in {\mathbb{N}}} $: \begin{definition} \label{def:computational indistinguishability} \end{definition} $ \{ P_{n} \}_{n \in {\mathbb{N}}} $ AND $ \{ Q_{n} \}_{n \in {\mathbb{N}}} $ ARE COMPUTATIONAL INDISTINGUISHABLE: for every probabilistic polynomial-time Turing machine M, for every positive polynomial $ p ( \cdot ) $ and for all sufficentely large n: \begin{equation} | Pr[ M ( 1^{n} , P_{n} ) = 1 ] \, - \, Pr[ M ( 1^{n} , Q_{n} ) = 1 ] | \; < \; \frac{1}{p(n)} \end{equation} \begin{definition} \label{def:Goldreich PRG} \end{definition} GOLDREICH PRG: a polynomial-time Turing machine G such that there exist a monotonically increasing function $ l : {\mathbb{N}} \rightarrow {\mathbb{N}} $ such that the following two probability ensembles, denoted by $ \{ G_{n} \}_{n \in {\mathbb{N}}} $ and $ \{ R_{n} \}_{n \in {\mathbb{N}}} $, are computationally indistinguishable: \begin{itemize} \item $ G_{n} $ is defined as the output of G on a uniformely-selected n-bits string \item $ R_{n} $ is defined as the uniform probability distribution on $ \Sigma^{l(n)} $ \end{itemize} \smallskip The link between definition\ref{def:Goldreich PRG} and Structural Complexity Theory passes through \textbf{one-way functions}, i.e. functions easy to compute but hard to invert: \begin{definition} \label{def:one-way function} \end{definition} ONE-WAY FUNCTION: a function $ f : \Sigma^{\star} \, \rightarrow \, \Sigma^{\star} $: \begin{itemize} \item easy to compute: f is computable in polynomial time \item hard to invert: for every probabilistic polynomial-time Turing machine M, for every positive polynomial $ p ( \cdot ) $ and for all sufficentely large n and $ \vec{x} $ uniformely distributed over $ \Sigma^{n} $ \begin{equation} Pr[ M ( 1^{n} , f( \vec{x} ) ) \in f^{- 1} ( f ( \vec{x} ) ) = 1 ] \; < \; \frac{1}{p(n)} \end{equation} \end{itemize} And here appears the following: \begin{conjecture} \label{con:fundamental conjecture of Structural Complexity Theory} \end{conjecture} FUNDAMENTAL CONJECTURE OF STRUCTURAL COMPLEXITY THEORY: \begin{equation} P \; \neq \; NP \end{equation} governing the following: \begin{theorem} \end{theorem} \begin{equation} \exists G \, \text{ Goldreich PRG } \; \Leftrightarrow \; \exists \, f \, \text{ one-way function } \; \Rightarrow \; \text{ Conjecture\ref{con:fundamental conjecture of Structural Complexity Theory} holds } \end{equation} \medskip The inter-relation between definition\ref{def:Martin Lof PRG} and definition\ref{def:Goldreich PRG} has not been analyzed yet. Ultimately it touches the issue of the link existing between Structural Complexity Theory and Algorithmic Information Theory that, though having been the subject of intensive investigation since the pioneristic Levin's analysis of the inter-relation among \textbf{perebor} (a russian term literally meaning \emph{"brute force"} and adopted from the late fifthies by the sovietic operations research community to denote the necessity of an exhaustive search of all the alternatives in certain search problems) and Kolmogororov's ideas (cfr. the sixth paragraph of the second part of \cite{Shasha-Lazerre-98}, the $ 7^{th} $ chapter of ), is still uncertain (cfr. \cite{Longpre-92}, the $ 7^{th} $ chapter of \cite{Li-Vitanyi-97} and the $ 10^{th} $ chapter of \cite{Balcazar-Diaz-Gabarro-90} ) \newpage \section{Equivalence between passage of a Martin L\"{o}f universal sequential statistical test and Chaitin randomness} \label{sec:Equivalence between passage of a Martin Lof universal statistical test and Chaitin randomness} In the last paragraph we have introduced the notion of a Martin-L\"{o}f test on strings only for the uniform distribution, the only one necessary in order to define a PRG. \smallskip Since, anyway, we will front, in the next sections, also not-uniform distributions it may be appropriate to give a more general definition. \smallskip Given a recursive probability distribution P on $ \Sigma^{\star} $: \begin{definition} \label{def:P-test for strings} \end{definition} MARTIN L\"{O}F TEST OF P-RANDOMNESS (P-TEST) : a function $ \delta : \Sigma^{\star} \, \rightarrow \, {\mathbb{N}} $: \begin{enumerate} \item the set $ V \; := \; \{ ( m , \vec{x} ) \, : \, \delta ( \vec{x}) > m \} $ is recursively enumerable \item \begin{equation} \sum_{\vec{x} \in \Sigma^{n}} \{ P( \vec{x} | | \vec{x} | = n \, : \, \delta ( \vec{x} ) \, \geq \, m \} \; \leq \; \frac{1}{ 2^{m}} \; \; \forall n \end{equation} \end{enumerate} \medskip Defined the \textbf{critical region of the test at level $ \frac{1}{2^{m} }$}, for any integer $ m \geq 1 $, as: \begin{equation} V_{m} \; := \; \{ \vec{x} \in \Sigma^{\star} \, : \, \delta ( \vec{x} ) \geq m \} \end{equation} it is immediate to see that for $ P = P_{unbiased} $ the definition\ref{def:P-test for strings} reduces to the definition\ref{def:uniform-test for strings}. \bigskip We would like, now, to extend this definition from $ \Sigma^{\star} $ to $ \Sigma^{\infty} $. Since an effective test can't be performed on an infinite sequence it is necessary to introduce an effective process of sequential approximations. \medskip So, given a recursive probability measure $ \mu $ on $ \Sigma^{\infty} $: \begin{definition} \label{def:sequential mu-test for sequences} \end{definition} SEQUENTIAL MARTIN L\"{O}F TEST OF $ \mu $-RANDOMNESS (SEQUENTIAL $ \mu $-TEST) : a function $ \delta : \Sigma^{\infty} \, \rightarrow \, {\mathbb{N}} \bigcup \{ \infty \} $: \begin{enumerate} \item \begin{equation} \delta ( \bar{x} ) \; = \; \sup_{n \in {\mathbb{N}}} \{ \gamma ( \vec{x}(n) ) \} \end{equation} where $ \vec{x}(n) \in \Sigma^{n} $ denotes the prefix of length n of the sequence $ \bar{x} $ while $ \gamma \, : \, { \Sigma^{\star} } \, \rightarrow \, {\mathbb{N}} $ is a \textbf{total enumerable function} (i.e. $ V \; := \; \{ ( m , \vec{y} ) : \gamma ( \vec{y} ) \geq m \} $ is a \textbf{recursively enumerable set} \item \begin{equation} \mu ( \{ \bar{x} \in \Sigma^{\infty} \, : \, \delta ( \bar{x} ) \geq m \} \; \leq \; \frac{1}{2^{m}} \; \; \forall m \geq 0 \end{equation} \end{enumerate} \medskip Given a \textbf{sequential $\mu$-test} $ \delta $ we have that a sequence $ \bar{x} \in \Sigma^{\infty} $ passes the test if $ \delta ( \bar{x} ) \, < \, \infty $ while it doesn't passes the test if $ \delta ( \bar{x} ) \, = \, \infty $. \smallskip The set of the sequences passing the test $ \delta $ are those that it declares random: \begin{definition} \label{def:mu-random sequences w.r.t. to a mu-test} \end{definition} SET OF THE $ \mu $-RANDOM SEQUENCES W.R.T. $ \delta $: \begin{equation} \mu - RANDOM ( \Sigma^{\infty} ; \delta ) \; := \; \{ \bar{x} \in \Sigma^{\infty} \, : \, \delta ( \bar{x} ) \, < \, \infty \} \end{equation} \medskip Clearly the definition\ref{def:mu-random sequences w.r.t. to a mu-test} depends on the particular sequential $ \mu $-test considered. \medskip This relativization can be, anyway, eliminated by the usual strategy of Algorithmic Information Theory: \begin{definition} \label{def:universal sequential mu-test for sequences} \end{definition} UNIVERSAL SEQUENTIAL MARTIN L\"{O}F TEST OF $ \mu $-RANDOMNESS (UNIVERSAL SEQUENTIAL $ \mu $-TEST) : a sequential $ \mu $ -test f such that for every other sequential $ \mu$ -test $ \delta $ there exist a constant $ c \geq 0 $ such that: \begin{equation} f ( \bar{x} ) \; \geq \; \delta ( \bar{x} ) - c \end{equation} \medskip A universal sequential $ \mu $-test is a sequential $ \mu $-test that is as strong as any other sequential $ \mu $-test. So it is reasonable to fix once and for all a universal sequential $ \mu $-test $ \delta_{0} ( \cdot | \mu ) $ and define: \begin{definition} \label{def:mu-random sequences} \end{definition} SET OF THE $ \mu $-RANDOM SEQUENCES: \begin{equation} \mu - RANDOM ( \Sigma^{\infty} ) \; := \; \mu - RANDOM ( \Sigma^{\infty} ; \delta_{0} ( \cdot | \mu ) ) \end{equation} \medskip To be a $ \mu $-random sequence is the $ \mu $-rule in $ \Sigma^{\infty} $ since: \begin{theorem} \label{th:foundation of the applicability of probability theory to reality} \end{theorem} FOUNDATION OF THE APPLICABILITY OF PROBABILITY THEORY TO REALITY \begin{equation} \mu [ \mu - RANDOM ( \Sigma^{\infty} ) ] \; = \; 1 \end{equation} \medskip The most important case from which, in a certain proper sense, all the others cases may be derived is when the measure $ \mu $ is the unbiased Lebesgue measure $ P_{unbiased} $ on $ \Sigma^{\infty} $. \begin{definition} \label{def:Martin-Lof random sequences} \end{definition} MARTIN-L\"{O}F RANDOM SEQUENCES: \begin{equation} MARTIN-L\ddot{O}F - RANDOM ( \Sigma^{\infty} ) \; := \; P_{unbiased} - RANDOM( \Sigma^{\infty} ) \end{equation} \smallskip We want now to present one of the more fundamental results of Algorithmic Information Theory: the Chaitin-Schnorr's Theorem. This requires, anyway, the introduction of some technicalities. Given a sequence $ \bar{x} \in \Sigma^{\infty} $ and a set of strings $ S \subset \Sigma^{\star} $ let us denote by $ S \, \Sigma^{ \infty } $ the set of all the sequences having the strings of S as prefixes, i.e.: \begin{equation} S \, \Sigma^{ \infty } \; := \; \{ \bar{x} \in \Sigma^{\infty} \, : \, \vec{x}(n) \in S \text{ for some natural } n \geq 1 \, \} \end{equation} lightening the notation for singletons by poning: \begin{equation} \vec{x} \Sigma^{\infty} \; := \{ \vec{x} \} \Sigma^{\infty} \; \; \vec{x} \in \Sigma^{\infty} \end{equation} We need the following: \begin{lemma} \label{lem:classical algorithmic randomness in terms of coverings} \end{lemma} \begin{multline} \bar{x} \in MARTIN-L\ddot{O}F - RANDOM ( \Sigma^{\infty} ) \; \Leftrightarrow \\ \forall Covering \in \Sigma^{\star} \times {\mathbb{N}} \text{ r.e. } : ( P_{unbiased} ( Covering_{n} \Sigma^{\infty} ) < \frac{1}{2^{n}} \, \forall n \geq 1 ) \; \\ \exists m \in {\mathbb{N}} \, : \, \bar{x} \notin Covering_{m} \Sigma^{\infty} \end{multline} where, as with the same notation of eq.\ref{eq:critical region for the uniform-test} that we will understand from here and beyond: \begin{equation} Covering_{n} \; := \; \{ \vec{x} \in \Sigma^{\star} \, : \, ( \vec{x}, n ) \in Covering \} \end{equation} \smallskip Indeed Lemma\ref{lem:classical algorithmic randomness in terms of coverings} is the starting point of a path leading to Solovay' s way of characterizing classical algorithmic randomness. The first step it to observe that one can always effectively pass from an arbitrary covering to a prefix-free one, as is stated by the following: \begin{lemma} \label{lem:effective passage from arbitrary to prefix-free covering} \end{lemma} For every r.e. set $ B \subset \Sigma^{\star} \times {\mathbb{N}}_{+} $, we can effectively find a r.e. set $ C \subset \Sigma^{\star} \times {\mathbb{N}}_{+} $ such that: \begin{align} C_{n} & \; \text{ is prefix-free} \; \; \forall n \in {\mathbb{N}}_{+} \\ B_{n} \Sigma^{\infty} & \; = \; C_{n} \Sigma^{\infty} \; \; \forall n \in {\mathbb{N}}_{+} \end{align} \medskip Lemma\ref{lem:classical algorithmic randomness in terms of coverings} and Lemma\ref{lem:effective passage from arbitrary to prefix-free covering} immediately imply the following: \begin{lemma} \label{lem:classical algorithmic randomness in terms of prefix-free coverings} \end{lemma} \begin{multline} \bar{x} \in MARTIN-L\ddot{O}F - RANDOM ( \Sigma^{\infty} ) \; \Leftrightarrow \\ \forall Covering \in \Sigma^{\star} \times {\mathbb{N}} \text{ r.e. } : ( Covering_{n} \text{ is prefix-free} \, \forall n ) \, and \,( P_{unbiased} ( Covering_{n} \Sigma^{\infty} ) < 2^{n} \, \forall n \geq 1 ) \; \\ \exists m \in {\mathbb{N}} \, : \, \bar{x} \notin Covering_{m} \Sigma^{\infty} \end{multline} that allows to show the equivalence between the passage of a universal sequential Martin-L\"{o}f test and Solovay randomness defined as: \begin{definition} \label{def:Solovay randomness} \end{definition} $ \bar{x} \in \Sigma^{\infty} $ IS SOLOVAY-RANDOM ( $ \bar{x} \in SOLOVAY-RANDOM( \Sigma^{\infty} ) ) $: \begin{multline} \forall X \subset \Sigma^{\star} \times {\mathbb{N}}_{+} \, r.e. \, : \sum_{n=1}^{\infty} P_{unbiased} ( X_{n} \Sigma^{\infty} ) \, < \, \infty \\ \exists N \in {\mathbb{N}} : \bar{x} \notin X_{n} \Sigma^{\infty} \, \, \forall n > N \end{multline} as stated by the following: \begin{theorem} \label{th:equivalence of Martin-Lof randomness and Solovay randomness} \end{theorem} \begin{equation} MARTIN-L\ddot{O}F - RANDOM ( \Sigma^{\infty} ) \; = \; SOLOVAY-RANDOM( \Sigma^{\infty} ) \end{equation} \begin{proof} \begin{itemize} \item $ SOLOVAY-RANDOM( \Sigma^{\infty} ) \; \subseteq \; MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) $ Clearly to prove the thesis is equivalent to show that: \begin{equation} \bar{x} \notin MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \; \Rightarrow \; \bar{x} \notin SOLOVAY-RANDOM( \Sigma^{\infty} ) \end{equation} Let us then assume that $ \bar{x} \notin MARTIN-L\ddot{O}F - RANDOM ( \Sigma^{\infty} ) $. By Lemma\ref{lem:classical algorithmic randomness in terms of prefix-free coverings} it follows that there exist a r.e. set $ X \subset \Sigma^{\star} \times {\mathbb{N}}_{+} $ such that: \begin{align} X_{n} & \; \text{ is prefix-free } \; \; \forall n \in {\mathbb{N}}_{+} \\ P_{unbiased} ( X_{n} \Sigma^{\infty} ) & \; < \; \frac{1}{2^{n}} \\ \bar{x} & \; \notin \; \bigcap_{n=1}^{\infty} X_{n} \Sigma^{\infty} \end{align} But then: \begin{equation} \sum_{n=1}^{\infty} P_{unbiased} ( X_{n} \Sigma^{\infty} ) \; \leq \; \sum_{n=1}^{\infty} \frac{1}{ 2^{n} } \; = \; 1 \; < \; \infty \end{equation} and conseguentially $ \bar{x} \, \notin SOLOVAY-RANDOM(\Sigma^{\infty} ) $ \item $MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \; \subseteq \; SOLOVAY-RANDOM( \Sigma^{\infty} ) $ Clearly to prove the thesis is equivalent to show that: \begin{equation} \bar{x} \notin SOLOVAY-RANDOM( \Sigma^{\infty} ) \; \Rightarrow \; \bar{x} \notin MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \end{equation} Let us then assume that $ \bar{x} \notin MARTIN-L\ddot{O}F - RANDOM ( \Sigma^{\infty} ) $. Conseguentially there exist a r.e. set $ X \subset \Sigma^{\star} \times {\mathbb{N}}_{+} $ such that: \begin{align} X_{n} & \; \text{ is prefix-free } \; \; \forall n \in {\mathbb{N}}_{+} \\ P_{unbiased} ( X_{n} \Sigma^{\infty} ) & \; < \; \frac{1}{2^{n}} \\ cardinality ( & \{ n \in {\mathbb{N}}_{+} \, : \, \bar{x} \, \in \, X_{n} \Sigma^{\infty} \} ) \; = \; \aleph_{0} \end{align} Given an arbitrary positive real number $ c \in {\mathbb{R}}_{+} $ let us introduce the set: \begin{equation} B \; := \; \{ ( \vec{y} , n ) \in \Sigma^{\star} \times {\mathbb{N}} \, : \, cardinality( \{ n \in {\mathbb{N}}_{+} : \vec{y} \in X_{n} \Sigma^{\star} \} ) \: > \: 2^{n+c} \} \end{equation} By construction: \begin{equation} P_{unbiased} ( B_{n} \Sigma^{\infty} ) \; < \; 2^{- n} \; \; \forall n \in {\mathbb{N}}_{+} \end{equation} Furthermore $ \bar{x} \in \bigcap_{n=1}^{\infty} B_{n} \Sigma^{\infty} $, i.e. for every natural $ n \geq 1 $ there exist a natural $ m \geq 1 $ such that: \begin{equation} cardinality( \{ n \in {\mathbb{N}}_{+} : \vec{x}(m) \in X_{n} \Sigma^{\star} \} \; > \; 2^{n+c} \end{equation} Just take $ m \, = \, \max \{ i_{1} , i_{2} , \cdots , i_{t} \}$, where $ t \; > \; 2^{n+c} $ and: \begin{equation} \bar{x} \; \in \; \bigcap_{j=1}^{t} X_{i_{t}} \Sigma^{\infty} \end{equation} \end{itemize} \end{proof} \medskip An other ingredient required for proving Chaitin-Schnorr's Theorem is (a slighty streghtened form of the) Chaitin-Levin's Theorem expressing the deep link existing between \textbf{prefix algorithmic entropy} and the \textbf{universal algorithmic probability} introduced by definition\ref{def:universal algorithmic probability}, namely the following: \begin{theorem} \label{th:strengthened Chaitin-Levin theorem} \end{theorem} \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; 0 \: \leq \: I ( \vec{x}) \, + \, \log_{2} P_{U} ( \vec{x}) \: \leq \: c \; \; \forall \vec{x} \in \Sigma^{\star} \end{equation} \begin{corollary} \label{cor:Chaitin-Levin theorem} \end{corollary} CHAITIN-LEVIN'S THEOREM \begin{equation} I( \vec{x}) \; \stackrel{ + }{=} \; - \log_{2} P_{U} ( \vec{x} ) \end{equation} \medskip The last ingredient required for proving Chaitin-Schnorr's Theorem is the following generalization of Theorem\ref{th:Kraft inequality} to arbitrary r.e. sets. \begin{theorem} \label{th:Kraft-Chatin theorem} \end{theorem} KRAFT-CHAITIN'S THEOREM \begin{hypothesis} \end{hypothesis} \begin{equation*} \phi \in C_{M}-C_{\Phi}-\Delta_{0}^{0}-\stackrel{ \circ } {MAP} ( {\mathbb{N}} , {\mathbb{N}} ) \; : \; HALTING(\phi) \text{ is an initial segment of } {\mathbb{N}}_{+} \end{equation*} \begin{thesis} \end{thesis} The following statements are equivalent: \begin{enumerate} \item We can effectively construct a function $ \theta \in C_{M}-C_{\Phi}-\Delta_{0}^{0}- \stackrel{ \circ } {MAP} ( {\mathbb{N}}_{+} , \Sigma^{\star}) $ such that: \begin{align} HALTING(\theta) & \; = \; HALTING(\phi) \\ | \theta ( n ) | & \; = \; \phi (n) \; \; \forall n \in HALTING( \phi) \\ Range(\theta) & \; \text{ is prefix-free} \end{align} \item \begin{equation} \sum_{n \in HALTING(\phi)} 2^{- \phi(n)} \; \leq \; 1 \end{equation} \end{enumerate} \medskip Let us finally afford our objective: \begin{theorem} \label{th:Chaitin-Schnorr theorem} \end{theorem} CHAITIN-SCHNORR'S THEOREM \begin{equation} MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \; = \; CHAITIN-RANDOM( \Sigma^{\infty} ) \end{equation} \begin{proof} \begin{itemize} \item $ MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \; \subseteq \; CHAITIN-RANDOM( \Sigma^{\infty} ) $ Clearly to prove the thesis is equivalent to show that: \begin{equation} \bar{x} \notin CHAITIN-RANDOM( \Sigma^{\infty} ) \; \Rightarrow \; \bar{x} \notin MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \end{equation} Let us assume that for every $ m > 0 $ there exists an $ n_{m} $ such that $ I( \vec{x} ( n_{m} ) ) \, < \, n_{m} $. By theorem\ref{th:strengthened Chaitin-Levin theorem} we know we can choose a natural number $ c > 0 $ such that: \begin{equation} \exists c \in {\mathbb{R}}_{+} \; : \; 0 \: \leq \: I ( \vec{x}) \, + \, \log_{2} P_{U} ( \vec{x}) \: \leq \: c \; \; \forall \vec{x} \in \Sigma^{\star} \end{equation} Let us introduce the set: \begin{equation} Covering \; := \; \{ ( \vec{y} ,t ) \in \Sigma^{\star} \times {\mathbb{N}}_{+} \: : \: I ( \vec{y} ) \, < \, | \vec{y} | - t - c - 1 \} \end{equation} Clearly the set Covering is r.e. and: \begin{multline} P_{unbiased} ( Covering_{t} \Sigma^{\infty} ) \; \leq \; \sum_{\vec{y} \in Covering_{t}} 2^{- |\vec{y}|} \; = \\ = \; \sum_{ \{ \vec{y} \in \Sigma^{\star} \, : \, I ( \vec{y} ) \, < \, | \vec{y} | - t - c - 1 \}} 2^{- |\vec{y}|} \; \leq \\ \leq \; \sum_{ \{ \vec{y} \in \Sigma^{\star} \, : \, I ( \vec{y} ) \, < \, | \vec{y} | - t - c - 1 \}} 2^{ - I ( \vec{y} ) - t - c - 1 } \end{multline} so that: \begin{multline} P_{unbiased} ( Covering_{t} \Sigma^{\infty} ) \; \leq \; \sum_{ \vec{y} \in \Sigma^{\star} } 2^{ - I ( \vec{y} ) - t - c - 1 } \; = \\ = \; 2^{t-c-1} \sum_{\vec{y} \in \Sigma^{\star}} 2^{ - I ( \vec{y} )} \; \leq \; 2^{- t - 1} \sum_{\vec{y} \in \Sigma^{\star}} P_{U} ( | \vec{y} | ) \; = \\ = \; 2^{t - 1} \; < \; 2^{ - t} \end{multline} We prove now that $ \bar{x} \in \bigcap_{t = 1}^{\infty} Covering_{t} \Sigma^{\infty} $. Indeed, given $ t > 0 $, construct $ m_{t} \, := \, n_{t + c + 1} $ and use the hypothesis: \begin{equation} I ( \vec{x} ( m_{t} ) ) \; = \; I ( \vec{x} ( n_{t + c + 1} )) \; < \; n_{t + c + 1} \, - \, ( t + c + 1 ) \; = \; m_{t} - t - c - 1 \end{equation} i.e. $ \vec{x}( m_{t} ) \, \in \, Covering_{t} $. By Lemma\ref{lem:classical algorithmic randomness in terms of coverings} $ \bar{x} \notin MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) $ \item $ CHAITIN-RANDOM( \Sigma^{\infty} ) \; \subseteq \; MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) $ To prove the thesis is equivalent to show that: \begin{equation} \bar{x} \notin MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) \; \Rightarrow \; \bar{x} \notin CHAITIN-RANDOM( \Sigma^{\infty} ) \end{equation} Let us assume that $ \bar{x} \notin MARTIN-L \ddot{O}F-RANDOM( \Sigma^{\infty} ) $. By Lemma\ref{lem:classical algorithmic randomness in terms of coverings} there exist a r.e. set $ Covering \, \subset \, \Sigma^{\star} \times {\mathbb{N}} $ such that: \begin{align} P_{unbiased} & ( Covering_{n} \Sigma^{\infty} ) \; < 2^{ - n} \; \; \forall n \in {\mathbb{N}}_{+} \\ \bar{x} & \in \bigcap_{n=1}^{\infty} Covering_{n} \Sigma^{\infty} \end{align} Furthermore, by Lemma\ref{lem:effective passage from arbitrary to prefix-free covering}, we may assume that $ Covering_{n} $ is prefix-free for all $ n \, geq \, 1 $. Then: \begin{multline} \sum_{n=2}^{\infty} \sum_{\vec{y} \in Covering_{n^{2}}} 2^{ - ( | \vec{y} | - n )} \; = \; \sum_{n=2}^{\infty} 2^{n} \sum_{\vec{y} \in Covering_{n^{2}}} 2^{ - | \vec{y} | } \: = \\ = \; \sum_{n=2}^{\infty} 2^{n} P_{unbiased} ( Covering_{n^{2}} \Sigma^{\infty} ) \; \leq \; \sum_{n=2}^{\infty} 2^{ n - n^{2} } \; \leq \; 1 \end{multline} By theorem\ref{th:Kraft-Chatin theorem} we get a Chaitin computer C satisfying the following requirement: \begin{equation} \forall n \geq 2 , \forall \vec{y} \in Covering_{n^{2}} \; \exists \vec{u} \in \Sigma^{ | \vec{y} | -n } \, : \, C( \vec{u} , \lambda ) \, = \, \vec{y} \end{equation} By the Invariance Theorem for Prefix Algorithmic Entropy, namely theorem\ref{th:invariance theorem for prefix algorithmic entropy}, there exists a positive constant c such that: \begin{equation} \label{eq:application of the invariance theorem} I( \vec{y} ) \; \leq \; | \vec{y} | - n + c \; \; \forall n \geq 2 , \forall \vec{y} \in Covering_{n^{2}} \end{equation} Next we prove that for all natural $ n \geq 1 $ there exist infinitely many m such that $ \vec{x}(m) \, \in \, Covering_{n^{2}} $. By hypothesis: \begin{equation} \vec{x} \; \in \; \bigcap_{k=1}^{\infty} Covering_{k} \Sigma^{\infty} \end{equation} so for every n we can find a natural $ m_{ n^{2} } $ with $ \vec{x} ( m_{n^{2}}) \, \in \, Covering_{n^{2}} $. We have to prove that we can choose these numbers $ m_{n^{2}} $ as large as we wish. Assume, for the sake of a contradiction, that $ m_{n^{2}} \; \geq \; N $, for all n and some fixed N. This means the existence of a string $ \vec{y} $ of length less than N such that $ \vec{y} \, in \, Covering_{n^{2}} $ , for all $ n \, \geq \, 1 $. Accordingly, for every $ n \geq 1 $ one has: \begin{equation} \vec{y} \Sigma^{\infty} \; \subset \; Covering_{n^{2}} \Sigma^{\infty} \end{equation} and: \begin{equation} 2^{ - n^{2}} \; > \; P_{unbiased} ( Covering_{ n^{2}}) \Sigma^{\infty} \; \geq \; P_{unbiased} ( \vec{y} \Sigma^{\infty} ) \; = \, 2^{ - | \vec{y} | } \; \geq \;2^{ - N } \end{equation} that is a contradiction. In conclusion, given $ d > 0 $ we pick $ i > d + c $ and $ m \geq 2 $ in order to get $ \vec{x}(m) \in Covering_{n^{2}} $: by eq.\ref{eq:application of the invariance theorem}: \begin{equation} I ( \vec{x} (m) ) \; \leq \; m - n + c \; < \; m - d \end{equation} \end{itemize} \end{proof} \medskip Summing up, theorem\ref{th:equivalence of Martin-Lof randomness and Solovay randomness} and theorem\ref{th:Chaitin-Schnorr theorem} show that Martin-L\"{o}f randomness, Solovay randomness and Chaitin randomness are equivalent notions, characterizing what is nowadays \emph{almost} universally considered as the the correct characterization of the concept of \textbf{$C_{\Phi}$-algorithmic randomness}. The above \emph{almost} is owed to to a problem we mentioned at the end of section\ref{sec:Brudno random sequences of cbits}, almost always misunderstood and that is the main source of a sort of incomunicability between the scientific community of mathematical physicists studying Dynamical Systems Theory and the scientific community of the logico-mathematicians and Theoretical-Computer scientists studying Algorithmic Information Theory: Theorem\ref{th:Brudno randomness is weaker than Chaitin randomness} stating that: \begin{equation} BRUDNO-RANDOM( \Sigma^{\infty}) \; \subset \; CHAITIN -RANDOM(\Sigma^{\infty}) \end{equation} whose proof, as promized, we report here: \begin{proof} Given a universal computer C, let us introduce another computer $ C' $ defined in the following way: \begin{equation} C'( \vec{x} ) \; := \; \begin{cases} \cdot_{i=1}^{ | \vec{y}| } {\mathcal{I}} ( y_{i} , 2^{i}, C ( \vec{z} )) & \text{if $ \exists \vec{y}, \vec{z} \in \Sigma^{\star} : \vec{x} = \cdot_{i=1}^{ | \vec{y}| } y_{i}^{2} \cdot 01 \cdot \vec{z} $ }, \\ C ( \vec{x} ) & \text{otherwise}. \end{cases} \end{equation} where, generally, $ {\mathcal{I}} ( a , n , \vec{b} ) $ denotes the string obtained inserting the letter a at the $ n^{th} $ place of the string $ \vec{b} $, i.e.: \begin{equation}\label{eq:insertion operator} {\mathcal{I}} ( a , n , \vec{b} ) \; := \; b_{1} \ldots b_{n - 1} \cdot a \cdot b_{n + 1} \ldots b_{| \vec{b} | } \; \; a \in \Sigma , \vec{b} \in \Sigma^{\star} , n \in {\mathbb{N}}_{+} : n \leq | \vec{b} | \end{equation} Clearly $ C' $ is a universal computer too. Given $ \bar{u} \in CHAITIN -RANDOM(\Sigma^{\infty}) $ let us consider the sequence $ \bar{u}' $ defined in the following way: \begin{equation} \bar{u}'_{i} \; := \; \begin{cases} 0 & \text{if $ i = 2^{k} , k \in { \mathbb{N}}$ }, \\ \bar{u}_{i} & \text{otherwise}. \end{cases} \end{equation} Since: \begin{equation} K_{C'} ( \vec{u}_{n} ) \; \leq \; K_{C'} ( \vec{u}'_{n} ) + 2 + \log_{2} n \end{equation} It follows that: \begin{equation} \bar{u}' \; \in \; BRUDNO-RANDOM(\Sigma^{\infty}) \end{equation} Let us now consider the Martin-L\"{o}f sequential test $ V \, \subset \, \Sigma^{\star} \times {\mathbb{N}}_{+} $ whose $ n^{th}$ section is given by: \begin{equation} V_{n} \; := \; \{ \vec{x} \in \Sigma^{\star} \, : \, x_{2^{k}} = 0 \, \, k = 0 , \cdots , i - 1 \} \end{equation} Since by construction one has that: \begin{equation} \bar{u}' \notin \bigcap_{n=1}^{\infty} V_{n} \end{equation} it follows that: \begin{equation} \bar{u}' \; \notin \; CHAITIN-RANDOM(\Sigma^{\infty}) \end{equation} \end{proof} \newpage \chapter{Classical algorithmic randomness as satisfaction of all the classical algorithmic typical properties} \label{chap:Classical algorithmic randomness as satisfaction of all the classical algorithmic typical properties} \section{Typical properties of a classical probability space} \label{sec:Typical properties of a classical probability space} We have seen in chapter\ref{chap:Classical algorithmic randomness as passage of all the classical algorithmic statistical tests} that classical algorithmic randomness may be characterized as the passage of all the classical algorithmic statistical tests, i.e. of all the effectively implementable testes designed to catch some kind of regularity: in this chapter we will show in which sense the absense of any kind of regularity may be interpreted as a condition of maximal conformism, i.e. as the ownership to all the overwhelming majorities. Let us consider a collectivity S made of $ N \in {\mathbb{N}} $ people. Given a property $ p( \cdot ) $ will say that it is a \textbf{majoritary} property of S if the people in S having such a property are more than those not having it, i.e. iff: \begin{equation} cardinality( \{ x \in S : p(x) \; holds \} ) \; > \; \frac{N}{2} \end{equation} We will say that $ p( \cdot ) $ is a \textbf{typical property} of S if the people in S having such a property are very more than those not having it, i.e. iff: \begin{equation} cardinality( \{ x \in S : p(x) \; holds \} ) \; \gg \; \frac{N}{2} \end{equation} Of course this last notion is only an informal one, owing to the informal nature of the ordering relation \textbf{very greater than}. \smallskip Let us now consider the case in which the collectivity S is infinite but countable, i.e. $ cardinality(S) \, = \, \aleph_{0} $; in this case the same notion of a majoritary property loses its meaning \footnote{Let us observe, by the way, that this inficiates the meaning of the generalization to infinite collectivities of the celebrated Nobel-prize for-Economics-winning Kenneth Arrow's theorem on the impossibility of democracy stating , in technical terms, that under the assumption that the decisive sets form an ultrafilter on the set of voters they form a principal filter too (and so there exist a dictator, i.e. a voter whose vote alone determines the result of any election). Such a generalization may be obtained in the same way the impossibility of applying the theorem stating the principality of any ultafilter on finite sets was overcome by Kurt Go\"{e}del in his mathematical formalization of Anselm of Aosta's ontological proof simply by adding the assumption that being God is a positive property: appealing to the theorem stating that an ultrafilter on a (finite or infinite set) containing the intersection of all its elements is principal \cite{Manin-98} \cite{Odifreddi-00b}}. In the case of an infinite and uncountable community, i.e. $ cardinality(S) \, \geq \, \aleph_{1} $, if S admits an unbaised probability measure $ P_{unbiased} $ the notion of a \textbf{typical property} of S may be rigorously defined as a property holding $ P_{unbiased} $ almost everywhere in S. So the characterization of classical algorithmic randomness as \textbf{absolute conformism}, i.e. as the ownership of all the typical properties, would seem to be precisely formalized. Such a formalization, anyway, results in an empty notion: asbolute conformism is impossible. The solution to such a bug consists in requiring only the ownerhip of all the \textbf{effectively-refutable} typical properties. The fact that, once again, Classical Measure Theory appears not to be self-consistent as to the characterization of classical algorithmic randomness has a great foundational relevance. \medskip Given a classical probability space $ CPS \; := \; ( \, M \, , \, \mu \, ) $: \begin{definition} \label{def:classical null set} \end{definition} $ S \; \subset \; M $ IS A NULL SET OF CPS: \begin{equation} \forall \epsilon > 0 \; \exists F_{ \epsilon } \in HALTING(\mu) \; : \; S \subset F_{ \epsilon } \; and \; \mu( F_{ \epsilon } ) < \epsilon \end{equation} Let us introduce the following notions: \begin{definition} \end{definition} UNARY PREDICATES ON M: \begin{equation} {\mathcal{P}} ( M ) \; \equiv \; \{ p( x ) \, : \, \text{ predicate about } x \in M \} \end{equation} \begin{definition} \label{def:typical properties of a classical probability space} \end{definition} TYPICAL PROPERTIES OF CPS: \begin{equation} {\mathcal{P}} ( CPS )_{TYPICAL} \; \equiv \; \{ \, p ( x ) \in {\mathcal{P}} ( M ) \, : \{ x \in M \, : \, p ( x ) \text{ doesn't hold } \} \; \text{is a null set} \} \end{equation} \begin{example} \end{example} TYPICAL PROPERTIES OF A DISCRETE CLASSICAL PROBABILITY SPACE If CPS is discrete-finite $ ( \, M \; = \; \{ a_{1} \, , \ldots \, a_{n} \} \, ) $ or discrete-infinite $ ( \, M \; = \; \{ a_{n} \}_{n \in \mathbb{N}} \, ) $ it is natural to assume that $ \mu ( \{ a_{i} \} ) > 0 \; \forall i $ since an element whose singleton has zero probability can be simply thrown away from the beginning. It follows, than, that CPS has no null sets and, conseguentially, typical properties are simply the holding properties. \begin{example} \label{ex:some typical property of the unbiased space of cbit's sequences} \end{example} SOME TYPICAL PROPERTY OF THE UNBIASED SPACE OF CBITS' SEQUENCES: Among the typical properties of the \textbf{unbaised space of binary sequences} $ ( \Sigma^{\infty} \, , \, P_{unbaised} ) $ there are the following: \begin{itemize} \item \textbf{Borel normality of order $ m \in {\mathbb{N}}$}: \begin{equation} p_{m-Borel} ( \bar{x} ) \; := \; << \, \lim_{n \rightarrow \infty} \frac{ N_{i}( \vec{x}(n))}{ \lfloor \frac{n}{m} \rfloor } \; = \; \frac{1}{2^{m}} \, >> \end{equation} where $ N_{i}( \vec{y}) \, i \in \Sigma $ denotes the number of occurence of the letter $ i \in \Sigma $ in the string $ \vec{y} \in \Sigma^{\star} $ \item \textbf{infinite recurrence} \begin{equation} p_{infinite \; recurrence} ( \bar{x} ) := << cardinality \{ n \in {\mathbb{N}} \; : \; \frac{ N_{1}( \vec{x} (n))}{n} \; = \; \frac{1}{2} \} \; = \; \aleph_{0} >> \end{equation} \item \textbf{iterated-logarithm property} \begin{equation} p_{iterated \; logarithm} ( \bar{x} ) := << \, \lim \sup_{n \rightarrow \infty} \frac{ \sum_{i=1}^{n} x_{i} \, - \, \frac{n}{2}}{ \sqrt{n \log \log n}} \; \leq \; \frac{1}{\sqrt{2}} \, >> \end{equation} \item \textbf{transcendence} \begin{equation} p_{trascendence} ( \bar{x} ) := << \, {\mathcal{N}} ( \bar{x}) \, \notin \, {\mathbb{A}} >> \end{equation} \item \textbf{irrecursivity} \begin{equation} p_{irrecursivity} ( \bar{x} ) := << \, {\mathcal{N}} ( \bar{x}) \, \notin \, \Delta_{0}^{0} ( [ 0 ,1 ) ) \, >> \end{equation} \item \textbf{irrationality} \begin{equation} p_{irrationality} ( \bar{x} ) := << \, {\mathcal{N}} ( \bar{x}) \, \notin \, {\mathbb{Q}} \, >> \end{equation} \item \textbf{ownership of all substrings} \begin{equation} p_{\text{ownership of all substrings}} ( \bar{x} ) := << \, \forall \vec{y} \in \Sigma^{\star} \, \exists n,m \in {\mathbb{N}}_{+} : \vec{x} (n,m) \, = \, \vec{y} \, >> \end{equation} \item \textbf{difference from $ \bar{y} \in \Sigma^{\infty} $} \begin{equation} p_{\text{difference from $ \bar{y}$}} ( \bar{x} ) := << \, \bar{x} \, \neq \, \bar{y} >> \end{equation} \end{itemize} \newpage \section{Impossibility of absolute conformism in Classical Probability Theory} Kolmogorov's original idea about the characterization of the \textbf{intrinsic randomness} of an \textbf{individual object} was to consider it as more random as more it is conformistic, in the sense of conforming itself to the collectivity belonging to all the overwhelming majorities, i.e. possessing all the typical properties \cite{Li-Vitanyi-97}, \cite{Calude-94} . Such an attitude results in the following: \begin{definition} \end{definition} SET OF THE KOLMOGOROV-RANDOM ELEMENTS OF $ ( \Sigma^{\infty} \, , \, P_{unbaised} ) $: \begin{multline} KOLMOGOROV-RANDOM [ ( \Sigma^{\infty} \, , \, P_{unbaised} ) ] \; \equiv \\ \{ \, x \, \in \, M \, : p ( x ) \; holds \; \; \forall p \in {\mathcal{P}} [ ( \Sigma^{\infty} \, , \, P_{unbaised} ) ]_{TYPICAL} \; \} \end{multline} But an immediate application of the Cantorian diagolization-proof's technique \cite{Odifreddi-89} lead to the following: \begin{theorem} \label{th:not existence of Kolmogorov random sequences of cbits} \end{theorem} NOT EXISTENCE OF KOLMOGOROV RANDOM SEQUENCES OF CBITS \begin{equation} KOLMOGOROV-RANDOM [ ( \Sigma^{\infty} \, , \, P_{unbaised} ) ] \; = \; \emptyset \end{equation} \begin{proof} Let us consider again the family of unary predicates $ p_{\text{difference from $ \bar{y}$}} $ over $ \Sigma^{\infty} $ depending on the parameter $ \bar{y} \in \Sigma^{\infty} $ introduced in the example\ref{ex:some typical property of the unbiased space of cbit's sequences} We already saw that they are all typical properties of $ ( \Sigma^{\infty} \, , \, P_{unbaised} ) $ \begin{equation} p_{\text{difference from $ \bar{y}$}} (\bar{x}) \; \in \; {\mathcal{P}} [ ( \Sigma^{\infty} \, , \, P_{unbaised} )]_{TYPICAL} \; \; \forall \bar{y} \in \Sigma^{\infty} \end{equation} Let us now observe that: \begin{equation} p_{\text{difference from $ \bar{x}$}} (\bar{x}) \; \text{doesn't hold} \; \; \forall \bar{x} \in \Sigma^{\infty} \end{equation} So $ p_{\text{difference from $ \bar{x}$}} $ is a typical property that is not satisfied by any element of $ \Sigma^{\infty} $, immediately implying the thesis \end{proof} \bigskip The theorem\ref{th:not existence of Kolmogorov random sequences of cbits} shows that we have to relax the condition that a random sequence of cbits possesses \textbf{all the typical properties} requiring only that it satisfies \textbf{a proper subclass of typical properties}. The right subclass was proposed by P. Martin L\"{o}f who observed that all the Classical Laws of Randomness, i.e. all the properties of Classical Probability Theory that are known to hold with probability one (such as the \textit{Law of Large Numbers}, the \textit{Law of Iterated Logarithm} and so on ) are \textbf{effectively-falsificable} in the sense that we can effectively test whether they are violated (though we cannot effectively certify that they are satisfied). This leads, assuming the Church-Turing's Thesis \cite{Odifreddi-89} and endowed $ \Sigma^{\infty} $ with the \textbf{product topology} induced by the \textbf{discrete topology} of $ \Sigma $, to introduce the following notions: \begin{definition} \end{definition} $S \; \subset \; \Sigma^{\infty} $ IS ALGORITHMICALLY-OPEN: \begin{equation} ( S \text{ is open } ) \; and \; ( S \, = \, X \Sigma^{\infty} \, {\mathbf{X \; recursively-enumerable}}) \end{equation} \begin{definition} \end{definition} ALGORITHMIC SEQUENCE OF ALGORITHMICALLY-OPEN SETS: a sequence $ \{ S_{n} \}_{n \geq 1} $ of algorithmically open sets $ S_{n} \; = \; X_{n} \Sigma^{\infty} $ : $ \exists X \; \subset \; \Sigma^{\star} \times {\mathbb{N}} $ \textbf{recursively enumerable} with: \begin{equation*} X_{n} \; = \; \{ \vec{x} \in \Sigma^{\star} \, : \, ( \vec{x} , n ) \in X \} \; \; \forall n \in {\mathbb{N}}_{+} \end{equation*} \smallskip Given the classical probability space $ CPS \, := \, ( \Sigma^{\infty} , P ) $: \begin{definition} \end{definition} $ S \; \subset \; \Sigma^{\infty} $ IS AN ALGORITHMICALLY-NULL SUBSET OF CPS: $ \exists \{ G_{n} \}_{n \geq 1} $ algorithmic sequence of algorithmically-open sets : \begin{equation*} S \; \subset \; \cap_{n \geq 1} G_{n} \end{equation*} and: \begin{equation*} alg - \lim_{n \rightarrow \infty} P ( G_{n} ) \; = \; 0 \end{equation*} i.e. there exist and increasing, unbounded, \textbf{recursive} function $ f \, : \, {\mathbb{N}} \rightarrow {\mathbb{N}} $ so that $ P ( G_{n} ) \; < \; \frac{1}{2^{k}} $ whenever $ n \; \geq \; f(k) $ \begin{definition} \end{definition} LAWS OF RANDOMNESS OF CPS: \begin{multline} \mathcal{L}_{randomness} (CPS) \; \equiv \; \{ \, p ( \bar{x} ) \in {\mathcal{P}} ( \Sigma^{\infty} ) \, : \\ \{ \bar{x} \in \Sigma^{\infty} \, : \, p ( \bar{x} ) \text{ doesn't hold } \} \; \text{is an algorithmically null set of CPS} \} \end{multline} \begin{example} \label{ex:some law of randomness of the unbiased space of cbit's} \end{example} SOME LAW OF RANDOMNESS OF THE UNBIASED SPACE OF CBITS' SEQUENCES: Let us consider again the typical properties of the classical probability space $ ( \Sigma^{\infty} , P_{unbiased} ) $ introduced in the example\ref{ex:some typical property of the unbiased space of cbit's sequences}. \textbf{Borel normality of order $ m \in {\mathbb{N}} $}, \textbf{infinite recurrence}, the \textbf{iterated-logarithm property}, \textbf{transcendence} and \textbf{irrationality} are all effectively-refutable and, hence, are all Laws of Randomness. To refute that a sequence has the property of \textbf{irrecursivity}, or of \textbf{ownership of all substrings}, or of \textbf{difference from $ \bar{y} \in \Sigma^{\infty} $} would, instead, require the inspection the analysis of an infinite number of of its digits. Hence all such typical properties are not effectively-refutable and, hence, are not laws of randomness. \bigskip We can now introduce the following: \begin{definition}\label{def:P-conformistically randomness} \end{definition} P-CONFORMISTICALLY RANDOM ELEMENTS OF CPS: \begin{equation} P-CONF-RANDOM(\Sigma^{\infty}) \; := \; \{ \bar{x} \in \Sigma^{\infty} \: p ( \bar{x} ) \text{ holds } \; \forall p \in {\mathcal{L}_{randomness}} (CPS) \} \end{equation} As usual the case of the unbaised measure deserves an ad hoc definition: \begin{definition}\label{def:conformistically random sequences} \end{definition} CONFORMISTICALLY-RANDOM SEQUENCES: \begin{multline} CONF-RANDOM(\Sigma^{\infty}) \; := \\ P_{unbaised}-CONF-RANDOM(\Sigma^{\infty}) \end{multline} \smallskip \begin{remark} \label{rem:why the diagonalization proof doesn't apply to P-conformistically randomness} \end{remark} WHY THE DIAGONALIZATION PROOF OF THEOREM\ref{th:not existence of Kolmogorov random sequences of cbits} DOESN'T APPLY TO P-CONFORMISTICALLY RANDOMNESS Let us observe that the diagonalization proof of theorem\ref{th:not existence of Kolmogorov random sequences of cbits} is based on the one-parameter family of typical properties $ p_{\text{difference from $ \bar{y}$}} \, , \,\bar{y} \in \Sigma^{\infty} $ . Since noone of these is a law of randomness the argument falls down. \newpage \section{Equivalence between Martin L\"{o}f-conformistical randomness and Chaitin randomness} Summing up, we have seen that Per Martin L\"{o}f introduced two approaches to the mathematical characterization of classical algorithmic randomness: \begin{itemize} \item the \textbf{statistical approach} discussed in chapter\ref{chap:Classical algorithmic randomness as passage of all the classical algorithmic statistical tests} and resulting in the definition of the set $ MARTIN L\ddot{O}F-RANDOM(\Sigma^{\infty}) $ whose equality with the set $ CHAITIN-RANDOM(\Sigma^{\infty}) $ is stated by theorem\ref{th:Chaitin-Schnorr theorem} \item the \textbf{logical approach} discussed in the previous sections and resulting in the definition of the set $ CONF-RANDOM(\Sigma^{\infty}) $ \end{itemize} In this section we will prove Martin-L\"{o}f's Theorem showing the the complete equivalence of these approaches. This requires the introduction of some technical ingredient, starting from the following: \begin{lemma} \label{lem:cardinality inequality of a sequential Martin-Lof test} \end{lemma} For every sequential $ P_{unbaised}$-test V and for every natural $ m \geq 1 $: \begin{equation} \sum_{\vec{x} \in \Sigma^{n} \bigcap V_{m}} 2^{- | \vec{x} |} \, < \, 2^{- m} \end{equation} \begin{proof} It follows immediately from the cardinality inequality in the definition of a sequential Martin-L\"{o}f test \end{proof} \smallskip Then we need the following: \begin{lemma} \label{lem:asymptotic condition for a sequential Martin-Lof test} \end{lemma} Let V be a sequential $ P_{unbaised}$-test. Then \begin{equation} \lim_{m \rightarrow \infty} P_{unbaised} ( V_{m} \Sigma^{\infty} ) \; = \; 0 \; \; constructively \end{equation} \begin{proof} Take V and define for every natural $ m \geq 1 $ the sets $ W_{m} \, := \, V_{m} \Sigma^{\infty}$. It is seen that for each $ m \geq 1 $, $ W_{m} \, = \, \bigcup_{n=2}^{\infty} X_{n}$ , where: \begin{equation} X_{n} \; := \; \bigcup_{\vec{x} \in \Sigma^{n} \bigcap V_{m}} \vec{x} \Sigma^{\infty} \end{equation} Furthermore, $ X_{n} \; \subset \; X_{n+1} $ and: \begin{multline} P_{unbaised} ( X_{n} ) \; = \; \sum_{\vec{x} \in \Sigma^{n} \bigcap V_{m}} P_{unbaised} ( \vec{x} \Sigma^{\infty} ) \\ = \; \sum_{\vec{x} \in \Sigma^{n} \bigcap V_{m}} 2^{- | \vec{x} | } \; = \; \frac{cardinality( \Sigma^{n} \bigcap V_{m})}{2^{n}} \\ < 2^{- m} \end{multline} in view of lemma\ref{lem:asymptotic condition for a sequential Martin-Lof test} and of the fact that the sets $ \{ \vec{x} \Sigma^{\infty} : \vec{x} \in \Sigma^{n} \bigcap V_{m} \} $ are mutually disjoint. So: \begin{equation} P_{unbiased} ( W_{m} ) \; = \; \lim_{n \rightarrow \infty} P_{unbiased} ( X_{n} ) \; \leq \; 2^{- m} \end{equation} Finally, put $ H( m ) \; := \; m +1 $ and notice that if $ m \; \geq \; H(k) $, then $ P_{unbiased} ( W_{m} ) \; \leq \; 2^{- k} $ \end{proof} The last ingredient required for proving Martin-L\"{o}f's Theorem is the following: \begin{lemma} \label{lem:algorithmic null set associated to a sequential Martin Lof test} Let V be a sequential $ P_{unbaised}$-test. Then $ \bigcap_{m=1}^{\infty} ( V_{m} \Sigma^{\infty} ) $ is an algorithmically-null subset of $ ( \Sigma^{\infty} \, , \, P_{unbaised} ) $ \end{lemma} \begin{proof} Take V and define for every natural $ m \geq 1 $ the sets $ W_{m} \, := \, V_{m} \Sigma^{\infty}$. Since V is r.e. it follows that the sequence $ \{ V_{m} \}_{m \in {\mathbb{N}}_{+}} $ is an algorithmic sequence of algorithmically opens sets. By lemma\ref{lem:asymptotic condition for a sequential Martin-Lof test} it follows the thesis \end{proof} \smallskip We have at last all the ingredients required to prove the following: \begin{theorem} \label{th:Martin-Lof theorem} \end{theorem} MARTIN-L\"{O}F'S THEOREM: \begin{equation} CONF-RANDOM(\Sigma^{\infty}) \; = \; CHAITIN-RANDOM(\Sigma^{\infty}) \end{equation} \begin{proof} Fix a universal sequential $ P_{unbaised}$-test U. Since: \begin{equation} \Sigma^{\infty} \, - \, CHAITIN-RANDOM( \Sigma^{\infty} ) \; = \; \bigcap_{m=1}^{\infty} U_{m} \Sigma^{\infty} \end{equation} we may apply lemma\ref{lem:algorithmic null set associated to a sequential Martin Lof test} to conclude that $ \Sigma^{\infty} \, - \, CHAITIN-RANDOM( \Sigma^{\infty} ) $ is an algorithmically-null set. Next let $ S \, \subset \, \Sigma^{\infty} $ be an arbitrary algorithmically-null set. We shall prove that: \begin{equation} S \; \subset \; \Sigma^{\infty} \, - \, CHAITIN-RANDOM( \Sigma^{\infty} ) \end{equation} To this aim let us consider an algorithmic sequence of algorithmically open sets $ ( G_{m} )_{m \geq 1} $ such that: \begin{equation} S \; \subset \; \bigcap_{m=1}^{\infty} G_{m} \end{equation} and: \begin{equation} P_{unbaised} ( G_{t} ) \; < \; 2^{-m} \; \; \forall t \, \geq \, H(m) \end{equation} where $ H : {\mathbb{N}} \, \mapsto \, {\mathbb{N}} $ is a fixed increasing, unbounded recursive function. Write: \begin{equation} G_{m} \; := \; X_{m} \Sigma^{\infty} \; = \; ( X_{m} \Sigma^{\star} ) \Sigma^{\infty} \end{equation} for all $ m \, \geq \, 1 $, where $ X_{m} \; \subset \; \Sigma^{\star} $ is an r.e. set. We have to construct a sequential $ P_{unbiased} $-test V such that: \begin{equation} \label{eq:sequential Martin-Lof test to construct} \bigcap_{m=1}^{\infty} V_{m} \Sigma^{\infty} \; = \; \bigcap_{m=1}^{\infty} G_{m} \end{equation} Put: \begin{equation} V_{m} \; := \; \bigcap_{i=1}^{H(m)} X_{i} \Sigma^{\star} \; \; m \in {\mathbb{N}}_{+} \end{equation} Clearly the set V defined as: \begin{equation} V \; := \; \{ ( \vec{x} , m ) \in \Sigma^{\star} \times {\mathbb{N}}_{+} \, : \, \vec{x} \in V_{m} \} \end{equation} is r.e., $ V_{m+1} \; \subset \; V_{m} $ and is such that if $ \vec{x} <_{p} \vec{y} $ and $ \vec{x} \in V_{m} $ then $ \vec{y} \in V_{m} $. Fixed $ n, m \in {\mathbb{N}}_{+} $: \begin{multline} cardinality( \Sigma^{n} \bigcap V_{m} ) \; \leq \; cardinality( X_{H(m)} \Sigma^{\star} \bigcap \Sigma^{n} \\ = \; 2^{n} \, cardinality( X_{H(m)} \Sigma^{\star} \bigcap \Sigma^{n}) \, 2^{- n} \; = \; 2^{n} P_{unbaised} ((( X_{H(m)} \Sigma^{\star}) \, \bigcap \, \Sigma^{n} ) \Sigma^{\infty} ) \\ \leq \; 2^{n} P_{unbaised} (( X_{H(m)} \Sigma^{\star} ) \Sigma^{\infty} ) \; \leq \; 2^{n - m} \end{multline} So V is a sequential $ P_{unbaised} $-test and, hence, eq.\ref{eq:sequential Martin-Lof test to construct} holds by virtue of the strict monotonicity of H. According to the universality of U one can find a natural c such that: \begin{equation} V_{m + c} \; \subset \; U_{m} \; \; \forall m \in {\mathbb{N}}_{+} \end{equation} Then: \begin{multline} S \; \subset \; \bigcap_{m=1}^{\infty} V_{m} \Sigma^{\infty} \; \subset \; \bigcap_{m=1}^{\infty} V_{m+c} \Sigma^{\infty} \\ \subset \bigcap_{m=1}^{\infty} U_{m} \Sigma^{\infty} \; = \; \Sigma^{\infty} \, - \, CHAITIN-RANDOM( \Sigma^{\infty} ) \end{multline} \end{proof} \smallskip \begin{remark} \label{rem:on why Classical Probability Theory applies to reality} \end{remark} ON WHY CLASSICAL PROBABILITY THEORY APPLIES TO REALITY We can now fully appreciate the conceptual relevance of theorem\ref{th:foundation of the applicability of probability theory to reality} (and the name we gave to it): it tells us that extracted at random a sequence according to the probability distribution $ \mu $ the occured sequence will satisfy all the $ \mu$ Laws of Randomness with certainty. We indeed have to bless such a theorem: it is only for his courtesy that it is possible to give certain mathematical predictions concerning the statistical behaviour of classically-non deterministic phenomena. This is particularly rilevant in the case in which $ \mu $ is the unbaised probability measure $ P_{unbiased} $: it is only because making infinite independent tosses of a fair coin we obtain with certainty a sequence without intrinsic regularity that we can find a mathematical regularity in classical-nondeterminism. This clarifies why, as we will discuss in chapter\ref{chap:The irreducibility of quantum probability both to classical determinism and to classical nondeterminism} is very raesonable to expect that an analogous situation must happen also in Quantum Probability Theory. \chapter{Classical algorithmic randomness as stability of the relative frequences under proper classical algorithmic place selection rules} \label{chap:Classical algorithmic randomness as stability of the relative frequences under proper classical algorithmic place selection rules} \section{Von Mises' Frequentistic Foundation of Probability} \label{sec:Von Mises' Frequentistic Foundation of Probability} The mirable features of the Kolmogorovian measure-theoretic axiomatization of Classical Probability Theory \cite{Kolmogorov-56} has led to consider it as the last word about Foundations of Classical Probability Theory, leading to the general attitude of forgetting the other different axiomatizations and, in particular, von Mises' Frequentistic one \cite{von-Mises-81}. Richard Von Mises' axiomatization of Classical Probability Theory lies on the mathematical formalization of the following two empirical laws: \begin{enumerate} \item \textbf{Law of Stability of Statistic Relative Frequencies} \begin{center} \textit{"It is essential for the theory of probability that experience has shown that in the game of dice, as in all other mass phenomena which we have mentioned, the relative frequencies of certain attributes become more and more stable as the number of observations is increased"} (cfr. pag.12 of \cite{von-Mises-81}) \end{center} \item \textbf{Law of Excluded Gambling Strategies} \begin{center} \textit{"Everybody who has been to Monte Carlo, or who has read descriptions of a gambling bank, know how many 'absolutely safe' gambling systems, sometimes of an enormously complicated character, have been invented and tried out by gamblers; and new systems are still suggested every day. The authors of such systems have all, sooner or later, had the sad experience of finding out that no system is able to improve their chance of winning in the long run,i.e. to affect the relative frequencies with which different colours of numbers appear in a sequence selected from the total sequence of the game. This experience forms the experimental basis of our definition of probability"} (cfr. pagg.25-26 of \cite{von-Mises-81}) \end{center} \end{enumerate} According to Von Mises Probability Theory concerns properties of collectivities, i.e. of sequences of identical objects. Considering each individual object as a letter of an alphabet $ \Sigma$, we can then say that Probability Theory concerns elements of the set $ \Sigma^{\infty} $ of the sequences of letters from $ \Sigma $ or, more properly, a certain subset $ {\mathcal{C}}ollectives\; \subset \; \Sigma^{\infty} $ whose elements are called \textbf{collectives}. \medskip Let us then introduce the set $ {{\mathcal{A}}ttributes} ( \Sigma ) $ of the \textbf{attributes} of $ \mathcal{C} $'s elements defined as the set of unary predicates about the generic $ C \; \in \; {\mathcal{C}}ollectives $. The mathematical formalization of the \textbf{Law of Stability of Statistic Relative Frequencies} results in the following: \begin{axiom} \label{ax:axiom of convergence} \end{axiom} AXIOM OF CONVERGENCE \begin{hypothesis} \end{hypothesis} \begin{equation*} C \; \in \; {\mathcal{C}}ollectives \end{equation*} \begin{equation*} A \; \in \; {{\mathcal{A}}ttributes} ( \Sigma ) \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} \exists \; \; \lim_{n \rightarrow \infty} \frac{ N( A | \vec{C}(n) ) }{n} \end{equation*} where $N( A | \vec{C}(n) )$ denotes the number of elements of the prefix $ \vec{C}(n) $ of C of length n for which the attribute A holds. Given an \textbf{attribute} $ A \in {\mathcal{A}}ttributes ( \Sigma ) $ of a \textbf{collective} $ C \in {\mathcal{C}}ollectives $ the axiom\ref{ax:axiom of convergence} make consistent the following definition: \begin{definition} \end{definition} VON MISES' FREQUENTISTIC PROBABILITY OF A IN C: \begin{equation}\label{eq:von Mises' definition of probability} P_{VM} ( A | C ) := lim_{n \rightarrow \infty} \frac{ N( A | \vec{C}(n) ) }{n} \end{equation} Let us then introduce the following basic definition: \begin{definition} \label{def:gambling strategy} \end{definition} GAMBLING STRATEGY: $ S : \Sigma^{\star} \stackrel {\circ}{\rightarrow} \{ 0 , 1 \} $ \smallskip Given a gambling strategy S: \begin{definition} \label{def:subsequence extraction function} \end{definition} SUBSEQUENCE EXTRACTION FUNCTION INDUCED BY S: $ EXT[S] : \Sigma^{\infty} \; \rightarrow \; \Sigma^{\infty} $ : \begin{equation} EXT[S] ( x_{1} x_{2} \cdots ) \; := \; \text{ordered concatenation} ( \{ x_{n} \, : \, S ( x_{1} \cdots x_{n-1} ) = 1 , n \in {\mathbb{N}}_{+} \} ) \end{equation} The name in the definition\ref{def:subsequence extraction function} is justified by the fact that obviously: \begin{equation} EXT[S] ( \bar{x} ) \leq_{s} \bar{x} \; \; \forall \bar{x} \in \Sigma^{\infty} \end{equation} where $ \leq_{s} $ is the following: \begin{definition} \end{definition} SUBSEQUENCE ORDERING RELATION ON $ \Sigma^{\infty} $ \begin{equation} \bar{x} \leq_{s} \bar{y} := \text{$ \bar{x} $ is a subsequence of $ \bar{y} $ } \end{equation} \begin{example} \label{ex:bet on the last result} \end{example} BET EACH TIME ON THE LAST RESULT Considered the binary alphabet $ \Sigma \; := \; \{ 0 ,1 \} $, let us analyze the following gambling strategy: \begin{equation} S ( x_{1} \cdots x_{n} ) \; := \; \begin{cases} \uparrow & \text{if $ n = 0 $}, \\ x_{n} & \text{otherwise} \end{cases} \; \; x_{1} \cdots x_{n} \in \Sigma^{n} , n \in \mathbb{N} \end{equation} and the \textit{subsequence extraction function} $EXT[S]$ it gives rise to. Clearly we have that: \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\vec{x}$ & $S( \vec{x} )$ \\ \hline $\lambda$ & $\uparrow$ \\ 0 & 0 \\ 1 & 1 \\ 00 & 0 \\ 01 & 1 \\ 10 & 0 \\ 11 & 1 \\ 000 & 0 \\ 001 & 1 \\ 010 & 0 \\ 011 & 1 \\ 100 & 0 \\ 101 & 1 \\ 110 & 0 \\ 111 & 1 \\ 0000 & 0 \\ 0001 & 1 \\ 0010 & 0 \\ 0011 & 1 \\ 0100 & 0 \\ 0101 & 1 \\ 0110 & 0 \\ 0111 & 1 \\ 1000 & 0 \\ 1001 & 1 \\ 1010 & 0 \\ 1011 & 1 \\ 1100 & 0 \\ 1101 & 1 \\ 1110 & 0 \\ 1111 & 1 \\ \hline \end{tabular} \smallskip Furthermore we have, clearly, that: \begin{align*} EXT[S] ( 0^{\infty} ) \; & = \; \lambda \\ EXT[S] ( 0^{\infty}) \; & = \; 1^{\infty} \\ EXT[S] ( 01^{\infty}\cdots ) \; & = \; 0^{\infty} \\ EXT[S] ( 10^{\infty} ) \; & = \; 0^{\infty} \\ EXT[S] ( \bar{x}_{Champernowne} ) \; & = \; 0101\cdots \end{align*} where $ \bar{x}_{Champernowne} $ is the Champernowne sequence defined as the lexicografic ordered concatenation of the binary strings: \begin{equation*} \bar{x}_{Champernowne} \; = \; 0100011011000001010011100101110111 \cdots \end{equation*} \begin{example} \label{ex:bet on the less frequent letter} \end{example} BET ON THE LESS FREQUENT LETTER Considered again the binary alphabet $ \Sigma \; := \; \{ 0 ,1 \} $, let us analyze the following gambling strategy: \begin{equation} S ( \vec{x} ) \; = \; \begin{cases} \uparrow & \text{if $ \vec{x} = \lambda $ or $ N_{0} ( \vec{x} ) = N_{1} ( \vec{x} )$} , \\ 1 & \text{if $N_{0} ( \vec{x} ) > N_{1} ( \vec{x} )$} , \\ 0 & \text{otherwise}. \end{cases} \end{equation} where $ N_{0} ( \vec{x} ) , N_{1} ( \vec{x} ) $ denote the number of, respectively, zeros and ones in the string $ \vec{x} $. We have that: \smallskip \begin{tabular}{|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\vec{x}$ & $S( \vec{x} )$ \\ \hline $\lambda$ & $\uparrow$ \\ 0 & 1 \\ 1 & 0 \\ 00 & 1 \\ 01 & $\uparrow$ \\ 10 & $\uparrow$ \\ 11 & 0 \\ 000 & 1 \\ 001 & 1 \\ 010 & 1 \\ 011 & 0 \\ 100 & 1 \\ 101 & 0 \\ 110 & 0 \\ 111 & 0 \\ 0000 & 1 \\ 0001 & 0 \\ 0010 & 1 \\ 0011 & $\uparrow$ \\ 0100 & 1 \\ 0101 & $\uparrow$ \\ 0110 & $\uparrow$ \\ 0111 & 1 \\ 1000 & 1 \\ 1001 & $\uparrow$ \\ 1010 & $\uparrow$ \\ 1011 & 0 \\ 1100 & $\uparrow$ \\ 1101 & 0 \\ 1110 & 0 \\ 1111 & 0 \\ \hline \end{tabular} \smallskip As to the extraction function of S: \begin{align*} EXT[S] ( 0^{\infty} ) \; & = \; 0^{\infty} \\ EXT[S] ( 1^{\infty} ) \; & = \; \lambda \\ EXT[S] ( 01^{\infty} ) \; & = \; 1^{\infty} \\ EXT[S] ( 10^{\infty} ) \; & = \; \lambda \\ EXT[S] ( \bar{x}_{Champernowne} ) \; & = \; 10011011\cdots \end{align*} Denoted by $ {\mathcal{S}}trategies ( {\mathcal{C}}ollectives ) $ the \textbf{set of gambling strategies} concerning $ {\mathcal{C}}ollectives $, we can formalize the \textbf{Law of Excluded Gambling Strategies} by the following: \begin{axiom} \label{ax:axiom of randomness} \end{axiom} AXIOM OF RANDOMNESS \begin{hypothesis} \end{hypothesis} \begin{equation*} S \, \in \, {{\mathcal{S}}trategies}_{admissible} ( {\mathcal{C}}ollectives ) \end{equation*} \begin{equation*} C \, \in \, {\mathcal{C}}ollectives \end{equation*} \begin{equation*} A \, \in \, {{\mathcal{A}}ttributes} ( \Sigma ) \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} P_{VM}( \,A \,| \, EXT[S] (C) \, ) \; = \; P_{VM} ( A | C ) \end{equation*} where $ {{\mathcal{S}}trategies}_{admissible} ( {\mathcal{C}}ollectives ) \; \subseteq \; {\mathcal{S}}trategies ( {\mathcal{C}}ollectives ) $ is the \textbf{set of admissible gambling strategies} whose mathematical characterization will be investigated in the next sections. \newpage \section{Classical Gambling in the framework of Classical Statistical Decision Theory} \label{sec:Classical Gambling in the framework of Classical Statistical Decision Theory} Classical Statistical Decision Theory \cite{French-Rios-Insua-00} concerns the following situation: a \textit{decision maker} have to make a single action $ a \in {\mathcal{A}}ctions $ from a space $ {\mathcal{A}}ctions $ of possible actions. Features that are unknown about the external world are modelled by an unknown state of nature $ s \in {\mathcal{S}}tates $ in a set $ {\mathcal{S}}tates $ of possible states of nature. The consequence $ c ( a , s ) \in {\mathcal{C}}onsequences $ of his choice depends both on the action chosen and on the unknwown state of nature. Before making his decision the decision maker may observe an outcome $ X = x $ of an experiment, which depends on the unknown state s. Specifically the observation X is drawn from a distribution $ P_{X} ( \cdot | s ) $. His objectives are encoded in a real valued \textit{utility function} $ u ( a , s ) $. Let us assume that the \textit{decision maker} knows the \textit{action space} $ {\mathcal{A}}ctions $, \textit{state space} $ {\mathcal{S}}tates $ and \textit{consequence space} $ {\mathcal{C}}onsequences $, along with the probability distribution and the \textit{utility function}. His problem is: \textbf{observe $ X = x $ and then choose an action $ d(x) \in {\mathcal{A}}ctions $, using the information that $ X = x $, to maximize, in some sense, $ u ( d ( x ) , s ) $ }. Every decision process may obviously be seen as a gambling situation: the \textit{action space} $ {\mathcal{A}}ctions$ may be seen as the set of possible bets of the decision maker, that we will call from here and beyond the \textit{gambler}, while the \textit{utility function} gives the \textit{payoff}. Let us consider, in particular, the following gambling situation: in the city's \textit{Casino} at each turn $ n \in \mathbb{N} $ the croupier tosses a fair coin. Before the $ n^{th} $ toss the gambler can choose among one of the possbile choices: \begin{itemize} \item to bet one fiche on \textit{head} \item to bet one fiche on \textit{tail} \item not to play at that turn \end{itemize} Leaving all the philosophy behind its original foundational purpose we can, now, from inside the standard Kolomogorovian measure-theoretic formalization of Classical Probability Theory, appreciate the very intuitive meaning lying behind Von Mises' axioms. Let us indicate by $ X_{n} $ the random variable on the binary alphabet $ \Sigma := \{ 0 , 1 \} $ (where we will assume from here and beyond, that $ head = 1 $ and $ tail = 0 $) corresponding to the $ n^{th} $ coin toss and by $ x_{n} \in \Sigma $ the result of the $ n^{th} $ coin toss. Let us, furthermore, denote by $ \bar{x} \; := \; ( x_{1} , x_{2} , , \cdots )\in \Sigma^{\infty} $ the sequence of all the results of the coin tosses and by $ \vec{x}(n) \in \Sigma^{n} $ its $ n^{th} $ prefix. By hypothesis $\{ X_{n} \}_{n \in \mathbb{N}} $ is a Bernoulli($ \frac{1}{2}$) discrete-time stochastic process over $ \Sigma $. A gambling strategy $ S : \Sigma^{\star} \stackrel {\circ}{\rightarrow} \{ 0 , 1 \} $ determines the gambler's decision at the $ n^{th} $ turn in the following way: \begin{itemize} \item if $ S( \vec{x}(n-1) ) \; = \; 1 $ he bets on \textit{head} \item if $ S( \vec{x}(n-1) ) \; = \; 0 $ he bets on \textit{tail} \item if $ S( \vec{x}(n-1) ) \; = \; \uparrow $ he doesn't bet at that turn \end{itemize} \medskip \begin{example} \end{example} APPLYING TO THE CASINO THE GAMBLING STRATEGY OF EXAMPLE\ref{ex:bet on the last result} Let us suppose that the first 10 coin tosses give the following string of results: $ \vec{x}(n) \; = \; 1101001001 $ Our evening to Casino may be told by the following table: \smallskip \begin{tabular}{|c|c|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... TOSS & RESULT OF THE TOSS & BET MADE ABOUT THAT TOSS & PAYOFF \\ \hline 1 & 1 & no bet & 0 \\ 2 & 1 & 1 & +1 \\ 3 & 0 & 1 & 0 \\ 4 & 1 & 0 & -1 \\ 5 & 0 & 1 & -2 \\ 6 & 0 & 0 & -1 \\ 7 & 1 & 0 & -2 \\ 8 & 0 & 1 & -3 \\ 9 & 0 & 0 & -2 \\ 10 & 1 & 0 & -3 \\ \hline \end{tabular} \smallskip As we see $ PAYOFF(10) \; = \; -3 $. \begin{example} \end{example} APPLYING TO THE CASINO THE GAMBLING STRATEGY OF EXAMPLE\ref{ex:bet on the less frequent letter} \smallskip Let us suppose again that the first 10 coin tosses give the following string of results: $ \vec{x}(n) \; = \; 1101001001 $. Our evening to Casino may be told by the following table: \smallskip \begin{tabular}{|c|c|c|c|} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... TOSS & RESULT OF THE TOSS & BET MADE ABOUT THAT TOSS & PAYOFF \\ \hline 1 & 1 & no bet & 0 \\ 2 & 1 & 0 & -1 \\ 3 & 0 & 0 & 0 \\ 4 & 1 & 0 & -1 \\ 5 & 0 & 0 & 0 \\ 6 & 0 & 0 & +1 \\ 7 & 1 & no bet & +1 \\ 8 & 0 & 0 & +2 \\ 9 & 0 & no bet & +2 \\ 10 & 1 & 1 & +3 \\ \hline \end{tabular} \smallskip As we see $ PAYOFF(10) \; = \; +3 $. \bigskip The probability distribution of the string $ \vec{x}(n) $ is the uniform distribution on $ \Sigma^{n} $ : \begin{equation} Prob [ \vec{x}(n) = \vec{y} ] \; = P_{unbaised \, , \, n} ( \vec{y} ) \; := \; \frac{1}{2^{n}} \; \; \forall \vec{y} \in \Sigma^{n} , \forall n \in \mathbb{N} \end{equation} When $ n \rightarrow \infty $ such a distribution tends to the unbiased probability measure $ P_{unbiased} $ on $ \Sigma^{\infty} $. \medskip Clearly the possible \textbf{attributes} of a letter on the binary alphabet are: \begin{itemize} \item $ a_{1} \; := \; << \text{to be 1} >> $ \item $ a_{0} \; := \; << \text{to be 0} >> $ \end{itemize} so that: \begin{equation} {{\mathcal{A}}ttributes} ( \Sigma ) \; = \; \{ a_{1} , a_{0} \} \end{equation} Whichever $ {\mathcal{C}}ollectives\; \subset \; \Sigma^{\infty} $ is the axiom\ref{ax:axiom of convergence} is, from inside the standard kolmogorovian measure-theoretic foundation, an immediate corollary of the Law of Large Numbers. As far as axiom\ref{ax:axiom of randomness} is concerned, anyway, the situation is extraordinarily subtler. Every \textbf{intrinsic regularity} of $ \vec{x}(n) $ could have been encoded by the gambler in a proper \textbf{winning strategy up to the $ n^{th} $ turn}. The same definition of what a \textbf{winning strategy} is requires some caution: we can, indeed, give two possible definitions of such a concept: \begin{definition}[AVERAGE-WINNING STRATEGY UP TO THE $ n^{th} $ TOSS] \label{def:average winning strategy} a strategy so that the expectation value of the payoff after the first n tosses \textbf{payoff(n) } is greater than zero \end{definition} The fact the a strategy is average-winning doesn't imply that the payoff after the $ n^{th} $ toss will be strictly positive with certainty: it happens if we are lucky. Let us now introduce a weaker notion of a winning strategy: \begin{definition}[LUCKY-WINNING STRATEGY UP TO THE $ n^{th} $ TOSS] \label{def:lucky winning strategy} a strategy so that the the probability that the payoff after the first n tosses \textbf{payoff(n) } is greater than zero is itself greater than zero \end{definition} For finite n every strategy is obviously lucky-winning. Let us now consider the limit $ n \rightarrow \infty $. By purely measure-theoretic considerations we may easily prove the following: \begin{theorem} \label{th:weak law of excluded gambling systems} WEAK LAW OF EXCLUDED GAMBLING STRATEGIES For $ n \rightarrow \infty $ the set of the \textit{average-winning strategies} tends to the null set \end{theorem} \begin{proof} Given a gambling strategy $ S : \Sigma^{\star} \stackrel {\circ}{\rightarrow} \{ 0 , 1 \} $ we have clearly that the conditional expectation of the payoff at the $ n^{th} $ turn conditioned to the payoff at the $ (n-1)^{th} $ turn is the sum of two addenda: \begin{itemize} \item the payoff at the $ (n-1)^{th} $ turn \item the expectation value of the gain at the $ (n-1)^{th} $ turn \end{itemize} This second addendum is clearly equal to zero if the adopted gambling strategy prescribes not to bet at the $ n^{th} $ turn. Otherwise its is itself given by the sum of two addenda: \begin{itemize} \item one related to the case in which heads turn up and given by the probability of this fact, obviously equal to $ \frac{1}{2}$, taken with positive sign if we betted on head and taken with negative sign if we betted on tail \item one related to the case in which tails turn up and given by the probability of this fact, obviously equal to $ \frac{1}{2}$, taken with positive sign if we betted on tail and taken with negative sign if we betted on head \end{itemize} But these last two addenda obviously compensate each other, so that the conditional expectation of the payoff at the $ n^{th} $ turn conditioned to the payoff at the $ (n-1)^{th} $ turn is simply given by the payoff at the $ (n-1)^{th} $ turn. This reasoning can be expressed in formulae as: \begin{multline} \label{eq:conditional payoff at step n} E[payoff(n) | payoff(n-1) ] \; = \; payoff(n-1) + \\ If[ S ( \vec{x}_{n-1} ) = \uparrow , 0 , \frac{1}{2} \\ If[ S ( \vec{x}_{n-1} )= 1 , 1 , -1 ] + \frac{1}{2} \\ If[ S ( \vec{x}_{n-1} ) = 0 , 1 , -1]] \; = \; payoff(n-1) \; \; \forall n \in \mathbb{N} \end{multline} (where I have adopted Mc Carthy's LISP conditional notation \cite{Mc-Carthy-60} popularized by Wolfram's \textit{Mathematica} \cite{Wolfram-96}). Furthermore: \begin{equation} \label{global payoff at step n} E[payoff(n)] \; = \; \sum_{k = -n+1}^{n-1} P[payoff(n-1) = k] \, E[payoff(n) | payoff(n-1) ] \; \; \forall n \in \mathbb{N} \end{equation} We will prove that $ \lim_{n \rightarrow \infty} E[ payoff(n) ] \; = \; 0 $ by proving by induction on n that $ E[ payoff(n) ] \; = \; 0 \; \; \forall n \in {\mathbb{N}} $. That $ E[payoff(1)] \; = \; 0 $ follows immediately by the fact that $ S ( \lambda ) \; = \; \uparrow \; \; \forall S $. We have, conseguentially, simply to prove that $ E[payoff(n-1)] \; = \; 0 \; \; \Rightarrow \; \; E[payoff(n)] \; = \; 0 \; \; \forall S $. This is, anyway, an obvious conseguence of the equations eq.\ref{eq:conditional payoff at step n} and eq.\ref{global payoff at step n} \end{proof} \bigskip Theorem\ref{th:weak law of excluded gambling systems} is not, anyway, a great assurance for Casino's owner: in fact it doesn't exclude that the gambler, if enough lucky, may happen to get a positive payoff for $ n \rightarrow \infty $. What will definitely assure him is the following: \begin{theorem} \label{strong law of excluded gambling strategies} STRONG LAW OF EXCLUDED GAMBLING STRATEGIES For $ n \rightarrow \infty $ the set of the \textit{lucky-winning strategies} tends to the null set \end{theorem} And here comes the astonishing fact: Theorem\ref{strong law of excluded gambling strategies} can't be proved with purely measure-theoretic concepts. Our approach will consist in taking von Mises' axiom\ref{ax:axiom of randomness} as a definition of the set of subsequences to which such an axiom applies. Let us then define the \textbf{set of collectives} $ {\mathcal{C}}ollectives\; \subset \; \Sigma^{\infty} $ as the set of sequences having not enough intrinsic regularity to allow, if they occur, a lucky-winning strategy. Clearly such a definition depends on the class $ {{\mathcal{S}}trategies}_{admissible} ( {\mathcal{C}}ollectives )$ of admissible gambling strategies. It would appear natural ,at first, to admit every gambling strategy. But such a choice would lead immediately to conclude that $ {\mathcal{C}}ollectives \; = \; \emptyset $ since given two gambling strategies $ S_{0} $ and $ S_{1} $ so that: \begin{equation} EXT[S_{i}] ( \bar{x} ) \; \text{ is made only of i } \; i=0,1 \; \forall \bar{x} \in \Sigma^{\infty} \end{equation} we would have clearly that: \begin{equation} P_{VM}( a_{i} \,| \, EXT[S_{1}] (\bar{x}) \, ) \; \neq P_{VM}( a_{i} \,| \, EXT[S_{2}] (\bar{x}) \; \; \forall \bar{x} \in \Sigma^{\infty} \end{equation} The history of the attempts of characterizing in a proper way the class of the admissible gambling strategies is very long and curious \cite{Van-Lambalgen-87}, \cite{Li-Vitanyi-97}, \cite{Gillies-00} and involved many people: Church, Copeland, D\"{orge}, Feller, Kamke, Popper, Reichenbach, Tornier, Waismann and Wald; I will report here only the conceptually more important contributions: in the thirties Abraham Wald showed that: \begin{theorem} \label{th:Wald theorem} \end{theorem} WALD'S THEOREM \begin{equation} ( cardinality ( {{\mathcal{S}}trategies}_{admissible} ( {\mathcal{C}}ollectives ) ) \, = \, \aleph_{0} ) \; \Rightarrow \;( {\mathcal{C}}ollectives \neq \emptyset ) \end{equation} In the fourties, basing on the observation that gambling strategies must be effectively followed, Alonzo Church proposed, according to the Church-Turing's Thesis \cite{Odifreddi-89}, to consider admissible a gambling strategy if and only if it is a \textbf{partial recursive function}. With such an assumption: \begin{equation} \label{eq:Church admissible gambling strategies} {{\mathcal{S}}trategies}_{admissible} ( {\mathcal{C}}ollectives ) \; := C_{\Phi}-C_{M}-\Delta_{0}^{0}-\stackrel{ \circ } {MAP}( \Sigma^{\star},\Sigma^{\star} ) \end{equation} it can be proved that: \begin{equation} P_{unbiased} ( {\mathcal{C}}ollectives ) \; = \; 1 \end{equation} immediately implying Theorem\ref{strong law of excluded gambling strategies}. Let us the introduce the following: \begin{definition} \label{Church random sequences} \end{definition} CHURCH RANDOM SEQUENCES: \begin{equation} CHURCH-RANDOM( \Sigma^{\star} ) \; := \; {\mathcal{C}}ollectives \; \text{ with the assumption of eq.\ref{eq:Church admissible gambling strategies} } \end{equation} \smallskip \begin{remark} \end{remark} MARTINGALES AND THE REASON WHY REAL CASINOS RESULT IN ACTIVE It is important to observe that Theorem\ref{strong law of excluded gambling strategies} was proved under the assumption that the gambler bets always at a fixed odd. Assuming a more general definition of a gambling strategy in which the odd betted each time is adjustable in function of a recursive function of the history up to that bet, it may be easily proved that winning gambling strategies do exist. An example is given by \textbf{martingales}: let us suppose that the gambler plays in the following way: \begin{itemize} \item he insists on betting always on \emph{head}, doubling the stake after a loss \item he stops to bet for ever after the first win \end{itemize} In analyzing such a gambling situation Daniel Bernoulli introduced the so called \textbf{Saint Petersburg paradox}: since the gambler bets 1 fiche that heads will turn up on the first throw, 2 fiches that heads will turn up on the second throw if it didn't turn up on the first, 4 fiches that heads will turn up on the third throw if it didn't turn up in the first two throws and so on, one could conclude the gabler's expected pay off is infinite: \begin{equation} \label{eq:Saint Petersburg paradox} \frac{1}{2} (1) \, + \, \frac{1}{4}(2) \, + \, \frac{1}{8} (4) \, + \cdots \; = \; \frac{1}{2} \, + \, \frac{1}{2} \, + \, \frac{1}{2} \, + \, \cdots \; = + \infty \end{equation} To see clearly where the mistake is, let us proceed by steps. First of all let us observe that, since \textbf{ownerhip of all subsequences} is a Law of Randomness, we have in particular that \textbf{ownerhip of the subsequence 1} is a Law of Randomness too. Hence heads will certainly turn up one day. Conseguentially it is legitimate to express the expected payoff as a sum on the first time heads turn up, as it was done in eq.\ref{eq:Saint Petersburg paradox}: \begin{equation} \lim_{ n \rightarrow \infty } E[payoff(n)] \; = \; \sum_{n=1}^{\infty} P_{unbaised \, n} ( 0^{n-1} 1 ) gain(n) \end{equation} But the gain corresponding to the situation in which heads turn up for the first time at the $ n^{th} $ throw must take into account of all the fiches he lost in the previous $ n - 1 $ turns. So: \begin{equation} gain(n) \; = \; 2^{n-1} \, - \, \sum_{k=1}^{n-1} 2^{k} \; = \; 2^{n} \, - \, ( 2^{n-1} - 1) \; = \; 1 \end{equation} Hence: \begin{equation} \lim_{ n \rightarrow \infty } E[payoff(n)] \; = \; \sum_{n=1}^{\infty} \frac{1}{2^{n}} \; = \; 2 \end{equation} But, if allowing to rule also the stakes, one can violate even the Weak Law of Excluded Gambling Systems, why don't Casinos go all in ruin? The reason is that a gambling strategy as the displayed martingale requires an unbounded budget, i.e. that the gambler cannot go broke. \newpage \section{The weakness of Church randomness with respect to Chaitin randomness} The Law of Excluded Classical Gambling System could be seen, at a foundational level, as the corner stone for a mathematical characterization of the concept of classical algorithmic randomness. With the intuitivelly compelling choice of eq.\ref{eq:Church admissible gambling strategies} for the class of admissible gammblig strategies, this results in the notion of Church randomness introduce in the previous section. It appears then natural to ask ourselves which inter-relation exists between the resulting notion of Church randomness and the notion of Martin-L\"{o}f Solovay Chaitin randomness we have arrived to recognize as the correct notion of classical algorithmic randomness. We stressed in the remark\ref{rem:on why Classical Probability Theory applies to reality} the importance of the fact that $ P_{unbaised} ( CHAITIN-RANDOM( \Sigma^{\infty} ) ) \; = \; 1 $. So we can appreciate the fact that: \begin{theorem} \label{th:the occured sequence of infinite independent tosses of a fair coin is certainly Church random} \end{theorem} THE OCCURED SEQUENCE OF INFINITE INDEPENDENT TOSSES OF A FAIR COIN IS CERTAINLY CHURCH-RANDOM \begin{equation} P_{unbaised} ( CHURCH-RANDOM( \Sigma^{\infty} ) ) \; = \; 1 \end{equation} \begin{proof} Given a generic $ S \in \Delta_{0}^{0} - \stackrel{ \circ } {MAP} (\Sigma^{\star} , \{0,1 \}) $ let us consider the unary predicate \textbf{failure of the gambling-system S} $ p_{failure \; gambling-system \; S} \, \in \, {\mathcal{P}} ( \Sigma^{\infty} ) $ defined as: \begin{multline} p_{failure \; gambling-system \; S} ( \bar{x} ) := << \lim_{ n \rightarrow \infty} \frac{ N_{i} (EXT[S] ( \vec{x} ) (n) )}{n} \; = \\ \lim_{ n \rightarrow \infty} \frac{ N_{i}( \vec{x} (n))}{n} \; \; i \in \Sigma >> \end{multline} The thesis follows immediately by the observation that: \begin{equation} p_{failure \; gambling-system \; S} \; \in \; {\mathcal{P}}_{TYPICAL} ( \Sigma^{\infty}) \; \; \forall S \in \Delta_{0}^{0} - \stackrel{ \circ } {MAP} (\Sigma^{\star} , \{0,1 \}) \end{equation} \end{proof} \smallskip Let us , now, observe that: \begin{theorem} \label{Chaitin randomness implies Church randomness} \end{theorem} \begin{equation} CHURCH-RANDOM( \Sigma^{\infty} ) \; \subseteq \; CHAITIN-RANDOM( \Sigma^{\infty} ) \end{equation} \begin{proof} The generic predicate $ p_{failure \; gambling-system \; S}$ is effectively-refutable. Together with theorem\ref{th:the occured sequence of infinite independent tosses of a fair coin is certainly Church random} this implies that: \begin{equation} p_{failure \; gambling-system \; S} \; \in \; {\mathcal{L}}_{RANDOMNESS} [ ( \Sigma^{\infty} \, , \, P_{unbiased} ) ] \; \; \forall S \in \Delta_{0}^{0}- \stackrel{ \circ } {MAP} (\Sigma^{\star} , \{0,1 \}) \end{equation} from which the the thesis follows immediately \end{proof} \smallskip Church randomness is, anyway, weaker than Martin-L\"{o}f-Solovay-Chaitin randomness as it was proved by J. Ville in 1939. Demanding to the wonderful Michael Van Lambalgen's dissertation thesis \cite{Van-Lambalgen-87} (in particular to the section2.6 for an hystorical analysis of the decline of von Mises axiomatization of Classical Probability Theory after the Geneva conference of 1937, and the collection of objection , both philosophical and formal, it received by Frechet, to section3.1 for a deep analysis of the philosophical differences between Church randomness and Martin-L\"{o}f-Solovay-Chaitin randomness and to the fourth chapter for the more advanced available analysis of the formal differences between such notions) for further information, let us simply report the statement of Ville's result: \begin{theorem} \label{th:Ville theorem} \end{theorem} VILLE'S THEOREM: \begin{align} \exists \, & \, \bar{x} \in CHURCH-RANDOM( \Sigma^{\infty} ) \; : \; p_{\text{infinite recurrence}} ( \bar{x} ) \text{ doesn't hold} \\ \exists \, & \, \bar{y} \in CHURCH-RANDOM( \Sigma^{\infty} ) \; : \; p_{\text{iterated logarithm}} ( \bar{y} ) \text{ doesn't hold} \end{align} \smallskip Since the \textbf{infinite recurrence property} and the \textbf{iterated logarithm property} are Laws of Randomness, Ville Theorem immediately implies that: \begin{corollary} \end{corollary} \label{cor:Church randomness is weaker than Chaitin randomness} \begin{equation} CHURCH-RANDOM( \Sigma^{\infty} ) \; \subset \; CHAITIN-RANDOM( \Sigma^{\infty} ) \end{equation} \part{The road for quantum algorithmic randomness} \label{part:The road for quantum algorithmic randomness} \chapter{The irreducibility of quantum probability both to classical determinism and to classical nondeterminism} \label{chap:The irreducibility of quantum probability both to classical determinism and to classical nondeterminism} \section{Why to treat sequences of qubits one has to give up the Hilbert-Space Axiomatization of Quantum Mechanics}\label{sec:Why to treat sequences of qubits one has to give up the Hilbert-Space Axiomatization of Quantum Mechanics} The problem of giving a mathematical foundation, i.e. a rigorous mathematical axiomatization, of Quantum Mechanics was first faced by John Von Neumann through his 1932's masterpiece \cite{Von-Neumann-83} in which he introduced Hilbert spaces, codifying the rule they play in Quantum Mechanics. So he introduced his, nowadays standard, Hilbert space axiomatization of Quantum Mechanics, where: \begin{definition} \label{def:Hilbert space axiomatization of Quantum Mechanics} \end{definition} HILBERT SPACE AXIOMATIZATION OF QUANTUM MECHANICS: any axiomatization of Quantum Mechanics assuming the following two axioms: \begin{axiom} \label{ax:Hilbert space axiom on states} \end{axiom} HILBERT-SPACE'S AXIOM ON STATES: The \textbf{pure states} of a \textbf{quantum mechanical systems} are \textbf{rays} in an Hilbert space $ {\mathcal{H}} $ \medskip \begin{axiom} \label{ax:Hilbert space axiom on observables} \end{axiom} HILBERT-SPACE'S AXIOM ON OBSERVABLES: The \textbf{observables} of a \textbf{quantum mechanical systems} are \textbf{self-adjoint operators} on $ {\mathcal{H}} $. The expected value of the observable $ \hat{O} $ in the state $ | \psi > $ is: \begin{equation} E_{ | \psi > } ( \hat{O} ) \; = \; \frac{ < \psi | \hat{O} | \psi > }{ < \psi | \psi > } \end{equation} \medskip The success and influence of the book \cite{Von-Neumann-83} was so great that the point of view therein exposed became suddenly the \emph{koin\'{e}} about the foundation of Quantum Mechanics, taught in all undergraduate courses. This had the curious effect of throwing a shadow on Von Neumann's successive intellectual path that led him to doubt not only about his 1932's Hilbert space axiomatization, but of the same fact that Quantum Mechanics may be formalized through an an Hilbert space formalization of some kind. \medskip To understand the corner-stone of Von Neumann's post-32 doubts let us consider the \textbf{Separability Issue}. Given the many subtilities involved it is useful to recall even the more elementary notions: \begin{definition} \end{definition} HILBERT SPACE: a complete inner-product space \smallskip Given an Hilbert space $ {\mathcal{H}} $ we shall say that \cite{Reed-Simon-80}: \begin{definition} \end{definition} $ {\mathcal{H}} $ IS SEPARABLE: it has a finite or countable orthonormal basis \medskip Given two Hilbert spaces $ {\mathcal{H}}_{1} $ and $ {\mathcal{H}}_{2} $ their tensor product, i.e. the Hilbert space $ {\mathcal{H}}_{1} \; \bigotimes \; {\mathcal{H}}_{2}$ is defined in the following way \cite{Reed-Simon-80}: \begin{enumerate} \item to any couple $ ( \, | \phi_{1} > \, , \, | \phi_{2} > \, ) $ with $ | \phi_{1} > \in {\mathcal{H}}_{1} $ and $ | \phi_{2} > \in {\mathcal{H}}_{2} $ one can associate the conjugate bilinear form $ | \phi_{1} > \, \bigotimes \, | \phi_{2} > $ defined on $ {\mathcal{H}}_{1} \times {\mathcal{H}}_{2} $ as: \begin{equation} ( \, | \phi_{1} > \, \bigotimes \, | \phi_{2} > ) ( | \psi_{1} > , | \psi_{2} > ) \; := \; < \phi_{1} | \psi_{1} > < \phi_{2} | \psi_{2} > \; \; | \psi_{1} > \in {\mathcal{H}}_{1} \, , \, | \psi_{2} > \in {\mathcal{H}}_{2} \end{equation} \item one considers the set $ {\mathcal{E}} $ of the finite linear combinations of such conjugate linear forms \item one defines on $ {\mathcal{E}} $ an inner product $ < \cdot | \cdot > $ by defining: \begin{equation} < \phi_{1} \bigotimes \phi_{2} | \phi_{3} \bigotimes \phi_{4} > \; := \; < \phi_{1} | \phi_{3} > < \phi_{2} | \phi_{4} > \end{equation} and extending by linearity to $ {\mathcal{E}} $ \item one defines $ {\mathcal{H}}_{1} \times {\mathcal{H}}_{2} $ as the completion of $ {\mathcal{E}} $ under such an inner product \end{enumerate} \medskip Such a definition of the tensor product of two Hilbert spaces trivially generalizes to define the tensor product $ {\mathcal{H}}_{1} \; \bigotimes \; \cdots \; \; \bigotimes \; {\mathcal{H}}_{n} $ of a finite number of Hilbert spaces. In particular one can consider the case in which the Hilbert spaces $ {\mathcal{H}}_{1} \, , \, \cdots \, , \, {\mathcal{H}}_{n} $ are equal: \begin{equation} {\mathcal{H}}_{i} \; = \; {\mathcal{H}} \; \; i \, = \, 1 , \cdots , n \end{equation} in which the above construction results in the following: \begin{definition} \end{definition} n-FOLD TENSOR PRODUCT OF THE HILBERT SPACE $ {\mathcal{H}} $: \begin{equation} {\mathcal{H}}^{\bigotimes n } \; := \; \bigotimes_{k=1}^{n} {\mathcal{H}} \end{equation} \medskip If, anyway, one tries to generalize such a procedure to define the $ \infty $-fold tensor product $ {\mathcal{H}}^{\bigotimes \infty } $ of an Hilbert space $ {\mathcal{H}} $ one immediately sees that the business doesn't work \cite{Thirring-83}: the squared norm of a vector $ | \psi > \; := \; | \psi_{1} > \, \bigotimes \cdots \, \bigotimes \, | \psi_{n} > \; \in \; {\mathcal{H}}^{\bigotimes n } $ is given by: \begin{equation} \label{eq:squared norm in the n-fold tensor product} \| \psi \|^{2} \; = \; < \psi | \psi > \; = \; \prod_{k=1}^{n} \, < \psi_{k} | \psi_{k} > \end{equation} Now if $ n \, = \, \infty $ the productory can, in general, diverge so one has to restrict only to those vectors for which the r.h.s. of eq.\ref{eq:squared norm in the n-fold tensor product} converges. Furthermore, on the remaining vectors, the productory can converge to zero even in those particular cases in which $ < \psi_{k} | \psi_{k} > \, > \, 0 \; \; \forall k \in {\mathbb{N}} $. In order to take the quotient space with respect to the zero vectors it is then necessary to form the equivalences classes not only of vectors with some factor zero, but also containing the vectors for which $ \; \prod_{k=1}^{n} \, < \psi_{k} | \psi_{k} > $ converges to zero. On such a quotient space the eq.\ref{eq:squared norm in the n-fold tensor product} defines a separating norm that can be used to complete it resulting in the required Hilbert space $ {\mathcal{H}}^{\bigotimes \infty } $, with the linear structure defined in the usual way. This does not yet, however, suffice to define the scalar product of different vector $ | \psi > $ and $ | \phi > $. Though only vectors such that: \begin{equation} < \psi_{k} | \psi_{k} > \, = \, < \phi_{k} | \phi_{k} > \, = \; 1 \; \; \forall k \in {\mathbb{N}} \end{equation} need to be considered, there are still two possibilities, namely: \begin{itemize} \item \textbf{CASE-I}: \begin{equation} \prod_{k=1}^{\infty} | < \psi_{k} | \phi_{k} > | \; \rightarrow \; c \, > \, 0 \end{equation} \item \textbf{CASE-II}: \begin{equation} \prod_{k=1}^{\infty} | < \psi_{k} | \phi_{k} > | \; \rightarrow \; 0 \end{equation} \end{itemize} where $ \rightarrow $ means unconditional convergence. In case-2 $ \prod_{k=1}^{\infty} < \psi_{k} | \phi_{k} > \; \rightarrow \; 0 $ as well, and the vectors may be considered orthogonal. In case-II, on the other hand, there is no guarantee that $ \prod_{k=1}^{\infty} < \psi_{k} | \phi_{k} > $ converges. If $ | < \psi_{k} | \phi_{k} > | \: = \: \exp ( i \theta_{k} ) | < \psi_{k} | \phi_{k} > | $, then their product is said to converge if not only $ \prod_{k=1}^{\infty} | < \psi_{k} | \phi_{k} > | $ but also $ \sum_{k} | \phi_{k} | $ converges. One now encounters the convention that vectors may be deemed orthogonal whenever $ \sum_{k} | \phi_{k} | \: \rightarrow \: \infty $ (we will indicate this situation as the \textbf{case-I.B}) Let us then agree on the following definition of the inner product: \begin{definition} \label{def:non zero inner product in the infinite tensor product Hilbert space} \end{definition} $ < \psi | \phi > \: = \: c \, \neq \, 0$ (\textbf{case-I.A}) \begin{equation} \lim_{n \rightarrow \infty} < \psi_{n} | \phi_{n} > \, = \, c \end{equation} \begin{definition} \label{def:zero inner product in the infinite tensor product Hilbert space} \end{definition} $ < \psi | \phi > \: = \: 0 $ (\textbf{case-II} or \textbf{case-I.B})) \begin{equation} \lim_{n \rightarrow \infty} < \psi_{n} | \phi_{n} > \, = \, 0 \end{equation} \bigskip Let us now observe that \textbf{separability} is a rather robust property, i.e. a property that preserves under many operations. Given an Hilbert space ${\mathcal{H}}$: \begin{theorem} \label{th:separability preservation under finite tensor product} \end{theorem} PRESERVATION OF SEPARABILITY UNDER FINITE-FOLD TENSOR PRODUCT \begin{equation} {\mathcal{H}} \; \text{ is separable } \; \; \Rightarrow \; \; ( {\mathcal{H}}^{ \bigotimes n} \; \text{ is separable } \; \; \forall n \in {\mathbb{N}} ) \end{equation} \medskip Given a sequence of Hilbert spaces $ \{ {\mathcal{H}}_{n} \}_{ n \in { \mathbb{N}}} $ we have furthermore the following: \begin{theorem} \label{th:separability preservation under infinite direct sum} \end{theorem} PRESERVATION OF SEPARABILITY UNDER INFINITE DIRECT SUM: \begin{equation} ( {\mathcal{H}}_{n} \; \text{ is separable } \; \forall n \in {\mathbb{N}}) \; \; \Rightarrow \; \; \bigoplus_{ n \in { \mathbb{N}}} \, {\mathcal{H}}_{n} \; \text{ is separable } \end{equation} These theorems are sufficient to guarantee the separability of almost all the Hilbert spaces appearing in Theoretical Physics: For the theorem\ref{th:separability preservation under finite tensor product} this is certainly the case when, in Nonrelativistic Quantum Mechanics, one considers a finite number of particles of spin s: since the Hilbert space for one particle is $ {\mathcal{H}} := L^{2} ( {\mathbb{R}}^{3} \, d \vec{x} ) \, \bigotimes \, {\mathbb{C}}^{2 s + 1} $, the n - particle Hilbert space is $ S_{n} {\mathcal{H}}^{\bigotimes n} $ if s is integer (i.e. if the particles are \textbf{bosons} ) and $ A_{n} {\mathcal{H}}^{\bigotimes n} $ if s is half-integer (i.e. if the particles are \textbf{fermions} ) where $ S_{n} $ and $ A_{n} $ are, respectively, the \textbf{n-simmetrization}, \textbf{n-antisimmetrization} operators. The underlying Hilbert space continues to remain separable even allowing an infinite number of particles as follows immediately introducing the following Hilbert spaces: \begin{definition} \end{definition} FOCK SPACE ASSOCIATED TO $ {\mathcal{H}} $: \begin{equation} {\mathcal{H}} ^{ \bigotimes \star } \; := {\mathcal{F}} ( {\mathcal{H}} ) \; := \; \bigoplus_{ n \in { \mathbb{N}}} {\mathcal{H}}^{\bigotimes n} \end{equation} \begin{definition} \end{definition} BOSONIC FOCK SPACE ASSOCIATED TO $ {\mathcal{H}} $: \begin{equation} {\mathcal{F}}_{S} ( {\mathcal{H}} ) \; := \; \bigoplus_{ n \in { \mathbb{N}}} S_{n} {\mathcal{H}}^{\bigotimes n} \end{equation} \begin{definition} \end{definition} FERMIONIC FOCK SPACE ASSOCIATED TO $ {\mathcal{H}} $: \begin{equation} {\mathcal{F}}_{A} ( {\mathcal{H}} ) \; := \; \bigoplus_{ n \in { \mathbb{N}}} A_{n} {\mathcal{H}}^{\bigotimes n} \end{equation} and observing that, for $ {\mathcal{H}} := L^{2} ( {\mathbb{R}}^{3} \, d \vec{x} ) \, \bigotimes \, {\mathbb{C}}^{2 s + 1} $, they are separable owing to theorem\ref{th:separability preservation under finite tensor product} and theorem\ref{th:separability preservation under infinite direct sum}. Let us now pass to Relativistic Quantum Mechanics, i.e. to Quantum Field Theory: for a free-field theory the separability of the proper Fock spaces follows again immediately from theorem\ref{th:separability preservation under finite tensor product} and theorem\ref{th:separability preservation under infinite direct sum}. For interacting field theories the situation is more complicated owing to the fact that a general mathematically-rigorous formalization of Quantum Field Theory, beside its exceptional developments \cite{Deligne-Etingof-Freed-Jeffrey-Kazhdan-Morgan-Morrison-Witten-99a},\cite{Deligne-Etingof-Freed-Jeffrey-Kazhdan-Morgan-Morrison-Witten-99b} and all the work of the Constructivists \cite{Glimm-Jaffe-87}, \cite{Jaffe-00}, is unfortunately still lacking \cite{Witten-95}. One could simply assert that Wightman Axioms constraint the underlying Hilbert space to be separable \cite{Reed-Simon-75} but such an answer would sound as a rather dogmatical one. A more convincing argument consists in considering that in the Lehmann-Symanzik-Zimmerman formalism the involved Hilbert spaces are only the asympotic In and Out Fock spaces \cite{Strocchi-93}. \medskip Unfortunately the robustness of \textbf{separability} is not complete. In fact: \begin{theorem} \label{th:separability not preservation under infinite tensor product} \end{theorem} NOT PRESERVATION OF SEPARABILITY UNDER INFINITE-FOLD TENSOR PRODUCT $ {\mathcal{H}}^{ \bigotimes \infty} $ \textbf{is not separable even if} $ {\mathcal{H}} $ \textbf{is separable} \begin{example} \label{ex:the Hilbert space of Quantum Information Theory} \end{example} THE HILBERT SPACES OF QUANTUM INFORMATION THEORY How much classical information is contained in a state: \begin{equation} | \psi > \; := \; \alpha | + > \, + \, \beta | - > \; \; \alpha , \beta \in {\mathbb{C}} \, : \, | \alpha |^{2} + | \beta |^{2} = 1 \end{equation} of a $ spin \frac{1}{2} $ system ? Since the bidimensional complex projective space has the continuum power: \begin{equation} cardinality( {\mathbb{C}} P^{2} ) \; = \;\aleph_{1} \end{equation} the specification of a point P on it requires the assignation of a whole sequence $ \bar{x}_{P} \in \Sigma^{\infty} $. In this way one is led to to argue that: \begin{equation} \label{eq:conclusion that one qubit is equal to infinite cbits} information( | \psi > ) \; = \; \infty \, bits \end{equation} But, given a $ spin \frac{1}{2} $ system prepared in the state $ | \psi > $, let us now suppose to make a measurement of the operator $ \hat{S}_{z} $. The information gained by the knowledge of the experimental outcome is only of one bit. So, from this reasoning, one is led to argue that: \begin{equation} \label{eq:conclusion that one qubit is equal to one cbits} information( | \psi > ) \; = \; 1 \, bit \end{equation} Obviously eq.\ref{eq:conclusion that one qubit is equal to infinite cbits} and eq.\ref{eq:conclusion that one qubit is equal to one cbits} are incompatible. This simple reasoning shows that the quantification of the informational content of the state $ | \psi > $ must be given in terms of a measure's unity not commensurable with that of \textbf{classical information}. This is a a conceptually extremelly deep concept: there doesn't exist a unique, mathematically charaterizable, notion of information, resulting in a measure's unity , the \textbf{bit}, in terms of which one can analyze the informational content of both classical and quantum physical systems: \textbf{quantum information} is not commensurable with \textbf{classical information}. Hence one has to give up the universal notion of \textbf{bit}, replacing it with the following couple of notions: \begin{itemize} \item the \textbf{cbit}, i.e. the measure's unity of \textbf{classical information} \item the \textbf{qubit}, i.e. the measure's unity of \textbf{quantum information} \end{itemize} The quantum-informational amount of the state $ | \psi > $ gives the operational definition of the \textbf{qubit}. A more formal definition will be given, anyway, in the remark\ref{rem:the noncommutative combinatory information and the definition of the qubit} in terms of the notion of \textbf{combinatorial quantum information}. As we will see in section\ref{sec:From the communicational-compression of the Quantum Coding Theorems to the algorithmic-compression in Quantum Computation} the incommensurability of \textbf{classical information} and \textbf{quantum information} is deeply linked with the No-Cloning Theorem \smallskip \begin{definition} \end{definition} ONE QUBIT HILBERT SPACE: \begin{equation} {\mathcal{H}}_{2} \; := \; {\mathbb{C}}^{2} \end{equation} Given an $ n \in {\mathbb{N}} $: \begin{definition} \label{def:space of the quantum strings of n qubits} \end{definition} n QUBITS HILBERT SPACE: \begin{equation} {\mathcal{H}}_{2}^{\bigotimes n} \; := \; {\mathbb{C}}^{2^{n}} \end{equation} \begin{definition} \label{def:space of the quantum strings of qubits} \end{definition} HILBERT SPACE OF QUBITS' STRINGS: \begin{equation} {\mathcal{H}}_{2}^{\bigotimes \star} \; := \; {\mathcal{F}} ( {\mathcal{H}}_{2}) \end{equation} \medskip On all these \textbf{separable} Hilbert spaces it is useful to introduce orthonormal complete bases, said the \textbf{computational basis} that embeds the strings of cbits in the quantum domain: \begin{definition} \label{def:computational basis} \end{definition} COMPUTATIONAL BASIS OF $ {\mathcal{H}}_{2} $: \begin{equation} {\mathbb{E}}_{2} \; := \; \{ | 0 > , | 1 > \} \; : \; | 0 > \, := \, \, \begin{pmatrix} 1 \\ 0 \ \end{pmatrix} \; , \; | 1 > \, := \, \begin{pmatrix} 0 \\ 1 \ \end{pmatrix} \end{equation} \smallskip \begin{remark} \label{rem:on the qubit operator} \end{remark} ON THE QUBIT OPERATOR The adoption of the \textbf{computational language} requires some caution. As to definition\ref{def:computational basis}, it is only a renaming of the usual language of spin 1/2 system: \begin{align} | 0 > & \; := \; | \uparrow_{z} > \\ | 1 > & \; := \; | \downarrow_{z} > \\ \end{align} The correspondence clearly continues considering the projectors: \begin{equation} | 0 > < 0 | \; = \; \begin{pmatrix} 1 & 0 \ \end{pmatrix} \: \begin{pmatrix} 1 \\ 0 \ \end{pmatrix} \; = \; \begin{pmatrix} 1 & 0 \\ 0 & 0 \ \end{pmatrix} \; = \; | \uparrow_{z} > < \uparrow_{z} | \end{equation} \begin{equation} | 1 > < 1 | \; = \; \begin{pmatrix} 0 & 1 \ \end{pmatrix} \: \begin{pmatrix} 0 \\ 1 \ \end{pmatrix} \; = \; \begin{pmatrix} 0 & 0 \\ 0 & 1 \ \end{pmatrix} \; = \; | \downarrow_{z} > < \downarrow_{z} | \end{equation} The problem arises if one tries to introduce a \textbf{qubit operator} having the binary alphabet $ \Sigma := \{ 0 ,1 \} $ as eigenvalues; in fact,owing to the vanishing of the addendum concerning the zero eigenvalue, one has obviously that: \begin{equation} \label{eq:wrong definition of the qubit operator} \hat{q} \; := \; 0 \, | 0 > < 0 | \: + \: 1 \, | 1 > < 1 | \: = \: | 1 > < 1 | \: = \; | \downarrow_{z} > < \downarrow_{z} | \end{equation} So, in order of introducing a qubit operator, one has to avoid the zero eigenvalue, e.g. assuming the spectrum of the \textbf{qubit operator} to be the binary alphabet $ \{ +1 , - 1 \} $, with the convention that the eigenvalue $ + 1 $ corresponds to zero and the eigenvalue $ - 1 $ corresponds to one. With these conventions one has that: \begin{equation} \label{eq:right definition of the qubit operator} \hat{q} \; := \; + 1 \, | 0 > < 0 | \: + \: ( - 1 ) \, | 1 > < 1 | \: = \: \hat{\sigma}_{z} \: = \: \begin{pmatrix} 1 & 0 \\ 0 & -1 \ \end{pmatrix} \end{equation} \smallskip Given any positive integer number $ n \geq 3 $: \begin{definition} \end{definition} COMPUTATIONAL BASIS OF $ {\mathcal{H}}_{2}^{\bigotimes n} $: \begin{equation} {\mathbb{E}}_{n} \; := \; \{ \: | \vec{x} > \, , \, \vec{x} \in \Sigma^{n} \: \} \end{equation} \begin{definition} \end{definition} COMPUTATIONAL BASIS OF $ {\mathcal{H}}_{2}^{\bigotimes \star} $: \begin{equation} {\mathbb{E}}_{\star} \; := \; \{ | \vec{x} > \, , \, \vec{x} \in \Sigma^{\star} \: \} \end{equation} The generic \textbf{string of qubits}, i.e. the generic vector of $ {\mathbb{H}}_{2}^{\bigotimes \star} $ is then given by a linear combination of the form $ \sum_{ \vec{x} \in \Sigma^{\star}} c_{\vec{x}} | \vec{x} > $. And what about \textbf{sequences of qubits}? We can indeed introduce the following notions: \begin{definition} \label{def:space of the quantum sequences of qubits} \end{definition} HILBERT SPACE OF QUBITS' SEQUENCES: \begin{equation} {\mathcal{H}}_{2}^{\bigotimes \infty} \; := \; \bigotimes_{n \in {\mathbb{N}}} {\mathcal{H}}_{2} \end{equation} By theorem\ref{th:separability not preservation under infinite tensor product} $ {\mathcal{H}}_{2}^{\bigotimes \infty} $ is not separable. \bigskip The \textbf{Separability Issue} consists in the following question: \smallskip \begin{center} \textbf{is it necessary to modify the axiom\ref{ax:Hilbert space axiom on states} adding the constraint that the Hilbert space $ {\mathcal{H}} $ is separable?} \end{center} \smallskip The thesis that the correct answer is affermative has been asserted by authoritative voices; for example Walter Thirring remembers that \cite{Thirring-01}: \begin{center} \textit{"For finite tensor products the dimension of the spaces is multiplicative and for infinite tensor product is is uncountable even if the individual spaces have only dimension $ \, = \, 2$. This casts some doubt on whether there is a mathematically valid description of infinite quantum systems. Schr\"{o}dinger once told me that the corresponding non-separable Hilbert space did not make sense to him. To determine N components of his $ \psi $-function one needs N experiments and in a non-separable space one would need an uncountable number of measurements and this is nonsense. However such an opinion means that Schr\"{o}dinger did not get the main message of Von Neumann's celebrated paper on infinite tensor products ."} \end{center} Keeping aside for a moment Thirring's last remark, let us observe that there exist also many other arguments supporting a positive answer to the \textbf{Separability Issue}: for example on an not-separable Hilbert space the Gram-Schmidt orthogonalization process can be adopted only appealing to the Axiom of Choice \cite{Reed-Simon-75}. \smallskip But let us now analyze Thirring's last remark: is Thirring right in claiming that Von Neumann's celebrated paper on infinite tensor products lead to give a negative answer to the Separability Issue? Our point of view, though predictively completelly equivalent to Thirring's authoritative one, is philosophically different and lead, as to the Separability Issue, to the opposite answer, as we will arrive to discuss at the end of this section. To show why, anyway, it may be useful to follow the reconstruction of Von Neumann's intellectual path on the Foundations of Quantum Physics made by Miklos Redei in \cite{Redei-98}, emerging from \label{Petz-Redei-95} and condensated in \cite{Redei-01}: \begin{center} Hilbert spaces $ \rightarrow $ orthocomplemented modular lattices $ \rightarrow W^{\star}$-algebras \end{center} that, as we will show, correspond conceptually to the path: \begin{center} Quantum Mechanics $ \rightarrow $ Quantum Logic $ \rightarrow $ Quantum Probability \end{center} This requires , anyway the introduction of a whole abstract algebraic machinery. \begin{definition} \label{def:partially ordered set} \end{definition} PARTIALLY ORDERED SET (POSET) a couple $ ( {\mathcal{L}} \, , \, \preceq ) $ such that $ {\mathcal{L}} $ is a set, while $ \leq $ is a partial ordering on $ {\mathcal{L}} $, i.e. a reflexive, transitive, antisimmetric relation on $ {\mathcal{L}} $ \smallskip Given a poset $ ( {\mathcal{L}} \, , \, \preceq ) $ and two element $ a , b \in {\mathcal{L}} $: \begin{definition} \end{definition} \begin{equation} a \, \prec \, b \; := \; ( a \preceq b) \, and \, a \neq b \end{equation} \begin{definition} \end{definition} \begin{equation} a \, \succ \, b \; := \; ( a \succeq b) \, and \, a \neq b \end{equation} Given a set $ S \subseteq {\mathcal{L}} $: \begin{definition} \label{def:upper bound} \end{definition} a IS AN UPPER BOUND OF S IN THE POSET $ ( {\mathcal{L}} \, , \, \preceq ) $: \begin{equation} b \; \preceq a \; \; \forall b \in S \end{equation} \begin{definition} \label{def:lower bound} \end{definition} a IS A LOWER BOUND OF S IN THE POSET $ ( {\mathcal{L}} \, , \, \preceq) $: \begin{equation} b \; \succeq a \; \; \forall b \in S \end{equation} \begin{definition} \label{def:least upper bound} \end{definition} a IS THE LEAST UPPER BOUND OF S IN THE POSET $ ( {\mathcal{L}} \, , \, \preceq ) $: \begin{equation} b \; \preceq a \; \; \forall b \text{ upper bound of S in } ( {\mathcal{L}} \, , \, \leq ) \end{equation} \begin{definition} \label{def:least lower bound} \end{definition} a IS THE LEAST LOWER BOUND OF S IN THE POSET $ ( {\mathcal{L}} \, , \, \preceq ) $: \begin{equation} b \; \succeq \; a \; \; \forall b \text{ lower bound of S in } ( {\mathcal{L}} \, , \, \leq ) \end{equation} \begin{definition} \label{def:lattice} \end{definition} LATTICE: a poset $ ( {\mathcal{L}} \, , \, \preceq ) $ such that: \begin{align} \forall \, & a , b \in {\mathcal{L}} \; , \; \exists a \bigvee b \, := \, \text{ least upper bound of $ \{ a , b \} $ in } {\mathcal{L}} \\ \forall \, & a , b \in {\mathcal{L}} \; , \; \exists a \bigwedge b \, := \, \text{ greatest lower bound of $ \{ a , b \} $ in } {\mathcal{L}} \\ \exists \, & \, 0_{{\mathcal{L}}} \in {\mathcal{L}} \; : \; 0_{{\mathcal{L}}} \, \preceq a \; \; \forall a \in {\mathcal{L}} \\ \exists \, & \, 1_{{\mathcal{L}}} \in {\mathcal{L}} \; : \; a \, \preceq 1_{{\mathcal{L}}} \; \; \forall a \in {\mathcal{L}} \end{align} \smallskip Given a lattice $ ( {\mathcal{L}} \, , \, \preceq ) $: \begin{definition} \label{def:atom in a lattice} \end{definition} $ a \in {\mathcal{L}} $ IS AN ATOM OF $ ( {\mathcal{L}} \, , \, \preceq ) $: \begin{equation} b \, \preceq \, a \; \Rightarrow \; b = a \, or \, b = 0_{{\mathcal{L}}} \end{equation} \begin{definition} \label{def:atomic lattice} \end{definition} $ ( {\mathcal{L}} \, , \, \preceq ) $ IS ATOMIC: \begin{equation} \forall b \in {\mathcal{L}} \; \exists b \in {\mathcal{L}} \; atom \; : \; a \, \preceq \, b \end{equation} \begin{definition} \label{def:logical dimension function on a lattice} \end{definition} LOGICAL DIMENSION FUNCTION ON $ ( {\mathcal{L}} \, , \, \preceq ) $: a function $ d \, : \, {\mathcal{L}} \; \mapsto \; [ 0 , + \infty ] $ such that: \begin{align} a \, & \preceq \, b \; \Rightarrow \; d(a) \, \leq \, d(b) \; \; \forall a , b \in {\mathcal{L}} \\ d(a) \, & \, + \, d(b) \; = \; d ( a \bigwedge b ) \,+ \, d ( a \bigvee b ) \; \; \forall a , b \in {\mathcal{L}} \end{align} \begin{definition} \label{def:distributive lattice} \end{definition} $ ( {\mathcal{L}} \, , \, \preceq ) $ IS DISTRIBUTIVE: \begin{equation} a \bigvee ( b \bigwedge c ) \; = \; ( a \bigvee b ) \, \bigwedge \, ( a \bigvee c ) \; \; \forall a , b , c \in {\mathcal{L}} \end{equation} \begin{definition} \label{def:modular lattice} \end{definition} $ ( {\mathcal{L}} \, , \, \preceq ) $ IS MODULAR: \begin{equation} a \preceq b \; \Rightarrow \; a \bigvee ( b \bigwedge c ) \; = \; ( a \bigvee b ) \, \bigwedge \, ( a \bigvee c ) \; \; \forall a , b , c \in {\mathcal{L}} \end{equation} \begin{definition} \label{def:orthocomplementation on a lattice} \end{definition} ORTHOCOMPLEMENTATION ON $ ( {\mathcal{L}} \, , \, \preceq ) $: a map $ \bot : {\mathcal{L}} \mapsto {\mathcal{L}} $ such that: \begin{align} ( a & ^ {\bot})^{ \bot } \; = \; a \; \; \forall a \in {\mathcal{L}} \\ a \, & \preceq \, b \; \Rightarrow \; b^{\bot} \, \preceq \, a^{\bot} \; \; \forall a , b \in {\mathcal{L}} \\ a \, & \bigwedge \, a^{\bot} \; = \; 0_{{\mathcal{L}}} \; \; \forall a \in {\mathcal{L}} \\ a \, & \bigvee \, a^{\bot} \; = \; 1_{{\mathcal{L}}} \; \; \forall a \in {\mathcal{L}} \end{align} \smallskip \begin{definition} \label{def:orthocomplemented lattice} \end{definition} ORTHOCOMPLEMENTED LATTICE: a therne $ ( ( {\mathcal{L}} \, , \, \preceq \, , \, \bot ) $ such that $ ( {\mathcal{L}} \, , \, \preceq ) $ is a lattice while $ \bot $ is an orthocomplementation on $ ( {\mathcal{L}} \, , \, \preceq ) $ \smallskip Given an orthocomplemented lattice $ ( ( {\mathcal{L}} \, , \, \preceq \, , \, \bot ) $ an two its elements $ a,b \in {\mathcal{L}} $ \begin{definition} \label{def:orthogonality in an orthocomplemented lattice} \end{definition} a IS ORTHOGONAL TO B: \begin{equation} a \, \bot \, b \; := \; a \, \preceq b^{\bot} \end{equation} Clearly, by the definition\ref{def:orthocomplementation on a lattice}, one has that orthogonality is a simmetric relation: \begin{equation} a \, \bot \, b \; \Leftrightarrow \; b \bot a \; \; \forall a,b \in {\mathcal{L}} \end{equation} \begin{definition} \label{def:orthomodular lattice} \end{definition} $ ( ( {\mathcal{L}} \, , \, \preceq \, , \, \bot ) $ IS ORTHOMODULAR: \begin{equation} a \preceq b \; and \; a \bot c \; \Rightarrow \; a \bigvee ( b \bigwedge c ) \; = \; ( a \bigvee b ) \, \bigwedge \, ( a \bigvee c ) \; \; \forall a , b , c \in {\mathcal{L}} \end{equation} Orthomodularity is a weakening of modularity that is a weakening of distributivity as is stated by the following: \begin{theorem} \end{theorem} \begin{align} distributivity & \; \Rightarrow \; modularity \; \Rightarrow \; orthomodularity \\ orthomodularity & \; \nRightarrow \; modularity \; \nRightarrow \; distributivity \end{align} \smallskip We will soon see the utility of the following: \begin{theorem} \label{th:on the finite dimension} \end{theorem} THEOREM ON THE FINITE DIMENSION: \begin{hypothesis} \end{hypothesis} \begin{equation*} ({\mathcal{L}} \, , \, \preceq ) \; \; lattice \end{equation*} \begin{equation*} \exists \, d \, \text{ logical dimension function } \; : \infty \notin Range(d) \end{equation*} \begin{thesis} \end{thesis} \begin{center} $ {\mathcal{L}} $ is modular \end{center} \smallskip In a fundamental 1936's paper with G. Birkhoff \cite{Birkhoff-Von-Neumann-95}, Von Neumann suggested the idea, yet implicitely advanced in the fifth section of the third chapter of \cite{Von-Neumann-83}, that the difference between Quantum Mechanics and Classical Mechanics could be ascribed to the fact the algebraic structure of the set of all the propositions concerning a quantum system violates the laws of Classical Logic, obeying a new kind of logic. This was the seed of the 65 year old research-field of Quantum Logic. It must be remarked that the original Birkhoff and Von Neumann's definition of a quantum logic was more restrictive than that choram-populi later assumed in such a research field; to distinguish the two notion we will speak, respectively, of weak and strong quantum logics. Anyway, exactly as the Theoretical Physicist's community foxilized on the 1932's snapshot of Von Neumann's intellectual path, the same happened to the Quantum Logicist's community that foxilized on the 1936's snapshot (mostly altering it), so don't catching all the reasons led Von Neumann to make the phase-transition: \begin{center} Quantum Logic $ \rightarrow $ Quantum Probability \end{center} that, as we will briefly point in section\ref{sec:On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics}, can be seen as the starting point of the of the open intellectual challenge summarized by the path: \begin{center} Quantum Mechanics as Nondistributive Logic $ \rightarrow $ Quantum Mechanics as Noncommutative Probability $ \rightarrow $ Quantum Mechanics as Noncommutative Geometry \end{center} \smallskip The difference between Classical Logic and Quantum Logic is enclosed in the different algebraic structure that the set of all the propositions concerning a physical system obey as far as the \textbf{conjunction} $ \bigwedge $, \textbf{disjunction} $ \bigvee $ and \textbf{negation} $ ' $ are concerned: \begin{definition} \label{def:classical logic} \end{definition} CLASSICAL LOGIC: a \textbf{distributive}, \textbf{orthocomplemented} lattice \smallskip \begin{definition} \label{def:strong quantum logic} \end{definition} STRONG QUANTUM LOGIC: a \textbf{modular}, \textbf{orthocomplemented} lattice \medskip \begin{definition} \label{def:weak quantum logic} \end{definition} WEAK QUANTUM LOGIC: an \textbf{orthomodular}, \textbf{orthocomplemented} lattice \begin{example} \label{ex:the classical logic of power-sets} \end{example} THE CLASSICAL LOGIC OF POWER-SETS Given an arbitrary set S let us introduce on its power-set $ 2^{S} $ the partial-ordering relation: \begin{equation} a \,\preceq \, b \; := \; a \, \subseteq \, b \; \; a,b \in 2^{S} \end{equation} It may be easily verified that $ ( 2^{S} , \preceq ) $ is a lattice, with: \begin{align} a \, & \bigwedge \, b \; := \; a \, \bigcap \, b \; \; \forall a , b \in 2^{S} \\ a \, & \bigvee \, b \; := \; a \, \bigcup \, b \; \; \forall a , b \in 2^{S} \\ 0_{2^{S}} & \; = \; \emptyset \\ 1_{2^{S}} & \; = \; 2^{S} \end{align} Introduced on $ ( 2^{S} , \preceq ) $ the orthocomplementation map $ \bot \, : \, 2^{S} \mapsto 2^{S} $: \begin{equation} a^{\bot} \; := \; S \, - \, a \; \; a \in 2^{S} \end{equation} $ ( 2^{S} \, , \, \preceq \, , \, \bot ) $ is a classical logic. If $ cardinality(S) \, \in \, {\mathbb{N}} $ the map $ d : 2^{S} \mapsto {\mathbb{R}}_{+} $ defined as: \begin{equation} d(S) \; := \; cardinality(S) \end{equation} is a logical dimension function. This is not the case, anyway, if $ cardinality(S) \, \geq \, \aleph_{0} $, even generalizing definition\ref{def:logical dimension function on a lattice} in order of allowing infinite cardinal values of a logical dimension function, as can be seen, for example, observing that: \begin{align} {\mathbb{Z}} \, & \, \preceq \, {\mathbb{Q}} \; but \; cardinality({\mathbb{Z}}) \, = \, cardinality({\mathbb{Q}}) \, = \, \aleph_{0} \\ {\mathbb{R} } & - {\mathbb{Q}} \, \preceq \, {\mathbb{R}} \; but \; cardinality({\mathbb{R} } - {\mathbb{Q}}) \, = \, cardinality({\mathbb{R}}) \, = \, \aleph_{1} \end{align} \medskip \begin{example} \label{ex:the strong quantum logic of the n-qubits Hilbert space} \end{example} THE STRONG QUANTUM LOGIC OF THE n-QUBITS HILBERT SPACE Given the n-qubits Hilbert space $ {\mathcal{H}}_{2}^{ \bigotimes n} $ let us consider its \textbf{projective geometry}, i.e. the set $ {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) $ of all its linear subspaces: \begin{equation} {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \; := \; \bigcup_{k=0}^{2^{n}} G_{k , 2^{n} } ({\mathbb{C}}) \end{equation} Introduced on $ {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) $ the partial ordering relation: \begin{equation} a \, \preceq \, b \; := \; a \, \subseteq \, b \; \; a , b \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \end{equation} the poset $ ( {\mathcal{L}} \, , \, \preceq ) $ is an atomic lattice, with: \begin{align} a \, & \bigwedge \, b \; := \; a \, \bigcap \, b \; \; a , b \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \\ a \, & \bigvee \, b \; := \; a \, \bigoplus \, b \; \; a , b \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \\ \end{align} whose atoms are the one-dimensional subspaces, namely the elements of $ G_{1 , 2^{n} } ({\mathbb{C}}) \, = \, {\mathbb{C}}P^{ 2^{n} -1 }$, said the \textbf{points} of the \textbf{projective geometry}, i.e. the \textbf{rays} of $ {\mathcal{H}}_{2}^{ \bigotimes n} $. It may be easily verified that the following map: \begin{equation} d(a) \; := \; dim(a) \; \; a \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \end{equation} is a logical dimension function. Since: \begin{equation} \infty \; \notin \; Range(d) \; = \; \{ 0 , 1 , \cdots , 2^{n} \} \end{equation} it follows by theorem\ref{th:on the finite dimension} that $ ( {\mathcal{L}} \, , \, \preceq ) $ is modular. So, introduced on $ ( {\mathcal{L}} ({\mathcal{H}}_{2}^{ \bigotimes n} ) \, , \, \preceq ) $ the orthocomplementation map: \begin{equation} a^{\bot} \; := \; \{ | \psi_{1} > \in {\mathcal{H}}_{2}^{ \bigotimes n} \, : \, < \psi_{1} | \psi_{2} > = 0 \; \forall | \psi_{2} > \, \in \, a \} \; \; a \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \end{equation} $ ( ( {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) \, , \, \preceq \, , \, \bot ) $ is a strong quantum logic. \medskip \begin{example} \label{ex:the weak quantum logic of the qubits' strings Hilbert space} \end{example} THE WEAK QUANTUM LOGIC OF THE QUBITS' STRINGS HILBERT SPACE Given the qubits' strings Hilbert space $ {\mathcal{H}}_{2}^{\bigotimes \star} $ let us consider the set $ {\mathcal{L}} ( {\mathcal{H}}_{2}^{\bigotimes \star} ) $ of all its closed linear subspaces. Introduced on $ {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes n} ) $ the partial ordering relation of example\ref{ex:the strong quantum logic of the n-qubits Hilbert space}: \begin{equation} a \, \preceq \, b \; := \; a \, \subseteq \, b \; \; a , b \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{\bigotimes \star} ) \end{equation} the poset $ ( {\mathcal{L}} \, , \, \preceq ) $ is again an atomic lattice with atoms the \textbf{rays} of $ {\mathcal{H}}_{2}^{ \bigotimes \star} $. As in example\ref{ex:the strong quantum logic of the n-qubits Hilbert space} the following map: \begin{equation} d(a) \; := \; dim(a) \; \; a \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{\bigotimes \star} ) \end{equation} is a logical dimension function. But since now: \begin{equation} \infty \; = \; dim( {\mathcal{H}}_{2}^{\bigotimes \star} ) \; \in \; Range(d) \; = \; \{ 0 , 1 , \cdots , \infty \} \end{equation} we can't apply theorem\ref{th:on the finite dimension} anymore. Indeed it may be proved that the lattice $ ( {\mathcal{L}} ( {\mathcal{H}}_{2}^{\bigotimes \star} ) $ is not modular but only orthomodular. So, introduced on $ ( {\mathcal{L}} ({\mathcal{H}}_{2}^{\bigotimes \star} ) \, , \, \preceq ) $ the orthocomplementation map: \begin{equation} a^{\bot} \; := \; \{ | \psi_{1} > \in {\mathcal{H}}_{2}^{ \bigotimes \star} \, : \, < \psi_{1} | \psi_{2} > = 0 \; \forall | \psi_{2} > \, \in \, a \} \; \; a \in {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes \star} ) \end{equation} $ ( ( {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes \star} ) \, , \, \preceq \, , \, \bot ) $ is a not a strong quantum logic but only a weak quantum logic. \smallskip The situation delineated by example\ref{ex:the strong quantum logic of the n-qubits Hilbert space} and example\ref{ex:the weak quantum logic of the qubits' strings Hilbert space} is a particular case of Hilbert lattices' theory. Given an arbitrary Hilbert space $ {\mathcal{H}} $: \begin{definition} \label{def:Hilbert lattice} \end{definition} HILBERT LATTICE OF $ {\mathcal{H}} $: the orthocomplemented lattice $ ( {\mathcal{L}} ( {\mathcal{H}}) \, , \, \preceq \, , \, \bot ) $ where, as usual, $ {\mathcal{L}} ( {\mathcal{H}}) $ is the set of all the closed linear subspaces of $ {\mathcal{H}} $, and: \begin{align} a \, & \preceq \, b \; := \; a \, \subseteq \, b \; \; a , b \in {\mathcal{L}} ( {\mathcal{H}}) \\ a^{\bot} & \; := \; \{ | \psi_{1} > \in {\mathcal{H}} \, : \, < \psi_{1} | \psi_{2} > = 0 \; \forall | \psi_{2} > \, \in \, a \} \; \; a \in {\mathcal{L}} ( {\mathcal{H}} ) \\ \end{align} \begin{theorem} \label{th:Hilbert lattices as quantum logics} \end{theorem} \begin{align} ( & {\mathcal{L}} ( {\mathcal{H}}) \, , \, \preceq \, , \, \bot ) \text{ is a weak quantum logic} \\ ( & {\mathcal{L}} ( {\mathcal{H}}) \, , \, \preceq \, , \, \bot ) \text{ is a strong quantum logic } \; \Leftrightarrow \; dim({\mathcal{H}}) < \infty \end{align} \smallskip Theorem\ref{th:Hilbert lattices as quantum logics} can be applied also to the qubit sequences' Hilbert space $ {\mathcal{H}}_{2}^{ \bigotimes \star} $ to infer that $ ( {\mathcal{L}} ( {\mathcal{H}}_{2}^{ \bigotimes \star} ) \, , \, \preceq \, , \, \bot ) $ is a weak quantum logic. But here comes the great conceptual shock, to prepare which, let us observe, first of all, that the \textbf{Separability Issue} and Thirring's claim that Schrodinger negative answer to it was owed to a not-comprehension of Von Neumann's paper (written with J. Murray) on infinite tensor product clashes with the Von Neumann's 1936-dated confession to Birkhoff (partially reprinted in the paragraph7.1.2 of \cite{Redei-98}): \begin{center} \textit{"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert spaces any more. After all Hilbert space (as far as quantum mechanical things are concerned) was obtained by generalizing Euclidean space, footing on the principle of 'conserving the validity al all formal rules'. $ \cdots $ Now we begin to believe that it is not the vectors which matter, but the lattice of all linear (closed) subspaces. Because \begin{enumerate} \item The vectors ought to represent the physical states, but they do it redundantly, up to a complex factor only, \item and besides, the states are merely a derived notion, the primitive (phenomenologically given) notions being the qualities which correspond to the linear closed subspaces \end{enumerate} But if we wish to generalize the lattice of all linear closed subspaces from a Euclidean space to infinitely many dimensions, then one does not obtain Hilbert space, but the configuration which Murray and I called the 'case $ II_{1}$'.(The lattice of all linear closed subspace of Hilbert space is our '$I_{\infty}$' case)"} \end{center} What is Von Neumann speaking about? To answer this question it is necessary to introduce some notion concerning $W^{\star}$-algebras; demanding to the immense literature (e.g. \cite{Sunder-87}, \cite{Kadison-Ringrose-97a}, \cite{Kadison-Ringrose-97b}, \cite{Sakai-98}, \cite{Bing-Ren-92} from the purely mathematical side, to \cite{Thirring-81}, \cite{Thirring-83}, \cite{Simon-93}, \cite{Bratteli-Robinson-87}, \cite{Bratteli-Robinson-97}, \cite{Connes-94}, \cite{Ruelle-99} for physically motivated treatment or to the reviews \cite{Petz-Redei-95}, \cite{Kadison-90} if you want to survive the experience) for any further $ C_{\Phi}$-information we will here briefly recall the basic facts: \begin{definition} \label{def:algebra} \end{definition} ALGEBRA: a couple $( \, A \, , \, \circ \, ) $ such that: \begin{itemize} \item A is as linear space on the complex field $ {\mathbb{C}}$ \item $ \circ \, A \, \times \, A \, \rightarrow \, A \; \; ( a , b ) \rightarrow a b \, \equiv \, a \circ b \, : $ \begin{align} ( a b ) c \; & = \; a ( b c ) \; \; \forall a,b,c \, \in \, A \\ ( a + b ) + c \; & = \; a + ( b + c ) \; \; \forall a,b,c \, \in \, A \\ \exists \, & I \in A \, : \, a I \, = \, I a \; \; \forall a \, \in \, A \end{align} \end{itemize} \smallskip \begin{definition} \label{def:involutive algebra} \end{definition} INVOLUTIVE ALGEBRA $ ( \star-ALGEBRA ) $: a couple $( \, A \, , \, \star \, ) $ such that: \begin{itemize} \item A is an algebra \item $ \star \, : \, A \, \rightarrow \, A $ is an involution on A, i.e.: \begin{align} (a^{\star})^{\star} \; & = \; a \; \; \forall \, a \, \in \, A \\ ( a + \lambda b )^{\star} \; & = \; a^{\star} + \lambda^{\star} b^{\star} \; \; \forall \, a , b \in \, A \, \forall \lambda \in { \mathbb{C}} \\ ( a b ) ^{\star} \; & = \; b^{\star} a ^{\star} \; \; \forall \, a , b \in A \end{align} \end{itemize} \smallskip Given a $ \star - algebra $ A: \begin{definition} \label{def:unitary group of an involutive algebra} \end{definition} UNITARY GROUP OF A: \begin{equation} {\mathcal{U}}(A) \; := \; \{ u \in A \, : \, u u^{\star} = u^{\star} u = I \} \end{equation} \begin{definition} \label{def:positive part of an involutive algebra} \end{definition} POSITIVE PART OF A: \begin{equation} A_{+} \; := \; \{ a \in A \, : \, a = b b^{\star} \, b \in A \} \end{equation} \begin{definition} \label{def:self-adjoint part of an involutive algebra} \end{definition} SELF-ADJOINT PART OF A: \begin{equation} A_{sa} \; := \; \{ a \in A \, : \, a^{\star} = a \} \end{equation} \begin{definition} \label{def:set of the projections of an involutive algebra} \end{definition} SET OF THE PROJECTIONS OF A: \begin{equation} {\mathcal{P}}(A) \; := \; \{ a \in A \, \, : \, a = a^{\star} = a^{2} \} \end{equation} \smallskip Given an algebra A: \begin{definition} \label{def:norm on an algebra} \end{definition} NORM ON A: a map $ \| \cdot \| \, : \, A \rightarrow {\mathbb{R}}_{+} $ such that: \begin{align} \| a \| \; \geq 0 \; & \; \forall a \in A \\ \| a \| \; = 0 \; \Leftrightarrow & \; a = 0 \; \; \forall a \in A \\ \| \lambda a \| \; = | \lambda | \| a \| \ \; & \; \forall a \in A \, \forall \lambda \in {\mathbb{C}} \\ \| a + b \| \leq \| a \| + & \| b \| \; \; \forall a , b \in A \end{align} \begin{definition} \label{def:normed algebra} \end{definition} NORMED ALGEBRA: a couple ( A , $ \| \cdot \| \; $ ) such that: \begin{itemize} \item A is an algebra \item $ \| \cdot \| $ is a norm on A : \begin{equation} \| a b \| \; \leq \; \| a \| \, \| b \| \; \; \forall a , b \in A \end{equation} \end{itemize} \begin{definition} \label{def:Banach algebra} \end{definition} BANACH ALGEBRA: a normed algebra $ ( A \, , \, \| \cdot \| ) $ such that A is complete w.r.t. $ \| \cdot \| $ \smallskip \begin{definition} \label{def:Banach involutive algebra} \end{definition} BANACH INVOLUTIVE ALGEBRA $ ( B^{\star}-ALGEBRA ) $: a couple ( A , $\star$ ) such that: \begin{itemize} \item A is a Banach algebra \item $\star$ is an involution on A such that: \begin{equation} \| a^{\star} \| \; = \; \| a \| \; \; \forall a \in A \end{equation} \end{itemize} \smallskip \begin{definition} \label{def:C-star algebra} \end{definition} $C^{\star}-ALGEBRA$: a Banach $\star-algebra$ A such that: \begin{equation} \| a^{\star} a \| \; = \; \| a \|^{2} \; \; \forall a \in A \end{equation} \smallskip Given a $C^{\star}$-algebra A: \begin{definition} \label{def:spectrum of an element of a C-star algebra} \end{definition} SPECTRUM OF $ a \in A $: \begin{equation} Sp(a) \; := \; {\mathbb{C}} \, - \, \{ z \in {\mathbb{C}} \, : \, \exists ( a - z ) ^{- 1} \in A \} \end{equation} \begin{definition} \label{def:part with discrete spectrum of a C-star algebra} \end{definition} PART WITH DISCRETE SPECTRUM OF A: \begin{equation} A_{p.s.d} \; := \; \{ a \in A \: : \: cardinality( Sp(a)) \, \leq \, \aleph_{0} \} \end{equation} \begin{definition} \label{def:linear functional on a C-star algebra} \end{definition} LINEAR FUNCTIONAL ON A: \begin{equation} \begin{split} \varphi \, : \, A & \rightarrow {\mathbb{C}} \\ \varphi ( \lambda a + \mu b ) \; & = \; \lambda \varphi ( a ) + \mu \varphi ( b ) \; \; \forall a,b \in A \, , \, \forall \lambda , \mu \in {\mathbb{C}} \end{split} \end{equation} \begin{definition} \label{def:dual of a C-star algebra} \end{definition} DUAL OF A: \begin{equation} A^{\star} \; := \; \{ \varphi \, : \, \text{ linear functional on A} \} \end{equation} The dual $ A^{\star} $ of a $ C^{\star}$-algebra A is itself a normed space w.r.t the following norm: \begin{definition} \label{def:norm of a linear functional on a C-star algebra} \end{definition} NORM OF $ \varphi \in A^{\star} $ \begin{equation} \| \varphi \| \; := \; \varphi ( I ) \end{equation} \begin{definition} \label{def:positive linear functionals on a C-star algebra} \end{definition} POSITIVE LINEAR FUNCTIONALS ON a: \begin{equation} A^{\star}_{+} \; := \; \{ \varphi ( a^{\star} \, a ) \; \geq \; 0 \; \; \forall a \in A \} \end{equation} \begin{definition} \label{def:states on a C-star algebra} \end{definition} STATES ON A: \begin{equation} S(A) \; = \; \{ \omega \in A^{\star}_{+} \, : \, \| \omega \| = 1 \, \} \end{equation} \begin{definition} \label{def:mixed state on a C-star algebra} \end{definition} THE STATE $\omega \in S(A)$ IS MIXED: \begin{multline} \exists \, \lambda \in (0,1) \, , \, \exists \omega_{1} , \omega_{2} \in S(A) : \\ \omega_{1} \neq \omega_{2} \; e \; \omega = \lambda \omega_{1} + ( 1 - \lambda ) \omega_{2} \end{multline} \begin{definition} \label{def:pure states on a C-star algebra} \end{definition} PURE STATES OF A: \begin{equation} \Xi (A) \; \equiv \; \{ \omega \, \in \, S(A) \, : \, \omega \, \text{ is not mixed } \} \end{equation} A useful property we shall use in the sequel is the following: \begin{theorem} \label{th:Cauchy-Schwarz inequality} \end{theorem} CAUCHY-SCHWARZ INEQUALITY: \begin{equation} | \omega( b^{\star} a ) | ^{2} \; \leq \; \omega( b^{\star} b ) \, \omega( a^{\star} a ) \; \; \forall a,b \in A \, , \, \forall \omega \in S(A) \end{equation} \smallskip Given a $ C^{\star}$-algebra A let us consider the set of all the linear functionals over $ A^{\star}$, namely the dual of the dual $ A^{\star \star} $. Since an element of A $ a \in A $ may be identified with the following linear function over $ A^{\star}$: \begin{equation} a ( \varphi ) \; := \; \varphi ( a ) \; \; \varphi \in A^{\star} \end{equation} it follows that: \begin{equation} A \; \subseteq \; A^{\star \star} \end{equation} \begin{definition} \end{definition} $ W^{\star}$-TOPOLOGY ON $ A^{\star} $: the coarsest topology on $ A^{\star} $ w.r.t. which all the elements of A (seen as linear functionals over $ A^{\star} $) are continuous \smallskip Given two $C^{\star}-algebras$ A and B : \begin{definition} \label{def:involutive morphism between C-star algebras} \end{definition} INVOLUTIVE MORPHISM ($\star$-MORPHISM) FROM A TO B: a map $ \tau : A \rightarrow B $ such that: \begin{align} \tau ( \lambda a + \mu b ) \; = & \; \lambda \tau ( a ) + \mu \tau ( b ) \; \; \forall a , b \in A \, , \, \forall \lambda , \mu \in {\mathbb{C}} \\ \tau ( a b ) \; = & \; \tau ( a ) \tau ( b ) \; \; \forall a , b \in A \\ \tau ( a^{\star} ) \; = & \; \tau ( a )^{\star} \; \; \forall a \in A \end{align} \begin{definition} \label{def:involutive isomorphism between C-star algebras} \end{definition} INVOLUTIVE ISOMORPHISM ($\star$-ISOMORPHISM) FROM A TO B: an involutive morphism $ \tau : A \rightarrow B $ that is bijective \smallskip Given a $C^{\star}$-algebra A and an Hilbert space ${\mathcal{H}} $: \begin{definition} \label{def:representation of a C-star algebra} \end{definition} REPRESENTATION OF A ON ${\mathcal{H}}$: an involutive morphism $ \pi \, : \, A \rightarrow B ( {\mathcal{H}} ) $ from A to $B ({\mathcal{H}})$ \smallskip Given a representation $ \pi $ of A on the Hilbert space ${\mathcal{H}}$: \begin{definition} \label{reducibility of a representation} \end{definition} $ \pi $ IS REDUCIBLE: \begin{equation} \exists \; V \subset {\mathcal{H}} \text{ linear subspace} \; : \; \pi ( V ) \: \subseteq \: V \end{equation} Given two representations $ \pi_{1}$ and $ \pi_{2}$ of A on the Hilbert spaces, respectively, ${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$: \begin{definition} \label{def:equivalent representations of a C-star algebra} \end{definition} $ \pi_{1}$ AND $ \pi_{2}$ ARE EQUIVALENT: \begin{multline} \exists \, U \, : \, {\mathcal{H}}_{1} \, \rightarrow {\mathcal{H}}_{2} \, isomorphism \, : \\ \pi_{2} ( a ) \; = \; U \pi_{1} ( a ) U^{-1} \; \; \forall a \in A \end{multline} \smallskip Any state $ \omega \in S(A) $ over a $C^{\star}$-algebra A gives rise to a particularly important representation of A we are going to introduce. Defined the following subset of A: \begin{equation} {\mathcal{N}} \; := \; \{ a \in A \, : \, \omega ( a^{\star} a ) = 0 \} \end{equation} let us define on the quotient space $ \frac{A}{ {\mathcal{N}}} $ the inner product: \begin{equation} \label{eq:GNS inner product} < a | b > \; := \; \omega (a^{\star} b ) \; \; [ a ] , [ b ] \in \frac{A}{ {\mathcal{N}}} \end{equation} Let us observe, furthermore, that the canonical embedding $ i \, : \, A \, \mapsto \, \frac{A}{ {\mathcal{N}}} $: \begin{equation} i ( a ) \; := \; [ a ] \; = \; \{ b \in A : b = a + c \, , \, c \in {\mathcal{N}} \} \end{equation} is a continuous application from A (endowed with the norm-topology) to $ \frac{A}{ {\mathcal{N}}} $ (endowed with the norm topology induced by the inner-product of eq.\ref{eq:GNS inner product}). We can then introduce the following: \begin{definition} \label{def:GNS representation} \end{definition} GELFAND-NAIMARK-SEGAL REPRESENTATION (GNS REPRESENTATION) OF A W.R.T. $ \omega $ the representation $ \pi_{\omega} $ of A on the Hilbert space $ {\mathcal{H}}_{\omega} $: \begin{itemize} \item $ {\mathcal{H}}_{\omega} $ is the completion of the inner product space $ ( \, \frac{A}{ {\mathcal{N}}} \, , \, < a | b > ) $ \item $ \pi_{\omega} $ is the continuous extension of the application: \begin{equation} [ \pi_{\omega} ( a ) ] | [ b ] > \; := \; a \, b \; \; [ b ] \in \frac{A}{ {\mathcal{N}}} \end{equation} \end{itemize} One has that: \begin{theorem} \label{th:basic properties of the GNS representation} \end{theorem} BASIC PROPERTIES OF THE GNS REPRESENTATION: \begin{enumerate} \item the vector: \begin{equation} | \Omega_{\omega} > \; := \; | [ I ] > \end{equation} is cyclic, i.e. $ \pi_{\omega} ( A) | \Omega_{\omega} > $ is dense in $ {\mathcal{H}}_{\omega} $ \item any representation $ \pi $ of A admitting a ciclic vector $ | \Phi > $ is equivalent to the GNS representation $ \pi_{\varphi} $ w.r.t. the state: \begin{equation} \varphi ( a ) \; := \; < \Phi | \pi ( a ) | \Phi > \; \; a \in A \end{equation} \item \begin{equation} \pi_{\omega} \text{ is irreducible } \; \Leftrightarrow \; \omega \in \Xi (A) \end{equation} \end{enumerate} \smallskip Let us now present the following fundamental: \begin{theorem} \label{th:Gelfand isomorphism at C-star algebraic level} \end{theorem} GELFAND'S ISOMORPHISM AT $ C^{\star}$-ALGEBRAIC LEVEL: \begin{hypothesis} \end{hypothesis} A abelian $ C^{\star}$-algebra $ C( X ( A )) \; C^{\star}$-algebra of the complex-valued continuous (w.r.t. the $ W^{\star}$-topology) on $ X ( A ) $ \begin{thesis} \end{thesis} A and $ C( X ( A )) $ are $ \star $-isomorphic \medskip Indeed also the converse property holds, and theorem\ref{th:Gelfand isomorphism at C-star algebraic level} may be considerably streghtened, resulting in the following: \begin{theorem} \label{th:category isomorphism at the basis of Noncommutative Topology} \end{theorem} CATEGORY EQUIVALENCE AT THE BASIS OF NONCOMMUTATIVE TOPOLOGY The \textbf{category} having as \textbf{objects} the \textbf{ Hausdorff compact topological spaces} and as \textbf{morphisms} the \textbf{continuous maps} on such spaces is equivalent to the category having as \textbf{objects} the \textbf{abelian $C^{\star}$-algebras} and as \textbf{morphisms} the \textbf{involutive morphisms} of such spaces. \bigskip \begin{remark} \label{rem:the metaphore by which we can speak about noncommutative sets from within ZFC} \end{remark} THE METAPHORE BY WHICH WE CAN SPEAK ABOUT NONCOMMUTATIVE SETS FROM WITHIN ZFC: Theorem\ref{th:category isomorphism at the basis of Noncommutative Topology} says, in particular, that an abelian $ C^{\star} $-algebra may be always seen as an algebra of function over a suitable topological space X. This suggest to introduce a metaphore, that we call the \textbf{noncommutative metaphore} from here and beyond, according to which one looks at a noncommutative algebra A as if it was an algebras of functions over an hypothetic \textbf{noncommutative set}: \begin{equation} \label{eq:the impossible equation} A \; =_{METAPHORE} \; C ( X_{NC} ) \end{equation} Of course this is only a metaphore, but it is the only way we can speak about \textbf{noncommutative sets} from within the formal system of \textbf{commutative set theory} (namely the formal system ZFC of Zermelo Fraenkel endowed with the Axiom of Choice) giving foundations to Mathematics. Since there is no possibility inside ZFC of formalizing eq.\ref{eq:the impossible equation} and so to speak directly of $ X_{NC} $ , we will never mention it and we will directly refer to A as a \textbf{noncommutative set}. It should be even supreflous to remark that, of course, from the same fact we can speak about it inside ZFC, A is also a \textbf{commutative set}. So it is fundamental, when we speak about A, to specify if we are looking at it as an ordinary \textbf{commutative set} or as a \textbf{noncommutative set}, i.e. as a way of speaking about the non formalizable-in-ZFC object $ X_{NC} $. Such a double nature, of course, reflects itself at different levels: \begin{itemize} \item at a logical level if we look at A as a \textbf{noncommutative set}, one can formalize its propostion calculus through the quantum logic QL(A). If instead one look at A as a \textbf{commutative set}, one can, of course, apply to it the ordinary classical (i.e. distributive) set-theoretical predicative calculus \item we will soon introduce the notion of \textbf{noncommutative cardinality} of a noncommutative set; according to such a notion we will arrive to characterize the noncommutative binary alphabet $ \Sigma_{NC} \, = \, M_{2} ( {\mathbb{C}}) $ by the condition: \begin{equation} cardinality_{NC} ( \Sigma_{NC} ) \; = \; 1 \end{equation} Anyway, obviously, looking at $ M_{2} ( {\mathbb{C}}) $ simply as a commutative set, we can consider its commutative cardinality: \begin{equation} cardinality ( M_{2} ( {\mathbb{C}}) ) \; = \; \aleph_{1}^{4} \; = \; \aleph_{1} \end{equation} \end{itemize} \medskip Theorem\ref{th:category isomorphism at the basis of Noncommutative Topology} introduces a \textbf{topological structure} over \textbf{noncommutative sets}. It is just the first of a collection of Category Equivalence Theorems that allow to introduce an high hierarchy of more and more refined structures on \textbf{noncommutative sets}, resulting in the wonderful conceptual tower, namely Noncommutative Geometry, built by that genius named Alain Connes \cite{Jaffe-91}. \smallskip \begin{definition} \label{def:W-star algebra} \end{definition} $W^{\star}$-ALGEBRA (or ALGEBRAIC SPACE): a $ C^{\star}-ALGEBRA $ A such that $ ( A^{\star} \, , \, \| \cdot \| ) $ is a Banach space \smallskip \begin{definition} \label{def:commutative space} \end{definition} COMMUTATIVE SPACE: a commutative algebraic space \smallskip \begin{definition} \label{def:noncommutative space} \end{definition} NONCOMMUTATIVE SPACE: a noncommutative algebraic space \smallskip \begin{example} \label{ex:the W-star algebra of the bounded operators on an Hilbert space} \end{example} THE $ W^{\star}$-ALGEBRA OF THE BOUNDED OPERATORS ON AN HILBERT SPACE Given an arbitrary Hilbert space $ {\mathcal{H}} $ the space $ B({\mathcal{H}}) $ of all the bounded linear operators on $ {\mathcal{H}} $ is a $ W^{\star}$-algebra \medskip \begin{definition} \label{def:automorphisms of a W-star algebra} \end{definition} AUTOMORPHISMS OF A: \begin{equation} AUT(A) \; := \; \{ \tau : A \rightarrow A \text{ involutive isomorphism of A } \} \end{equation} \begin{definition} \label{def:inner automorphisms of a W-star algebra} \end{definition} INNER AUTOMORPHISMS OF A: \begin{equation} INN(A) \; := \; \{ \tau \in AUT(A) \, : \exists u \in {\mathcal{U}}(A) \, , \, \tau (a) = u a u^{\star} \, \, \forall a \in A \} \end{equation} \begin{definition} \label{def:outer automorphisms of a W-star algebra} \end{definition} OUTER AUTOMORPHISMS OF A: \begin{equation} OUT(A) \; := \; \frac{AUT(A)}{INN(A)} \end{equation} Given two algebraic spaces A and B: \begin{definition} \label{def:positive maps among two W-star algebras} \end{definition} POSITIVE MAPS FROM A TO B: \begin{equation} P ( A , B ) \; := \; \{ \, \tau \, : \, A \rightarrow B \, linear \, : \, \tau ( A_{+} ) \; \subseteq \; B_{+} \, \} \end{equation} \begin{definition} \label{def:completely positive maps among two W-star algebras} \end{definition} COMPLETELY POSITIVE MAPS FROM A TO B: \begin{equation} CP ( A, B ) \; \equiv \; \{ \, \tau \in P (A,B) \, : \, \tau \otimes {\mathbb{I}}_{n} \in P ( A,B ) \; \; \forall n \in {\mathbb{N}} \} \end{equation} \begin{definition} \label{def:CPU-maps among two W-star algebras} \end{definition} COMPLETELY POSITIVE UNITAL MAPS (CPU-MAPS OR CHANNELS) FROM A TO B: \begin{equation} CPU ( A,B ) \; \equiv \; \{ \, \tau \in CP (A,B) \, : \, \tau ( {\mathbb{I}} ) \, = \, {\mathbb{I}} \} \end{equation} In particular: \begin{definition} \label{def:positive maps on a W-star algebra} \end{definition} POSITIVE MAPS ON A: \begin{equation} P ( A ) \; := \; P( A , A) \end{equation} \begin{definition} \label{def:completely positive maps on a W-star algebra} \end{definition} COMPLETELY POSITIVE MAPS ON A: \begin{equation} CP ( A ) \; \equiv \; CP( A , A) \end{equation} \begin{definition} \label{def:CPU-maps on a W-star algebra} \end{definition} COMPLETELY POSITIVE UNITAL MAPS (CPU-MAPS OR CHANNELS) ON A: \begin{equation} CPU ( A ) \; := \; CPU( A , A) \end{equation} One has the following: \begin{theorem} \label{th:on the relationship between positivity and complete-positivity} \end{theorem} ON THE RELATIONSHIP BETWEEN POSITIVITY AND COMPLETE-POSITIVITY: \begin{enumerate} \item \begin{equation} A \; commutative \; \Rightarrow \; CP(A) \, = \, P(A) \end{equation} \item \begin{equation} A \; noncommutative \; \Rightarrow \; CP(A) \, \subset \, P(A) \end{equation} \end{enumerate} The analysis of channels is particularly simplified by the following: \begin{theorem} \label{th:Kraus-Stinespring's theorem} \end{theorem} KRAUS-STINESPRING'S THEOREM: \begin{hypothesis} \end{hypothesis} \begin{center} $ A \subseteq B( {\mathcal{H}}) $ Von Neumann algebra acting on the Hilbert space $ {\mathcal{H}} $ $ \alpha \, : \, A \rightarrow A $ normal linear map \end{center} \begin{thesis} \end{thesis} \begin{multline} \alpha \in CPU(A) \; \; \Leftrightarrow \; \; \exists \, \{ V_{i} \}_{ i \in I} \in {\mathcal{B}}( {\mathcal{H}}) \} \; : \\ \alpha (a ) \; = \; \sum_{ i \in I} V_{i}^{\star} a V_{i} \; \forall a \in A \\ \sum_{ i \in I} V_{i}^{\star} V_{i} \; = \; {\mathbb{I}} \end{multline} where the convergence is in the weak topology. \smallskip \begin{remark} \label{rem:identificability of involutively-isomorphic W-star algebras} \end{remark} IDENTIFICABILITY OF $\star$-ISOMORPHIC $W^{\star}$-ALGEBRAS: Since, from an algebraic viewpoint, $\star$-isomorphic $ W^{\star}$-algebras are identical, they can be identified. So, for example, the $W^{\star}$-algebra $ B ({\mathcal{H}}_{n}) $ of all the bounded linear operators on an n-dimensional Hilbert space $ {\mathcal{H}}_{n} $, with $ n \, \in \, {\mathbb{N}} $ may be identificated with the $W^{\star}$-algebra $ M_{n} ({\mathbb{C}}) $ of all the $ n \, \times \, n $ matrices with complex entries. So, in particular, the algebra $ B({\mathcal{H}}_{2}^{ \bigotimes n}) $ of all the bounded linear operators on the n-qubits Hilbert space $ {\mathcal{H}}_{2}^{ \bigotimes n} $ may be identified with the $W^{\star}$-algebra $ M_{2^{n}} ({\mathbb{C}}) $ \smallskip Given a $ C^{\star}-algebra $ we will, obviously, say that: \begin{definition} \label{def:abelian C-star algebra} \end{definition} A IS ABELIAN: \begin{equation} [ a , b ] \; := \; a b \, - \, b a \; = \; 0 \; \; \forall a,b \in A \end{equation} Given an abelian $ C^{\star}-algebra $ A: \begin{definition} \label{def:characters of an abelian C-star algebra} \end{definition} \begin{equation} X(A) \; := \; \{ \pi \, : \, \text{ representation of A on } {\mathbb{C}} \} \end{equation} \smallskip Given a $ C^{\star}-algebra $ A and a linear subspace $ B \subseteq A $: \begin{definition} \label{sub-C-star algebra of a C-star algebra} \end{definition} B IS A SUB-$C^{\star}$-ALGEBRA OF A: B is a $ C^{\star}-algebra $ w.r.t. to the restriction to B of the $ C^{\star}-algebraic $ structure of A \smallskip Given an Hilbert space $ \mathcal{H} $ ed and a $ sub- C^{\star}-algebra \; A \, \subseteq \, B({\mathcal{H}}) $ of $ B({\mathcal{H}}) $: \begin{definition} \label{def:commutant} \end{definition} COMMUTANT OF A: \begin{equation} A' \; = \; \{ a \in A \, : \, [ a , b ] \, = \, 0 \; \; \forall b \in B({\mathcal{H}}) \} \end{equation} \begin{definition} \label{def:centre} \end{definition} CENTRE OF A: \begin{equation} {\mathcal{Z}}(A) \; := \; A \cap A' \end{equation} \begin{definition} \label{def:Von Neumann algebra} \end{definition} A IS A VON NEUMANN ALGEBRA: \begin{equation} A'' \; := \; (A')' \; = \; A \end{equation} \smallskip The two notions of a $ W^{\star}$-algebra and of a Von Neumann algebra introduced, respectively, in definition\ref{def:W-star algebra} and definition\ref{def:Von Neumann algebra} would seem to have anything in common: the first is a purely abstract algebraic notion while the latter is a concrete notion concerning operators on an Hilbert space. So it may appear rather shocking that these notions are indeed equivalent (remember remark\ref{rem:identificability of involutively-isomorphic W-star algebras}) , as is stated by the following: \begin{theorem} \label{th:Sakai theorem} \end{theorem} SAKAI'S THEOREM: \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; C^{\star}-algebra \end{equation*} \begin{thesis} \end{thesis} \begin{center} A is $ \star$-isomorphic to a Von Neumann algebra $ \Leftrightarrow $ A is a $ W^{\star}-algebra $ \end{center} Up to now all these operator-algebraic stuff would seem no to have anything in common with Quantum Logic. Anyway it may be proved that: \begin{theorem} \label{th:the quantum logic of a Von Neumann algebra} \end{theorem} ON THE QUANTUM LOGIC OF A VON NEUMANN ALGEBRA: \begin{hypothesis} \end{hypothesis} \begin{align*} {\mathcal{H}} \, & \, \text{ Hilbert space } A \, & \, \subseteq \, B({\mathcal{H}}) \text{ Von Neumann algebra} \\ \end{align*} \begin{thesis} \end{thesis} \begin{align*} QL (A) \; := \; ( {\mathcal{P}}(A) \, , \, \preceq \, , \, \bot ) \text{ is a quantum logic } \\ {\mathcal{P}}(A)'' & \; = \;A \end{align*} where $ \preceq $ and $ \bot $ are, respectively, the partial-ordering relation and the orthocomplementation inherited from $ {\mathcal{L}}( {\mathcal{H}}) $. Theorem\ref{th:the quantum logic of a Von Neumann algebra} tells us that, substantially, a Von Neumann algebra is generated by the quantum logic it gives rise to. Indeed, as we will now show, the underlying quantum logics substantially govern the classification of Von Neumann algebras. Given a Von Neumann algebra $ A \subseteq B ({\mathcal{H}}) $: \begin{definition} \label{def:factor} \end{definition} A IS A FACTOR: \begin{equation} {\mathcal{Z}}(A) \; = \; \{ \lambda I \, , \, \lambda \in {\mathbb{C}} \} \end{equation} Factors are, substantially, the building blocks of Von Neumann algebras: any Von Neumann algebra A may, actually, be expressed as a direct integral of factors: \begin{multline} \label{eq:factor decomposition of a Von Neumann algebra} A \; = \; \int_{{\mathcal{Z}}(A)}^{\otimes} A_{\lambda} \, d \nu ( \lambda ) \\ {\mathcal{Z}}(A_{\lambda}) \; = \; \{ {\mathbb{C}} \, {\mathbb{I}} \} \; \; \forall \lambda \in {\mathcal{Z}}(A) \end{multline} Hence the analysis of a Von Neumann algebra may be reduced to the analysis of its building blocks. Given, now, a generic Von Neumann algebra $ A \subseteq B(\mathbb{H}) $ and two its projections $ a , b \in {\mathcal{P}}(A) $: \begin{definition} \label{def:equivalence of projections in a Von Neumann algebra} \end{definition} a AND b ARE EQUIVALENT IN A: \begin{multline} a \, \sim_{A} \, b \; := \; \exists o \in A \, : \, ( o | \psi > = 0 \, \forall | \psi > \in Range(a)^{\bot} ) \\ and \; ( \| o | \psi > \| = \| | \psi > \| \, \, \forall | \psi > \in Range(a) \} \end{multline} \begin{remark} \label{rem:intuitive meaning of the equivalence of projections in a Von Neumann algebra} \end{remark} INTUITIVE MEANING OF $ \sim_{A} $: EQUALITY OF THE DIMENSION RELATIVE TO A Definition\ref{def:equivalence of projections in a Von Neumann algebra} may appear rather counter-intuitive. Its meaning is that the existence of a partial (since it acts so only on Range(a) being identically null on its complement) isometry between Range(a) and Range(b) that belongs to A may be interpreted, informally speaking, as the fact that the \textbf{dimension relative to A} of the subspace a projects to is equal to the \textbf{dimension relative to A} of the subspace b projects to \smallskip The equivalence relation $ \sim_{A} $ over $ {\mathcal{P}}(A) $ may be used to introduce a new partial ordering on projections (different from that of the quantum logic of A) \begin{equation}\label{eq:total ordering of projections in a Von Neumann algebra} a \, \trianglelefteq \, b \; := \; \exists c \in {\mathcal{P}}(A) \: : \: a \, \sim_{A} \, c \, \leq \, b \end{equation} \begin{remark} \label{rem:intuitive meaning of the total ordering of projections in a Von Neumann algebra} \end{remark} INTUITIVE MEANING OF $ \trianglelefteq $: ORDERING ACCORDING TO THE RELATIVE DIMENSION The intuitive meaning of the partial ordering $ \trianglelefteq $ is induced by that of the equivalence relation $ \sim_{A} $. So the condition $ a \, \trianglelefteq \, b $ means,intuitively, that the \textbf{dimension relative to A} of a is less or equal to the \textbf{dimension relative to A} of b. But here comes the following: \begin{theorem} \label{th:total ordering of a factor's projections} \end{theorem} TOTAL ORDERING W.R.T. $ \trianglelefteq $ OF A FACTOR'S EQUIVALENCE CLASSES OF PROJECTIONS \begin{equation} {\mathcal{Z}}(A) = {\mathbb{C}} I \; \Rightarrow \; ( a \, \trianglelefteq \, b \; or \; b \, \trianglelefteq \, a \; \; \forall a , b \in \frac{{\mathcal{P}}(A)}{\sim_{A}} ) \end{equation} whose importance is owed to an immediate conseguence of its: \begin{corollary} \label{cor:order type is an invariant for factors} \end{corollary} ORDER TYPE OF $ \frac{{\mathcal{P}}(A)}{\sim_{A}} $ IS AN INVARIANT FOR FACTORS \begin{equation} A, B \, \star-isomorphic \, factors \; \Rightarrow \; Type-order( \frac{{\mathcal{P}}(A)}{\sim_{A}} ) ) \, = \, Type-order( \frac{{\mathcal{P}}(B)}{\sim_{B}} ) ) \end{equation} \smallskip To determine the order type of a $ \frac{{\mathcal{P}}(A)}{\sim_{A}} $ of a factor A a key concept is the finiteness of projections: \begin{definition} \label{def:finite projection on a lattice} \end{definition} $ a \in {\mathcal{P}}(A) $ IS FINITE: \begin{equation} a \, \sim_{A} \, b \, \trianglelefteq \, a \; \Rightarrow \: a \, = \, b \; \; \forall b \in {\mathcal{P}}(A) \end{equation} We can, at last, formalize the intuitive statements concerning the \textbf{relative dimension} of remark\ref{rem:intuitive meaning of the equivalence of projections in a Von Neumann algebra} and remark\ref{rem:intuitive meaning of the total ordering of projections in a Von Neumann algebra}introducing the following fundamental notion: \begin{definition} \label{def:relative dimension w.r.t. a factor} \end{definition} RELATIVE DIMENSION W.R.T. A: a map $ d \, : \, {\mathcal{P}}(A) \, \mapsto \, [ 0 , + \infty ] $ such that: \begin{align} d(a) \, & \, = \, 0 \; \Leftrightarrow \; a \, = \, 0 \; \; \forall a \in {\mathcal{P}}(A) \\ a \, & \bot \, b \; \Rightarrow d ( a + b ) \, = \, d(a) + d(b) \; \; \forall a \in {\mathcal{P}}(A) \\ d(a) \, & \, < d(b) \; \Leftrightarrow \; a \, \trianglelefteq \, b \; \; \forall a,b \in {\mathcal{P}}(A) \\ d(a) \, & \, + \infty \; \Leftrightarrow \; a \text{ is finite } \; \; \forall a \in {\mathcal{P}}(A) \\ d(a) \, & \, = \, d(b) \; \Leftrightarrow \; a \, \sim_{A} \, b \; \; \forall a,b \in {\mathcal{P}}(A) \\ d(a) \, & \, + \, d(b) \; = \; d ( a \bigwedge b ) \, + \, d ( a \bigvee b ) \; \; \forall a,b \in {\mathcal{P}}(A) \end{align} We will denote, from here and beyond, the relative dimension w.r.t. a factor A by $ d_{A} $. \smallskip The astonishing fact is that: \begin{theorem} \label{th:unicity of the relative dimension w.r.t. a factor} \end{theorem} UNICITY OF THE RELATIVE DIMENSION W.R.T. A FACTOR: \begin{equation} d_{1} \, , \, d_{2} \text{ relative dimensions w.r.t. A } \; \Rightarrow \; \exists c \, \in \, {\mathbb{R}}_{+} \, : \, ( d_{1}(a) \, = \, c \, d_{2}(a) \; \forall a \in {\mathcal{P}}(A) ) \end{equation} The importance of theorem\ref{th:unicity of the relative dimension w.r.t. a factor} lies in that it implies that the order type of $ \frac{{\mathcal{P}}(A)}{\sim_{A}} $ can be read off the order type of $ d_{A} $'s range. Murray and Von Neumann determined the possible ranges of $ d_{A} $ (suitably normalized) resulting in the following classification: \begin{definition} \label{def:discrete finite factor} \end{definition} A IS OF TYPE FINITE, DISCRETE $( Type (A) \, = \, I_{n} )$ \begin{equation} Range( d_{A} ) \; = \; \{ 0 , 1 , \ldots , n \} \; \; n \in {\mathbb{N}}_{+} \end{equation} \begin{definition} \label{def:discrete infinite factor} \end{definition} A IS OF TYPE INFINITE, DISCRETE ($ Type (A) \, = \, I_{\infty} $): \begin{equation} Range( d_{A} ) \; = \; \{ 0 , 1 , \ldots , + \infty \} \end{equation} \begin{definition} \label{def:continuous finite factor} \end{definition} A IS OF TYPE FINITE, CONTINUOUS ($ Type (A) \, = \, II_{1} $): \begin{equation} Range( d_{A} ) \; = \; [ 0 , 1 ] \end{equation} \begin{definition} \label{def:continuous infinite factor} \end{definition} A IS OF TYPE INFINITE, CONTINUOUS ($ Type (A) \, = \, II_{\infty} $): \begin{equation} Range( d_{A} ) \; = \; [ 0 , + \infty ] \end{equation} \begin{definition} \label{def:purely infinite factor} \end{definition} A IS OF TYPE PURELY INFINITE ($ Type (A) \, = \, III $) \begin{equation} Range( d_{A} ) \; = \; \{ 0 , + \infty \} \end{equation} \smallskip \begin{remark} \end{remark} THE ORDER TYPE OF $ \frac{{\mathcal{P}}(A)}{\sim_{A}} $ DOESN'T ALLOW A COMPLETE CLASSIFICATION OF FACTORS It is important, at this point, to remark that the the above classification of factors based on the order type of the algebraic dimension function's range is not complete. The complete classification, furnished almost fourty years later by the great Alain Connes, will be briefly introduced in section\ref{sec:On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics} \smallskip It is, now, conceptually important to observe that the relative dimension with respect to factors, underlying their Murray-Von Neumann classification, is a Quantum-Logic's notion: \begin{theorem} \label{th:relative dimension w.r.t. a factor is a lattice dimension function} \end{theorem} RELATIVE DIMENSION W.R.T. A FACTOR IS A MATTER OF QUANTUM LOGIC $ d_{A} $ is a lattice dimension function over the weak quantum logic QL(A) \begin{corollary} \label{cor:the quantum logic of a finite factor is strong} \end{corollary} THE QUANTUM LOGIC OF A FINITE FACTOR IS STRONG: \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; factor \; : \; Type(A) \, \in \, \{ I_{n} \}_{n \in {\mathbb{N}}} \, \bigcup \{ II_{1} \} \end{equation*} \begin{thesis} \end{thesis} \begin{center} QL(A) is a strong quantum logic \end{center} \begin{proof} Since $ d_{A} $ assumes only finite values this happens, obviously, also to its restriction to $ {\mathcal{P}}(A) $ that, by theorem\ref{th:relative dimension w.r.t. a factor is a lattice dimension function}, is an algebraic dimension function over QL(A). By theorem\ref{th:on the finite dimension} the thesis immediately follows \end{proof} We have, at last, almost all the ingredients required to understand the confession of Von Neumann to Birkhoff. The last ingredients required are those deriving from the following: \begin{theorem} \label{th:characterization of discrete factors} \end{theorem} CHARACTERIZATION OF DISCRETE FACTORS: \begin{equation} Type(A) \: \in \: \{ I_{n} \}_{n \in \{0 , 1 , \cdots , \infty \}} \; \Leftrightarrow \; \exists {\mathcal{H}} \text{ Hilbert space } : \, A \, = B ( \mathcal{H}) \end{equation} \begin{proof} A is $ \star$-isomorphic to a $ B ( {\mathcal{H}}) $, for a proper Hilbert space $ {\mathcal{H}} $, iff the quantum logic of A is an Hilbert Lattice. But a logic dimension function of an Hilbert lattice may be easily defined as the dimensionality, as linear subspaces, of its elements and, conseguentially, can assume only integer values. By theorem\ref{th:unicity of the relative dimension w.r.t. a factor}and theorem\ref{th:relative dimension w.r.t. a factor is a lattice dimension function} it immediately follows the thesis \end{proof} \begin{corollary} \label{cor:discreteness of a factor is equivalent to the atomicity of its quantum logic} \end{corollary} ATOMICITY OF A FACTOR'S QUANTUM LOGIC \begin{equation} QL(A) \text{ is atomic } \; \Leftrightarrow \; Type(A) \: \in \: \{ I_{n} \}_{n \in \{0 , 1 , \cdots , \infty \}} \end{equation} \begin{proof} By theorem\ref{th:characterization of discrete factors} we have that QL(A) is an Hilbert lattice iff A is discrete. The thesis immediately follows \end{proof} \smallskip Given a $ W^{\star}$-algebra A: \begin{definition} \label{def:trace on a W-star algebra} \end{definition} TRACE ON A: A linear map $ \tau \, : \, A_{+} \, \mapsto \, [ 0 , + \infty ] $ such that: \begin{equation} \tau \: \circ \: \alpha \; = \; \tau \; \; \forall \alpha \in INN(A) \end{equation} Given a trace $ \tau $ on A: \begin{definition} \label{def:finite trace on a W-star algebra} \end{definition} $ \tau $ IS FINITE: \begin{equation} \infty \; \notin \; Range( \tau ) \end{equation} A very important property of finite factors is stated by the following: \begin{theorem} \label{th:extensibility of the relative dimension w.r.t. a finite factor to a finite trace} \end{theorem} EXTENSIBILITY OF THE RELATIVE DIMENSION W.R.T. A FINITE FACTOR TO A FINITE TRACE \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; factor \; : \; Type(A) \: \in \: \{ I_{n} \}_{n \in {\mathbb{N}}} \, \bigcup \{ II_{1} \} \end{equation*} \begin{thesis} \end{thesis} \begin{equation} \exists ! \tau_{A} \text{ finite trace on A } \; : \; \tau_{A} | _{{\mathcal{P}}(A)} \, = \, d_{A} \end{equation} \begin{theorem} \label{th:extensibility of the finite state to a state} \end{theorem} \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; factor \; : \; Type(A) \: \in \: \{ I_{n} \}_{n \in {\mathbb{N}}} \, \bigcup \{ II_{1} \} \end{equation*} \begin{thesis} \end{thesis} \begin{equation} \exists ! \omega_{A} \in S(A) \; : \; \omega_{A}|_{A_{+}} \, = \, \tau_{A} \end{equation} \smallskip \begin{example} \label{ex:the finite trace on the n-qubits' W-star algebra} \end{example} THE FINITE TRACE ON THE n-QUBITS' $ W^{\star} $-ALGEBRA We know that: \begin{equation} B( \mathcal{H}_{2}^{ \bigotimes n}) \; = \; \bigotimes_{k=1}^{n} M_{2} ( {\mathbb{C}} ) \; = \; M_{2^{n}} ( {\mathbb{C}} ) \end{equation} The finite trace on the $ n \times n $ matrix algebra $ M_{n} ( {\mathbb{C}} ) $ is simply the normalized matricial trace: \begin{equation} \tau_{n} \; := \; \frac{1}{n} Tr_{n} \; = \; \bigotimes_{k=1}^{n} \tau_{2} \end{equation} so that the finite trace on $ B( \mathcal{H}_{2}^{ \bigotimes n}) $ is $ \tau_{2^{n}} $. \smallskip Let us now consider an arbitrary $ W^{\star}$-algebra A. We want to characterize the condition stating that A is $ C_{\Phi}-NC_{M}$-computable, i.e. is approximable with arbitrary precision by finite-dimensional matrix algebras. Mathematically formalized such a constraint results in the following: \begin{definition} \label{def:hyperfinite W-algebra} \end{definition} A IS HYPERFINITE: there exists an increasing sequence $ \{ A_{n} \}_{n \in {\mathbb{N}}} $ of $ W^{\star} $-algebras of A such that: \begin{equation} A \; = \; ( \bigcup_{n \in {\mathbb{N}}} A_{n} )'' \end{equation} \begin{example} \label{ex:the hyperfinite finite continuous factor} \end{example} THE HYPERFINITE $ II_{1} $ FACTOR R Given the one dimensional lattice $ {\mathbb{Z}} $ let us attach to the $ n^{th} $ lattice site the 1-qubit $ W^{\star}$-algebra: \begin{equation} A_{n} \; := \: M_{2} ({\mathbb{C}}) \; \; n \in {\mathbb{Z}} \end{equation} Given an arbitary set of sites $ \Lambda \, \subseteq \, {\mathbb{Z}} $ let us define: \begin{equation} A_{\Lambda} \; := \; \bigotimes_{n \in \Lambda} A_{n} \end{equation} Clearly we have that: \begin{equation} \label{eq:increasing nature of the local algebras} \Lambda_{1} \, \subseteq \, \Lambda_{2} \; \Rightarrow \; A_{\Lambda_{1}} \, \subseteq \, A_{\Lambda_{2}} \end{equation} \begin{equation}\label{eq:commutativity of the local algebras of disjoint regions} \Lambda_{1} \, \bigcap \, \Lambda_{2} \: = \: \emptyset \; \Rightarrow \; [ A_{\Lambda_{1}} , A_{\Lambda_{2}} ] \, = \, 0 \end{equation} Let us now consider the state $ \tau_{\Lambda} \, \in \, S ( A_{\Lambda} )$ defined as: \begin{equation} \tau_{\Lambda} \; := \; \bigotimes_{n \in \Lambda} \tau_{2} \end{equation} Clearly we have, in particular, that the state: \begin{equation} \tau_{ \{ 0 , 1 , \cdots , n \}} \; = \; \tau_{2^{n}} \end{equation} is nothing but the finite tracial state on the n-qubit $ W^{\star}$-algebra $ B( \mathcal{H}_{2}^{ \bigotimes n}) $ we saw in the example\ref{ex:the finite trace on the n-qubits' W-star algebra} We are at last ready to introduce one of the main actors of this dissertation, namely the $ W^{\star}$-algebra: \begin{equation} \label{eq:the hyperfinite finite continuous factor} R \; := \; \pi_{ \tau_{{\mathbb{Z}}}} ( A_{{\mathbb{Z}}} ) '' \end{equation} By eq.\ref{eq:commutativity of the local algebras of disjoint regions} and the linearity of the GNS-representations we infer that R is itself a factor. Eq.\ref{eq:increasing nature of the local algebras} implies that R is hyperfinite. We already know that the n-qubit $W^{\star}$-algebra is a type $ I_{2^{n}}$ factor, i.e. that, taking into account the normalization coefficient $ \frac{1}{2^{n}} $ of $ \tau_{2^{n}} $(not considered, for convenience, in definition\ref{def:discrete finite factor}), we have that: \begin{equation} Range ( d_{B ( \mathcal{H}_{2}^{ \bigotimes n})}) \; = \; \{ 0 \, , \, \frac{1}{2^{n}} \, , \, \frac{2}{2^{n}} \, , \, \cdots \, 1 \} \end{equation} Since in the limit $ n \rightarrow \infty $ the dyadic rationals fill the interval $ [ 0 \, , \, 1 ] $ it follows that: \begin{equation} Range( \tau_{\mathbb{Z}} |_{ {\mathcal{P}} ( A_{\mathbb{Z}})}) \; = \; [ 0 \, , \, 1 ] \end{equation} implying that R is a $ II_{1}$-factor. Since, as we will explain more clearly in section\ref{sec:On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics}, the hyperfinite $ II_{1}$-factor is unique (obviously, remembering remark\ref{rem:identificability of involutively-isomorphic W-star algebras}, up to $ \star $-isomorphism) it is precisely R. \medskip We can, at last, face Von Neumann's confession to Birkhoff, clarifying why the right the right noncommutative space of qubits' sequences is not $ B ( {\mathcal{H}}_{2}^{\bigotimes \infty} ) $ but R. \smallskip This requires, first of all, to understand that the equivalence relation of definition \ref{def:equivalence of projections in a Von Neumann algebra} is nothing but the noncommutative analogue of the Theory of Cardinal Numbers, i.e. the the theory of the \textbf{noncommutative cardinal numbers} describing the infinity's degree of \textbf{noncommutative sets}. In the words of Von Neumann: \begin{center} \textit{"$ \cdots $ the whole algorithm of Cantor theory is such that the most of it goes over in this case. One can prove various theorems on the additivity of equivalence and the transitivity of equivalence, which one would normally expect, so that one can introduce a theory of alephs here, just as in set theory. $ \cdots $ I may call this dimension since for all matrices of the ordinary space, is nothing else but dimension" (Unpublished, cited in \cite{Redei-98})} \textit{"One can prove most of the Cantoreal properties of finite and infinite, and, finally, one can prove that given a Hilbert space and a ring in it , a simple ring in it, either all linear sets except the null sets are infinite (in which case this concept of alephs gives you nothing new), or else the dimensions, the equivalence classes, behave exactly like numbers and there are two qualitatively different cases. The dimensions either behave like integers, or else they behave like all real numbers. There are two subcases, namely there is either a finite top or there is not" (Unpublished, cited in \cite{Redei-98})} \end{center} Given a factor A, let us uniformize the commutative and noncommutative terminology introducing the following notation: \begin{definition} \end{definition} A HAS NONCOMMUTATIVE CARDINALITY EQUAL TO $ n \in {\mathbb{N}}$: \begin{equation} cardinality_{NC}(A) \, = \, n \; := \; Type(A) \, = \, I_{n} \end{equation} \begin{definition} \end{definition} A HAS NONCOMMUTATIVE CARDINALITY EQUAL TO $ \aleph_{0} $: \begin{equation} cardinality_{NC}(A) \, = \,\aleph_{0} \; := \; Type(A) \, = \, I_{\infty} \end{equation} \begin{definition} \end{definition} A HAS NONCOMMUTATIVE CARDINALITY EQUAL TO $ \aleph_{1} $: \begin{equation} cardinality_{NC}(A) \, = \,\aleph_{1} \; := \; Type(A) \, \in \, \{ II_{1} , II_{\infty} \} \end{equation} A HAS NONCOMMUTATIVE CARDINALITY EQUAL TO $ \aleph_{2} $: \begin{equation} cardinality_{NC}(A) \, = \,\aleph_{2} \; := \; Type(A) \, = \, III \end{equation} The definition of noncommutative cardinality may then be extended to arbitary $ W^{\star}$-algebras rqeuiring its additiviy w.r.t. the factor decomposition. Let us then observe that in the commutative case (see theorem\ref{th:cardinalities of strings and sequences}) the passage from strings to sequences implies an increasing of one \textbf{commutative cardinal number}; this implies that: \begin{enumerate} \item it doesn't exist a bijection $ b : \Sigma^{\star} \mapsto \Sigma^{\infty} $, i.e.: \begin{equation} cardinality( \Sigma^{\star} ) \; \neq \; cardinality( \Sigma^{\infty}) \end{equation} \item it does exist an injection $ i : \Sigma^{\star} \mapsto \Sigma^{\infty} $, i.e.: \begin{equation} cardinality( \Sigma^{\star} ) \; \leq \; cardinality( \Sigma^{\infty}) \end{equation} \item the degree of infinity of $ \Sigma^{\infty} $ is that immediately successive to the the degree of infinity of $ \Sigma^{\star} $ \footnote{Such a constraint that there does not exist intermediate degrees of infinity requires the assumption of the following: \begin{axiom} \label{ax:continuum hypothesis} \end{axiom} CONTINUUM HYPOTHESIS: \begin{equation} 2^{\aleph_{0}} \; = \; \aleph_{1} \end{equation} that is well known to be \textbf{consistent} but \textbf{independent} from the formal system ZFC giving foundation to Mathematics \cite{Odifreddi-89}}, i.e.: \begin{equation} \nexists \; S \: : \; cardinality( \Sigma^{\star} ) \: < \: cardinality(S) \: < \: cardinality( \Sigma^{\infty} ) \end{equation} \end{enumerate} Denoted by $ \Sigma_{NC}^{\star} $ the \textbf{$ W^{\star}$-algebra of qubits's strings} and by $ \Sigma_{NC}^{\infty} $ the \textbf{$ W^{\star}$-algebra of qubits' sequences}, theorem\ref{th:category isomorphism at the basis of Noncommutative Topology} requires that the same conditions hold for noncommutative cardinality: \begin{enumerate} \item \begin{equation} cardinality_{NC}( \Sigma_{NC}^{\star} ) \; \neq \; cardinality_{NC}( \Sigma_{NC}^{\infty}) \end{equation} \item \begin{equation} cardinality_{NC}( \Sigma_{NC}^{\star} ) \; \leq \; cardinality_{NC}( \Sigma_{NC}^{\infty}) \end{equation} \item \begin{equation} \nexists \; A \, factor \; \: : \; cardinality_{NC}( \Sigma_{NC}^{\star} ) \: < \: cardinality_{NC}(A) \: < \: cardinality_{NC}( \Sigma_{NC}^{\infty} ) \end{equation} \end{enumerate} Since: \begin{equation} \Sigma_{NC}^{\star} \; = \; B ( \mathcal{H}_{2}^{\bigotimes \star} ) \end{equation} we have, conseguentially, that: \begin{equation} cardinality_{NC}( \Sigma_{NC}^{\infty}) \, = \, cardinality_{NC}( \Sigma_{NC}^{\star}) + 1 \, = \, \aleph_{1} \end{equation} Since we already know that $ cardinality_{NC} (B(\mathcal{H}_{2}^{\bigotimes \infty} ) ) \; = \; \aleph_{0} $ it follows that: \begin{equation} \Sigma_{NC}^{\infty} \; = \; R \; \neq \; B(\mathcal{H}_{2}^{\bigotimes \infty} ) \end{equation} \smallskip \begin{remark} \label{rem:difference between the raising of commutative cardinality ad the raising of noncommutative cardinality} \end{remark} DIFFERENCE BETWEEN THE RAISING OF COMMUTATIVE CARDINALITY AND THE RAISING OF NONCOMMUTATIVE CARDINALITY The fact that the passage from a separable to a non-separable Hilbert space involves a kind of passage from the \textbf{discrete} to the \textbf{continuum} may be highly misleading: introduced the: \begin{definition} \label{def:compuational rigged basis of the Hilbert space of qubits' sequences} \end{definition} COMPUTATIONAL RIGGED-BASIS OF $ {\mathcal{H}}_{2}^{\bigotimes \infty} $: \begin{align} {\mathbb{E}}_{\infty} & \; := \; \{ | \bar{x} > \, , \, \bar{x} \in \Sigma^{\infty} \: \} \\ < \bar{x} & | \bar{y} > \; = \; \delta ( \bar{x} - \bar{y} ) \; \; \bar{x} , \bar{y} \in \Sigma^{\infty} \\ \int_{ \Sigma^{\infty} } & d P_{unbiased} | \bar{x} > < \bar{x} | \; = \; \hat{{\mathbb{I}}} \end{align} theorem\ref{th:cardinalities of strings and sequences} may be restated as: \begin{align} \label{al:cardinalities of the computational bases} cardinality( & {\mathbb{E}}_{\star}) \; = \; \aleph_{0} \\ cardinality( & {\mathbb{E}}_{\infty}) \; = \; \aleph_{1} \end{align} So eqs.\ref{al:cardinalities of the computational bases} are simply the reformulation in an Hilbert space setting on the constraint imponing the raising of one cardinal number in passing from the \textbf{commutative cardinality} of the \textbf{commutative space} of cbits' stings to the \textbf{commutative space} of cbits' sequences. It is not the constraint imponing the raising of one cardinal number in passing from the \textbf{noncommutative cardinality} of the \textbf{noncommutative space} of cbits' strings to the \textbf{noncommutative space} of cbits' sequences, namely the following: \begin{theorem} \label{th:on the noncommutative cardinality of strings and sequences of qubits} \end{theorem} ON THE NONCOMMUTATIVE CARDINALITIES OF STRINGS AND SEQUENCES OF QUBITS: \begin{align*} cardinality_{NC}(\Sigma_{NC}^{\star}) \; & = \; \aleph_{0} \\ cardinality_{NC}(\Sigma_{NC}^{\infty}) \; & = \; \aleph_{1} \end{align*} \smallskip \begin{remark} \label{rem:the phenomenon of continuous dimension from a logical point of view} \end{remark} THE PHENOMENON OF CONTINUOUS DIMENSION FROM A LOGICAL POINT OF VIEW The phenomenon of continuous dimension involved in the passage from noncommutative cardinality $ \aleph_{0} $ to noncommutative cardinality $ \aleph_{1} $ has an intuitive logic meaning: the lost of \textbf{atomicity} of the underlying quantum logic states the disapperance of the \textbf{atomic propositions} as stated by corollary\ref{cor:discreteness of a factor is equivalent to the atomicity of its quantum logic}. We want here to give a more intuitive picture of what does it means. In the quantum logic of $ QL(B ( \mathcal{H}_{2}^{\bigotimes \star} )) $ the propositions of the form: \begin{multline} p_{\int_{\Sigma^{\infty}} c (\bar{x})| \bar{x} > } \; := \; \int_{\Sigma^{\infty}} c (\bar{x})^{\star} < \bar{x} | \; \int_{\Sigma^{\infty}} c (\bar{x}) | \bar{x} > \end{multline} are \textbf{atomic proposition}, i.e. correspond to the elementary statements from which all the others are generated through the logical connectives. This is not the case in $ QL ( \Sigma_{NC}^{\infty} ) $; one could think that the rule of elementary propositions is therein played by projections of the form: \begin{equation} p_{k \, , \, \vec{n}} \; := \; \bigotimes_{i \in {\mathbb{Z}}} a_{i \, , \, \vec{n}} \end{equation} \begin{equation} p_{i \, , \, \vec{n}} \; := \; \begin{cases} \frac{1}{2} ( I_{2} \, + \, \vec{\sigma} \cdot \, \vec{n} ) & \text{if $ i = k $}, \\ I_{2} & \text{otherwise}. \end{cases} \end{equation} (where $ I_{2} $ denotes the bidimensional identity matrix). that, interpreting $ \Sigma_{NC}^{\infty} $ as the $ W^{\star}$-algebra of a quantum spin-1/2 chain at infinite temperature, correspond to the statement $ << $ the spin in the $ k^{th} $ lattice site points in the direction $ \vec{n} \: >> $. Anyway $ p_{k \, , \, \vec{n}} $ is not atomic as can be inferred by the fact that the $ \aleph_{1}$ noncommutative cardinality of $ \Sigma_{NC}^{\infty} $ implies the existence of a projection $ p^{1}_{k \, , \, \vec{n}} \, \in \, {\mathcal{P}}( \Sigma_{NC}^{\infty}) $ such that: \begin{equation} d_{\Sigma_{NC}^{\infty}} ( p^{1}_{k \, , \, \vec{n}} ) \; = \; \frac{d_{\Sigma_{NC}^{\infty}} ( p_{k \, , \, \vec{n}} )}{2} \end{equation} and so: \begin{equation} p^{1}_{k \, , \, \vec{n}} \; \trianglelefteq \; p_{k \, , \, \vec{n}} \end{equation} But then there exist a projection $ p^{2}_{k \, , \, \vec{n}} \, \in \, {\mathcal{P}}( \Sigma_{NC}^{\infty}) $ such that: \begin{equation} p^{2}_{k \, , \, \vec{n}} \, \trianglelefteq \, p_{k \, , \, \vec{n}} \; and \; p^{2}_{k \, , \, \vec{n}} \, \sim_{\Sigma_{NC}^{\infty}} \, p^{1}_{k \, , \, \vec{n}} \end{equation} and hence: \begin{equation} d_{\Sigma_{NC}^{\infty}} ( p^{2}_{k \, , \, \vec{n}} ) \; = \; d_{\Sigma_{NC}^{\infty}} ( p^{1}_{k \, , \, \vec{n}} ) \; = \; \frac{d_{\Sigma_{NC}^{\infty}} ( p_{k \, , \, \vec{n}} )}{2} \end{equation} The projection $ p^{2}_{k \, , \, \vec{n}} $ is thus a non-zero projection strictly smaller than $ p_{k \, , \, \vec{n}} $, so that, conseguentially, $ p_{k \, , \, \vec{n}} $ is not an atom. \medskip Let us finally, as promized, discuss Walter Thirring's reasoning that lead him to give a negative answer to the \textbf{Separability Issue} despite of Shr\"{o}dinger opposite position \cite{Thirring-01}: \begin{center} \textit{"However such an opinion means that Shr\"{o}dinger did not get the main message of Von Neumann's celebrated paper on infinite tensor products. There he shows that the corresponding operator algebras are highly reducibly represented in this vast non-separable space and there are many (inequivalent) subrepresentations which act on a separable subspace"} \end{center} What Thirring is speaking about is the analysis he explicitly reports in the section1.4 of \cite{Thirring-83}: the cases \textbf{case-I.A} and \textbf{case-I} discussed in view of definition\ref{def:non zero inner product in the infinite tensor product Hilbert space} and definition\ref{def:zero inner product in the infinite tensor product Hilbert space} give rise to the following two equivalence relations inside $ \mathcal{H}_{2}^{\bigotimes \infty}$: \begin{definition} \label{def:strong equivalence relation on the qubits' sequences Hilbert space} \end{definition} $ | \psi > , | \phi > \; \in \; \mathcal{H}_{2}^{\bigotimes \infty} $ ARE STRONGLY-EQUIVALENT: \begin{equation} | \psi > \, \sim_{S} \, | \phi > \; := \; \prod'_{n} < \psi_{n} | \phi_{n} > \, \rightarrow \, c \neq 0 \end{equation} \begin{definition} \label{def:weak equivalence relation on the qubits' sequences Hilbert space} \end{definition} $ | \psi > , | \phi > \; \in \; \mathcal{H}_{2}^{\bigotimes \infty} $ ARE WEAKLY-EQUIVALENT: \begin{equation} | \psi > \, \sim_{W} \, | \phi > \; := \; \prod'_{n} < \psi_{n} | \phi_{n} > \, \rightarrow \, c > 0 \end{equation} where the symbol $ \prod' $ means that any finite number of factors 0 are to be left out. Both the quotient spaces $ \frac{\mathcal{H}_{2}^{\bigotimes \infty}}{\sim_{S}} $ and $ \frac{\mathcal{H}_{2}^{\bigotimes \infty}}{\sim_{W}} $ are linear spaces. Furthermore one has that: \begin{equation} cardinality ( \frac{\mathcal{H}_{2}^{\bigotimes \infty}}{\sim_{W}} ) \; > \; \aleph_{0} \end{equation} \begin{equation} [ | \psi > ]_{W} \, \neq \, [ | \phi > ]_{W} \; \Rightarrow \; < \psi | \phi > \, = \, 0 \; \; \forall | \psi > , | \phi > \in {\mathcal{H}}_{2}^{\bigotimes \infty} \end{equation} \begin{equation} [ [ | \psi > ]_{W} ]_{S} \, \neq \, [ [ | \phi > ]_{W} ]_{S} \; \Rightarrow \; < \psi | \phi > \, = \, 0 \; \; \forall | \psi > , | \phi > \in {\mathcal{H}}_{2}^{\bigotimes \infty} : [ | \psi > ]_{W} \, = \, [ | \phi > ]_{W} \end{equation} Adopting the notation used in example\ref{ex:the hyperfinite finite continuous factor}, the key point is that \begin{theorem} \label{th:on the high reducibility of the representations of the chain spin algebra over the qubits' sequences' Hilbert space} \end{theorem} ON THE HIGH REDUCIBILITY OF THE REPRESENTATIONS OF $ A_{{\mathbb{Z}}} $ ON $ {\mathcal{H}}_{2}^{\bigotimes \infty} $ \begin{hypothesis} \end{hypothesis} $ \pi $ representation of $ A_{{\mathbb{Z}}} $ over $ {\mathcal{H}}_{2}^{\bigotimes \infty} $ \begin{thesis} \end{thesis} \begin{equation} [ | \psi > ] _{S} \text{ is separable } \; \; \forall | \psi > \in {\mathcal{H}}_{2}^{\bigotimes \infty} \end{equation} \begin{equation} \pi ( [ | \psi > ] _{S} ) \; \subseteq \; [ | \psi > ] _{S} \; \; \forall | \psi > \in {\mathcal{H}}_{2}^{\bigotimes \infty} \end{equation} \begin{equation} \text{ the sub-representations arising from different weak equivalence classes are inequivalent} \end{equation} According to Thirring theorem\ref{th:on the high reducibility of the representations of the chain spin algebra over the qubits' sequences' Hilbert space} implies that one has to answer negatively to the Separability issue. According to him, one can consistentely assume that qubits sequences are described by rays of the not-separable Hilbert space $ {\mathcal{H}}_{2}^{\bigotimes \infty}$. Simply, as always happens in the limit of infinite degrees of freedom, different representations of the observables' algebra may become inequivalent so that one has to select the representation suitable to the physical situation he is studying. Since such a representation lives on a separable subspace of $ {\mathcal{H}}_{2}^{\bigotimes \infty}$ everything seems ok. This way of recovering an Hilbert space axiomatization of Quantum Mechanics is, anyway, apparent: one start a priori with the observable's algebra and returns to the usual Hilbert space formalism only a posteriori, after a suitable representation is used. Such an attitude to the algebraic approach (precisely codified in the section1.8 of \cite{Strocchi-85}), though FAPP completelly equivalent to the one we support, is philosophically disappealing since it founds on axioms based on posteriori derived quantities. According to theorem\ref{th:Sakai theorem} one can, in a completelly equivalent way, forget concrete algebras of operators on Hilbert spaces, forget Hilbert spaces themselves, and speak only about $ W^{\star}$-algebras. Futhermore there is no reason to represent these $ W^{\star}$-algebras, returning in this way to an Hilbert space setting. Such a view point corresponds, substantially, to give a positive answer to the Separability Issue, to infer from that that an Hilbert space axiomatization in terms of the not-separable Hilbert space $ {\mathcal{H}}_{2}^{\bigotimes \infty} $ is not acceptable and, conseguentially, to conclude that, as to qubits' sequences, one has to give up the idea to remain inside the boundaries of an Hilbert space axiomatization. \medskip We would like to end this section clarifying more properly the concept of qubit, through the following: \begin{remark} \label{rem:the noncommutative combinatory information and the definition of the qubit} \end{remark} THE NONCOMMUTATIVE COMBINATORY INFORMATION AND THE DEFINITION OF THE QUBIT It should be clear, at this point, the the \textbf{qubit} may be defined, in a conceptually more satisfying way, in the following way: Given a \textbf{noncommutative set} A let us introduce the following notion: \begin{definition} \label{def:noncommutative combinatory information} \end{definition} NONCOMMUTATIVE COMBINATORY INFORMATION: \begin{equation} I_{NC \; combinatory} (A) \; := \; \log_{2} cardinality_{NC} (A) \end{equation} By theorem\ref{th:on the noncommutative cardinality of strings and sequences of qubits} we have, via axiom\ref{ax:continuum hypothesis}, that: \begin{equation} I_{NC \; combinatory} ( \Sigma_{NC}^{\star} ) \; = \; \log_{2} ( \aleph_{0} ) \end{equation} \begin{equation} I_{NC \; combinatory} ( \Sigma_{NC}^{\infty} ) \; = \; \log_{2} ( \aleph_{1} ) \; = \; \aleph_{0} \end{equation} Then we can at last define precisely the qubit as: \begin{definition} \label{def:qubit} \end{definition} QUBIT: \begin{equation} \text{ 1 qubit } \; := \; I_{NC \; combinatory} (\Sigma_{NC}) \end{equation} where: \begin{equation} \Sigma_{NC} \; := \; {\mathcal{B}} ( {\mathcal{H}}_{2} ) \end{equation} is the \textbf{noncommutative binary alphabet} \newpage \section{On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics} \label{sec:On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics} In the previous section, precisely in the remark\ref{rem:the metaphore by which we can speak about noncommutative sets from within ZFC} we have introduced the Noncommutative Metaphore allowing to speak about noncommutative sets from inside the ZFC axiomatization of commutative set theory. By theorem\ref{th:category isomorphism at the basis of Noncommutative Topology} we have then given foundations to Noncommutative Topology, i.e. the analysis of topological structure on noncommutative sets. The next floor in the construction of the noncommutative tower is the introduction of Noncommutative Measure Theory \cite{Streater-95},\cite{Streater-00a}. \begin{definition} \label{def:algebraic probability space} \end{definition} ALGEBRAIC PROBABILITY SPACE: $ ( \, A \, , \, \omega \, ) $, where: \begin{itemize} \item A is a Von Neumann algebra \item $ \omega $ is a state on A \end{itemize} The notion of \textbf{algebraic probability space} is a noncommutative generalization of the notion of \textbf{classical probability space} as is implied by the following: \begin{theorem} \label{th:Gelfand isomorphism at W-star algebraic level} \end{theorem} GELFAND'S ISOMORPHISM AT $ W^{\star}$-ALGEBRAIC LEVEL: \begin{enumerate} \item a generic \textbf{classical probability space} $ ( \; X \, , \, \mu \; ) $ may be equivalententely seen as the algebraic probability space $ ( \, L^{\infty} ( X , \mu ) \, , \, \omega_{\mu} \,) $, with: \begin{equation} \begin{split} \omega_{\mu} ( A ) & \in S ( A ) \\ \omega_{\mu} ( a ) & \; := \; \int_{X} a ( x ) d \mu (x) \end{split} \end{equation} \item given a generic \textbf{abelian algebraic probability space} $ ( \, A \, , \, \omega \, ) $ there exist a \textbf{classical probability space} $ ( \; X \, , \, \mu \; ) $ and a $ \star $ - isomorphism $ {\mathcal{I}}_{GELFAND} \, : A \rightarrow L^{\infty} ( X , \mu \,) $, namely the \textbf{Gelfand isomorphism}, under which the state $ \omega \in S(A) $ corresponds to the state $ \omega_{\mu} \in L^{\infty} ( X , \mu )$. \end{enumerate} \medskip \begin{definition} \label{def:quantum probability space} \end{definition} NONCOMMUTATIVE PROBABILITY SPACE a non-abelian algebraic probability space. \smallskip Given a finite factor A: \begin{definition} \label{def:unbaised algebraic probability space} \end{definition} UNBIASED ALGEBRAIC PROBABILITY SPACE ON A: the algebraic probability space $ ( A \, , \, \tau_{unbiased} ) $, where: \begin{equation} \tau_{unbiased} \; := \; \tau_{A} \end{equation} \medskip Let us now clarify an important point. Given a Von Neumann algebra $ A \; \subseteq \; B({\mathcal{H}}) $: \begin{definition} \label{def:normal states on a Von Nuemann algebra} \end{definition} NORMAL STATES ON A: \begin{multline} S(A)_{n} \; := \; \{ \omega \in S(A) \; : \; \sup_{\alpha} \omega( a_{\alpha} ) \; = \; \omega ( \sup_{\alpha} a_{\alpha} ) \\ \forall \text{ bounded increasing net } \{ a_{\alpha} \} \text{ in A } \} \end{multline} The importance of normal states is owed to a key feature of them, to formalize which let us first introduce the following sequence of norms on $ B ({\mathcal{H}}) $: \begin{definition} \label{def:sequence of operatorial norms} \end{definition} $n^{th}$ OPERATORIAL NORM ON $ B ({\mathcal{H}}) $ \begin{equation} \| a \|_{n} \; := \; (Tr | a |^{n}) ^{\frac{1}{n}} \end{equation} where: \begin{equation} \label{eq:modulus of a bounded operator} | a | \; := \; \sqrt{a ^{\star} a } \end{equation} is the \textbf{modulus} of the operator a, whose name is owed to its rule in the polar decomposition $ a \; = \; U | a | $ by which any bounded operator on $ {\mathcal{H}} $ can be expressed as a product of a partial isometry U times the modulus of a, as in the usual polar decomposition $ z \; = \; e^{ i arg(z) } | z | $ of complex numbers. The definition\ref{def:sequence of operatorial norms} contains the usual norm $ \| \cdot \| $ of the $ W^{\star} $-algebraic structure of $ B ({\mathcal{H}}) $ as the $ n = \infty $ case: \begin{equation} \| a \|_{\infty} \; = \; \| a \| \; \; a \in B ({\mathcal{H}}) \end{equation} Introduced the subspace of the finite-rank bounded operators on $ {\mathcal{H}} $: \begin{equation} {\mathcal{E}}( {\mathcal{H}} ) \; := \; \{ a \in B ({\mathcal{H}}) \, : \, dim ( Range(a) ) < + \infty \} \end{equation} let us introduce the following sequence of operators' classes: \begin{definition} \label{def:classes of bounded operator} \end{definition} n-CLASS BOUNDED OPERATORS ON $ {\mathcal{H}} $: \begin{equation} {\mathcal{C}}_{n}( {\mathcal{H}} ) \; := \; completion( {\mathcal{E}}( {\mathcal{H}} ) , \| \cdot \|_{n} ) \end{equation} In particular, the operators in $ {\mathcal{C}}_{1}( {\mathcal{H}} ) $ are called \textbf{trace class}, those in $ {\mathcal{C}}_{2}( {\mathcal{H}} ) $ \textbf{Hilbert-Schmidt}, while the operators in $ {\mathcal{C}}( {\mathcal{H}} ) \; := \; {\mathcal{C}}_{\infty}( {\mathcal{H}} ) $ are called \textbf{compact} or \textbf{infinitesimal}, this last name being owed to the rule they play, as we will see, in Noncommutative Differential Calculus. Let us then introduce the following notion: \begin{definition} \label{density operators} \end{definition} DENSITY OPERATORS ON $ {\mathcal{H}} $: \begin{equation} {\mathcal{D}}( {\mathcal{H}} ) \; := \; \{ \rho \in {\mathcal{C}}_{1}( {\mathcal{H}} ) \bigcap ({\mathcal{B}}( {\mathcal{H}}))_{+} \, : \, Tr \rho =1 \} \end{equation} The key feature of the normal states over a Von Neumann algebra $ A \; \subseteq \; B({\mathcal{H}}) $ may then be stated as follows: \begin{theorem} \label{th:on the density operators of normal states} \end{theorem} ON THE DENSITY OPERATORS OF NORMAL STATES: \begin{equation} \omega \in S(A)_{n} \; \Leftrightarrow \; ( \exists \rho_{\omega} \in {\mathcal{D}}( {\mathcal{H}} ) \; : \; \omega( a ) \, = \, Tr ( \rho_{\omega} a ) \, \; \; \forall a \in A ) \end{equation} and is important essentially owing to the following: \begin{theorem} \label{th:normality of the states of noncommutative spaces with finite noncommutative cardinality} \end{theorem} NORMALITY OF THE STATES OF NONCOMMUTATIVE SPACES WITH FINITE NONCOMMUTATIVE CARDINALITY \begin{equation} cardinality_{NC} (A) \, \in \, {\mathbb{N}} \; \Rightarrow \; S(A) _{n} \, = \, S(A) \end{equation} \smallskip \begin{example} \label{ex:n qubits' noncommutative probability spaces} \end{example} n QUBITS' NONCOMMUTATIVE PROBABILITY SPACES Given an n qubit probability space $ ( B ({\mathcal{H}}_{2}^{\bigotimes n} ) \, , \, \omega ) $ theorem\ref{th:normality of the states of noncommutative spaces with finite noncommutative cardinality} implies that everything can be rephrased in terms of the more popular couple $ ( {\mathcal{H}}_{2}^{\bigotimes n} \, , \, \rho_{\omega} ) $. So, in this case when can make any noncommutative-probabilistic analysis avoiding all the algebraic machinery,e.g. according the lines developed in \cite{Parthasarathy-92}. This applies, in particular, for the n qubit unbaised noncommutative probability space $ ( B ({\mathcal{H}}_{2}^{\bigotimes n} ) \, , \, \tau_{unbiased}) $. Given an \textbf{algebraic random variable} $ a \in A $ over the \textbf{algebraic probability space} $ ( A , \omega )$: \begin{definition} \end{definition} $ n^{th} $ MOMENT OF a: \begin{equation} M_{n} (a) \; := \; \omega ( a ^{n} ) \end{equation} Of particular rilevance are the following: \begin{definition} \end{definition} EXPECTATION VALUE OF a: \begin{equation} E(a) \; := \; M_{1} (a) \end{equation} \begin{definition} \end{definition} VARIANCE OF a: \begin{equation} Var(a) \; := \; \sqrt{ E(a ^{2}) - ( E(a) )^{2} } \end{equation} The information contained in the moments'- sequence of a may be usefully incorporated in the following: \begin{definition} \end{definition} CHARACTERISTIC FUNCTION OF a: $ ZQ_{a} \; : \; Convergence-circle(a) \; \rightarrow \; {\mathbb{C}} $ \begin{equation} ZQ_{a}(t) \; := \; \sum_{n=0}^{\infty} M_{n} (a) \frac{ t^{n}}{ n !} \end{equation} where $ \; Convergence-circle(a) \; := \{ z \in {\mathbb{C}} \, : \, | z | \leq R_{Convergence-circle(a)} \} $ is the circle in the complex plane with centre the origin inside which the sum converges. When $ R_{Convergence-circle(a)} \; > \; 0 $ we have that: \begin{equation} M_{n} (a) \; = \; \frac{d^{n} ZQ_{a}(t) }{ d t^{n}} ( t=0 ) \end{equation} \medskip Given two \textbf{algebraic random variables} a e b on the \textbf{algebraic probability space} $ ( A , \omega )$: \begin{definition} \end{definition} a and b are INDEPENDENT: \begin{equation} E( a^{n} b^{m} ) \; = \; E( a^{n} ) E( b^{m} ) \; \; \forall n , m \in {\mathbb{N}} \end{equation} \medskip Given two sets $ Q_{1} $ and $ Q_{2} $ of algebraic random variables on \textbf{algebraic probability space} $ ( A , \omega )$: \begin{definition} \end{definition} $ Q_{1} $ and $ Q_{2} $ ARE INDEPENDENT: $ q_{1} $ e $ q_{2} $ are independent $ \forall q_{1} \in Q_{1} \, , \, \forall q_{2} \in Q_{2} $ \bigskip We have the following: \begin{theorem} \label{th:dependence of noncommutating random variables} \end{theorem} DEPENDENCE OF NONCOMMUTATING RANDOM VARIABLES: \begin{equation} a \, and \, b \; \; independent \; \; \Rightarrow \; \; [ a , b ] \, = \, 0 \end{equation} \bigskip Given a self-adjoint algebraic random variable $ a \in A_{sa} $: \begin{definition} \end{definition} CLASSICAL PROBABILITY MEASURE OF a: the classical probability measure $ \mu_{a} $ induced by $ \omega $ on the \textbf{spectrum} Sp(a) of a \begin{definition} \end{definition} RESULT OF A MEASUREMENT OF a: The classical random variable $ v_{a} $ on the \textbf{spectrum} Sp(a) of a having $ \mu_{a} $ as classical probability distribution. \medskip Given two noncommutative probability spaces $ ( A_{1} \, , \, \omega_{1} ) $ and $ ( A_{2} \, , \, \omega_{2} ) $ let us introduce the following notion: \begin{definition} \end{definition} TENSORIAL PRODUCT OF $ ( A_{1} \, , \, \omega_{1} ) $ AND $ ( A_{2} \, , \, \omega_{2} ) $: \begin{equation} ( A_{1} \, , \, \omega_{1} ) \, \bigotimes \, ( A_{2} \, , \, \omega_{2} ) \; := \; ( A_{1} \bigotimes A_{2} \, , \, \omega_{1} \cdot \omega_{2} ) \end{equation} where: \begin{equation} \omega_{1} \cdot \omega_{2} ( a_{1} \bigotimes a_{2} ) \; := \; \omega_{1} ( a_{1} ) \omega_{2} ( a_{2} ) \; \; \forall a_{1} \in A_{1} \, , \, \forall a_{2} \in A_{2} \end{equation} \medskip Clearly we have the following: \begin{theorem} \label{th:automatic independence on tensorial products} \end{theorem} AUTOMATIC INDEPENDENCE ON TENSORIAL PRODUCTS: $ a_{1} \bigotimes {\mathbb{I}} $ and $ {\mathbb{I}} \bigotimes a_{2} $ are \textbf{independent algebraic random variables} on $ ( A_{1} \, , \, \omega_{1} ) \, \bigotimes \, ( A_{2} \, , \, \omega_{2} ) \; \; \forall a_{1} \in A_{1} \, , \, \forall a_{2} \in A_{2} $ Given an \textbf{algebraic probability space} $ ( A \, , \, \omega ) $: \begin{definition} \end{definition} $ ( A \, , \, \omega ) $ IS FACTORIZABLE: \begin{equation} \exists A_{1} , A_{2} \text{ $W^{\star}$-algebras } \, , \, \omega_{1} \in S( A_{1} ) \, , \, \omega_{2} \in S( A_{2} ) \; : \; ( A \, , \, \omega ) \, = \, ( A_{1} \, , \, \omega_{1} ) \, \bigotimes \, ( A_{2} \, , \, \omega_{2}) \end{equation} \begin{definition} \end{definition} $ ( A \, , \, \omega ) $ IS ENTANGLED: it is not factorizable ed A is not an $ I_{2} $ - factor. \bigskip Let us consider, now, the following problem: \textbf{given a noncommutative probability space is it possible to approximate it through a classical probability space up to a given perturbative order ?}. \bigskip To assign an \textbf{algebraic random variable} a on the \textbf{algebraic probability space} $ ( A \, , \omega \, ) $ is equivalent to assign the moments'- sequence $ \{ M_{n} (a) \}_{n \in {\mathbb{N}}} $; \smallskip Given a \textbf{noncommutative probability space} $ ( A \, , \omega \, ) $ and a collection $ Q \; := \; \{ q_{1} \, , \, \cdots q_{n} \} $ of \textbf{noncommutative random variables} on it, one could think of trying to approximate the quantum random variables contained in Q by classical random variables reproducing correctly the moments up to a certain order. This may be formalized in the following way: \begin{definition} \label{def:approssimazione classica ad un dato ordine} \end{definition} CLASSICAL APPROXIMATION OF Q UP TO THE $ n^{th} $ ORDER: a bijection $ Ap : Q \mapsto C $, where C is a collection of classical random variables on a suitable classical probability space $ ( M \, , \, P ) $ such that: \begin{equation} E( q_{1}^{i_{1}} \, \cdots \, q_{n}^{i_{n}}) \; = \; \int_{M} d P Ap(q_{1})^{i_{1}} \, \cdots Ap(q_{n})^{i_{n}} \; \; i_{1} , \cdots i_{n} \in {\mathbb{N}} \; : \, \sum_{k=1}^{n} i_{k} \; \leq \; n \end{equation} Given a classical approximation $ Ap : Q \mapsto C $ of Q up to the $n^{th}$ order, let us introduce the following notion: \begin{definition} \end{definition} CHARACTERISTIC FUNCTION ASSOCIATED TO Ap: $ ZC_{Ap} \, : \, {\mathbb{C}}^{n} \; \rightarrow \; {\mathbb{C}} $ \begin{equation} ZC_{Ap} ( t_{1} , \cdots , t_{n} ) \; := \; \int_{M} d P e^{ \sum_{i=1}^{n} t_{i} Ap(q_{i})} \end{equation} We will have, then, clearly that: \begin{equation} E( q_{1}^{i_{1}} \, \cdots \, q_{n}^{i_{n}}) \; = \; \frac{d^{\sum_{k=1}^{n} i_{k}} ZC_{Ap} ( t_{1} , \cdots , t_{n} ) } {d t_{1}^{i_{1}} \, \cdots \, d t_{n}^{i_{n}}} ( t_{1} = 0, \cdots , t_{n} = 0 ) \; \; i_{1} , \cdots i_{n} \in {\mathbb{N}} \: : \: \sum_{k=1}^{n} i_{k} \; \leq \; n \end{equation} It appears , then, clear that a classical approximation up to the $ n^{th} $ order of Q involves precisely the consideration of a series-expansion of the associated characteristic function up to the $ n^{th} $ order. \medskip Let us introduce, now, the following fundamental notion: \begin{definition} \end{definition} Q IS IRRIDUCIBLE TO CLASSICAL PROBABILITY UP TO THE $ n^{th} $ ORDER : it doesn't exist a classical approximation of Q up to the $ n^{th} $ order. \bigskip Demanding to \cite{Ohya-Petz-93}, \cite{Benatti-93} for any suppletive notion, let us now briefly recall here the basic notions concerning the theory of noncommutative dynamical systems. Given an \textbf{algebraic probability space } $ ( \, A \, , \, \omega \, )$: \begin{definition} \label{endomorphism of an algebraic probability space} \end{definition} ENDOMORPHISMS OF $ ( \, A \, , \, \omega \, ) $: \begin{equation} END( \, A \, , \, \omega \, ) \; := \; \{ \tau \, : \, A \rightarrow A \; \text{ surjective $ \star $ - morphism of A} \; : \; \omega \, \in \, S_{\tau} ( A ) \} \end{equation} where $ S_{\tau} ( A ) $ is the set of the $ \tau $ - invariant states on A. \begin{definition} \label{automorphism of an algebraic probability space} \end{definition} AUTOMORPHISMS OF $ ( \, A \, , \, \omega \, ) $: \begin{equation} AUT( \, A \, , \, \omega \, ) \; := \; \{ \tau \, : \, A \rightarrow A \text{ bijective endomorphism of } ( \, A \, , \, \omega \, ) \} \end{equation} \begin{definition} \label{algebraic dynamical system} \end{definition} ALGEBRAIC DYNAMICAL SYSTEM : $ ( \, A \, , \, \omega \, , \, \tau ) $ such that: \begin{itemize} \item $ ( \, A \, , \, \omega \, ) $ is an algebraic probability space \item $\tau$ is an endomorphism of $ ( \, A \, , \, \omega \, ) $ \end{itemize} \begin{remark} \end{remark} ON THE PASSAGE FROM THE HEISENBERG-PICTURE TO THE SCHR\"{O}DINGER-PICTURE OF DYNAMICS: We have implicitely assumed Heisenberg's picture of dynamics (in which states are fixed while observable evolve with time). The passage to the Schr\"{o}dinger picture (in which observables are fixed while states evolve with time) is, anyway, straightforward: given a $ \star $-morphism C from a $ W^{\star}$-algebra B to a $ W^{\star}$-algebra A: \begin{definition} \label{def:dual of an involutive morphism} \end{definition} DUAL OF C: the map $ C_{\star} \, : S(A) \, \rightarrow \, S(B) $: \begin{equation} ( C_{\star} \alpha ) (b) \; := \; \alpha ( C b ) \; \; \forall \alpha \in S(A) \, , \, \forall b \in B \end{equation} \medskip Given an algebraic dynamical system $ ( \, A \, , \, \omega \, , \, \tau ) $: \begin{definition} \label{reversible algebraic dynamical system} \end{definition} $ ( \, A \, , \, \omega \, , \, \tau ) $ IS REVERSIBILE $\tau$ is an automorphism of $ ( \, A \, , \, \omega \, ) $. \medskip \begin{definition} \label{def:quantum dynamical system} \end{definition} $ ( \, A \, , \, \omega \, , \, \tau ) $ IS NONCOMMUTATIVE $ ( \, A \, , \, \omega \; ) $ is noncommutative \smallskip The notion of \textbf{(reversibile) algebraic dynamical system} is a noncommutative generalization of the notion of \textbf{(reversibile) classical dynamical system}. In fact: \begin{enumerate} \item $( \, X \, , \, {\mathcal{F}} \, , \, \mu \, , \, T \, ) $ (reversibile) classical dynamical system $\Rightarrow \; ( \, L^{\infty} ( X , \mu ) , \omega_{\mu} , \tau_{T}) $ (reversibile) algebraic dynamical system where: \begin{equation} \label{eq:endomorfismo associato a una mappa classica} \begin{split} \tau_{T} \; \; & \text{automorphism of $L^{\infty} ( X , \mu )$} \\ \tau_{T} & ( a ) \; \equiv \; a \cdot T^{-1} \end{split} \end{equation} \item given a (reversibile) algebraic dynamical system $ ( \, A , \omega , \tau \, )$ with A abelian $W^{\star}$-algebra , then equation eq.\ref{eq:endomorfismo associato a una mappa classica} univoquely individualizes an endomorphism (automorphism) T of the assoociated classical probability space. \end{enumerate} \medskip Such a result may be enunciated in the abstract language of Categories' Theory as the following: \begin{theorem} \label{th:category isomorphism at the basis of Noncommutative Probability} \end{theorem} CATEGORY EQUIVALENCE AT THE BASIS OF NONCOMMUTATIVE PROBABILITY The \textbf{category} having as \textbf{objects} the \textbf{classical probability spaces} and as \textbf{morphisms} the \textbf{endomorphisms (automorphisms)} of such spaces is equivalent to the \textbf{category} having as objects the \textbf{abelian algebraic probability spaces} and as\textbf{ morphisms} the \textbf{endomorphisms (automorphisms)} of such spaces. \medskip Let us now analyze the symmetries of algebraic dynamical systems: on discussing derivations on a $ C^{\star}$-algebra A we have already met strongly-continuous one-parameter subgroups of AUT(A). Given, in general, a Lie group G: \begin{definition} \label{def:set of the automorhisms' groups of a C-star algebra representing a Lie group} \end{definition} SET OF THE AUTOMORHISMS' GROUPS OF A REPRESENTING G: \begin{equation} GR-AUT( G \, , \, A) \; := \, \{ \{ \alpha_{g} \}_{g \in G} \, \text{ strongly-continuous subgroup of AUT(A) } \} \end{equation} \begin{definition} \label{def:set of the inner automorhisms' groups of a C-star algebra representing a Lie group} \end{definition} SET OF THE INNER AUTOMORHISMS' GROUPS OF A REPRESENTING G: \begin{equation} GR-INN( G \, , \, A) \; := \, \{ \{ \alpha_{g} \}_{g \in G} \, \in \, G-AUT( G \, , \, A) \; : \alpha_{g} \in INN(A) \, \forall g \in G \} \end{equation} \begin{definition} \label{def:set of the outer automorhisms' groups of a C-star algebra representing a Lie group} \end{definition} SET OF THE OUTER AUTOMORHISMS' GROUPS OF A REPRESENTING G: \begin{equation} GR-OUT( G \, , \, A) \; := \, \frac{GR-AUT( G \, , \, A)}{GR-INN( G \, , \, A)} \end{equation} Mackey's notion of \textbf{system of imprimitivity} as well as its unsharp generalization, i.e the notion of \textbf{generalized system of imprimitivity} often also called \textbf{system of covariance} (cfr. e.g the $ 3^{th}$ chapter of \cite{Prugovecki-92} and the section2.3 of \cite{Holevo-99}), may be generalized to the Quantum Probability's framework in the following way: \begin{definition} \label{def:covariance system on a W-star algebra w.r.t. a Lie Group} \end{definition} COVARIANCE SYSTEM ON A W.R.T. G: a couple $ ( E \, , \, \{ \alpha_{g} \}_{g \in G} ) $ such that: \begin{itemize} \item $ E \, \in \, \stackrel{\circ}{MAP} ( X , A_{+} ) $ is a POVM on A \item \begin{equation} \{ \alpha_{g} \}_{g \in G} \; \in \; GR-AUT( G \, , \, A) \end{equation} \item \begin{equation} \alpha_{g} \, E( B ) \; = \; E ( g \, B ) \; \; \forall B \in HALTING(E) \, , \, \forall g \in G \end{equation} \end{itemize} \smallskip \begin{remark} \label{rem:quantum physics versus noncommutative sets} \end{remark} QUANTUM PHYSICS VERSUS NONCOMMUTATIVE SETS He have seen in section\ref{sec:Why to treat sequences of qubits one has to give up the Hilbert-Space Axiomatization of Quantum Mechanics} how Von Neumann's investigations on the foundations of Quantum Mechanics led him to implicitely introduce Noncommutative Set Theory. This was the starting point of a foundational school aimed at explicating the transition from Classical Physics to Quantum Physics in terms of the ansatz: \begin{equation*} \text{commutative spaces} \; \rightarrow \; \text{noncommutative spaces} \end{equation*} The school looking at the modification of Probability Theory involved in such an ansatz as the root of the quantum peculiarity is called Quantum Probability. Such a position is exemplified by the following words of Raymond F. Streater \cite{Streater-00a}: \begin{center} \textit{"It took some time before it was understood that quantum theory is a generalization of probability, rather than a modification of the laws of mechanics. This was not helped by the term quantum \emph{mechanics}; more, the Copenhagen interpretation is given in terms of probability, meaning as understood at the time. Bohr has said that the interpretation of microscopic measurements must be done in terms of classical terms, because the measuring instruments are large, and are therefore describe by classical laws. It is true, that the springs and cogs making up a measuring instrument themselves obey classical laws; but this does not mean that the \emph{information} held on the instrument, in the numbers indicated by the dials, obey classical statistics. If the instruments faithfully measures an atomic variable, then the numbers indicated by the dials should be analyzed by quantum probability, however large the instruments is"} \end{center} Such a viewpoint pervaded Richard Feynman's thought\cite{Feynman-Hibbs-65}: \begin{center} \textit{"But far more fundamental was the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace. The quantum-mechanical laws of the physical world approach very closely the laws of Laplace as the size of the objects involved in the experiment increases. Therefore the laws of probabilities which are conventionally applied are quite satisfactory in analyzing the behaviour of the roulette wheel but no the behaviour of a single electron or a photon of light"} \end{center} being at the heart of his path-integral formalism based on the observation that the \textbf{Law of Composed Probabilities}: \begin{equation} \label{eq:law of composed probabilities} P ( x_{1} | x_{3} ) \; = \; \sum_{x_{2} \in {\mathcal{E}} } P ( x_{1} | x_{2} ) P ( x_{2} | x_{3} ) \end{equation} (with $ {\mathcal{E}} $ denoting the space of events) doesn't hold in Quantum Probability where it is replaced by the \textbf{Law of Composed Probability-amplitudes}: \begin{equation} \label{eq:law of composed probabilities-amplitudes} AP ( x_{1} | x_{3} ) \; = \; \sum_{x_{2}\in {\mathcal{E}} } AP ( x_{1} | x_{2} ) AP ( x_{2} | x_{3} ) \end{equation} that, owing to the link between \textbf{probabilities} and \textbf{probabilities-amplitudes}: \begin{equation} P ( x_{1} | x_{3} ) \; = \; | AP ( ( x_{1} | x_{3} ) |^{2} \end{equation} implies that: \begin{multline} P ( x_{1} | x_{3} ) \; = \; | \sum_{x_{2}\in {\mathcal{E}} } AP ( x_{1} | x_{2} ) AP ( x_{2} | x_{3} ) |^{2} \; = \\ ( \sum_{x_{2}\in {\mathcal{E}} } AP ( x_{1} | x_{2} ) AP ( x_{2} | x_{3} ) ) ^{\star} \, ( \sum_{x_{2}\in {\mathcal{E}} } AP ( x_{1} | x_{2} ) AP ( x_{2} | x_{3} ) ) = \\ \sum_{x_{2} \in {\mathcal{E}} } P ( x_{1} | x_{2} ) P ( x_{2} | x_{3} ) \; + CT(x_{1} | x_{3}) \end{multline} where the non-null \textbf{correction-term}: \begin{equation} CT(x_{1} | x_{3}) \; \neq \; 0 \end{equation} is the basis of the interference between different paths contributing to a path-integral. It was always such a viewpoint that led Feynman \cite{Feynman-99} to geniously perceive that the difference between Quantum Probability from Classical Probability implies the irreducibility of Quantum Computational Complexity Theory to Classical Computational Complexity Theory , catching the essence of Quantum Computation, as we will discuss more completely in section\ref{sec:Irreducibility of Quantum Computational Complexity Theory to Classical Computational Complexity Theory}. \bigskip To see why not only the Measure Theory, but also the geometry of noncommutative sets play a fundamental role for Quantum Physics, let us analyze the metric aspects of Quantum Information Theory. At this regard our point of view is rather different from the usual one, being based on the following: \begin{remark} \label{rem:it is wrong to apply commutative geometry to noncommutative sets} \end{remark} IT IS WRONG TO APPLY COMMUTATIVE GEOMETRY TO NONCOMMUTATIVE SETS The metric aspect of Classical Information Theory is required in order of formalizing the concept of \textbf{distance} among \textbf{classical probability distributions}. This may be done in terms of some conceptually appealing \textbf{metric} one can introduce on the \textbf{space of classical probability distributions} Clearly the same situation appears in Quantum Information Theory, where one needs to quantify the \textbf{distance} among \textbf{quantum probability distributions}. This has led to introduce suitable \textbf{metrics} on the space of \textbf{quantum probability distributions} generalizing the classical ones in a nice way. In \textbf{Information Geometry}, furthermore, one goes further introducing a suitable \textbf{riemannian metric} on the \textbf{space of classical probability distributions} such to formulate many Classical Statistics' issues in a purely \textbf{riemannian-geometric} context. So it has appeared natural to mimic such an attitude in Quantum Information Theory, giving rise to the discipline of \textbf{Quantum Information Geometry}, in which one introduces a suitable \textbf{riemannian metric} on the \textbf{space of quantum probability distributions} properly generalizing the classical one, again recasting many Quantum Statistics's issues in a \textbf{riemannian-geometric} context. According to us, anyaway, these approaches are unsatisfactory, in that they ultimatively apply the usual \textbf{Commutative Geometry} to \textbf{noncommutative spaces}: since the space of \textbf{space of quantum probability distributions} is a \textbf{noncommutative space} its metric properties should be formalized in terms of \textbf{noncommutative metrics}. The same reasoning applies to \textbf{Quantum Information Geometry}: since the \textbf{space of quantum probability distributions} is a \textbf{noncommutative space} one should introduce on it a \textbf{noncommutative riemannian-geometric structure} rather than a \textbf{commutative riemannian-geometric structure}. In the sequel we will see how this leads to an application of Alain Connes's Noncommutative Geometry in all its power and beauty. \medskip Given the set of n elements $ M \; := \; \{ 1 , \cdots , n \} $ let us denote by $ \mathcal{D} ( M ) $ the set of the probability distributions over M (endowed with the Borel-$\sigma$-algebra derived from the discrete topology). Since: \begin{equation} {\mathcal{D}} (M) = \{ \vec{p} = ( p_{1} , \cdots , p_{n} ) \in {\mathbb{R}}^{n} \, : \, \sum_{i=1}^{n} p_{i} = 1 \, , \, p_{i} \geq 0 \; i = 1 , \cdots , n \} \end{equation} we have that $ {\mathcal{D}} (M) $ is an $ (n-1)$-simplex of $ {\mathbb{R}}^{n} $. A first reasonable distance over $ \mathcal{D} ( M ) $ it is natural to take in consideration is the following: \begin{definition} \label{def:naife classical trace distance} \end{definition} CLASSICAL TRACE DISTANCE ON $ \mathcal{D} ( M ) $: \begin{equation} D_{T} ( \vec{p}^{(A)} , \vec{p}^{(B)} ) \; := \; \frac{1}{2} \sum_{i \in M} ( | p^{(A)}_{i} - p^{(B)}_{i} | ) \end{equation} The intuitive meaning of the definition\ref{def:naife classical trace distance} is clarified by the following\cite{Nielsen-Chuang-00}: \begin{theorem} \label{th:physical meaning of the naife classical trace distance} \end{theorem} CLASSICAL TRACE DISTANCE AS DISTANCE OF THE CLASSICAL PROBABILITY OF ANTIPODAL EVENTS: \begin{equation} D_{T} ( \vec{p}^{(A)} , \vec{p}^{(B)} ) \; = \; \max_{e \in 2^{M}} | p^{A} (e) \, - \, p^{B} (e) | \end{equation} The natural quantum corrispective of definition\ref{def:naife classical trace distance} could seem the following: given an n-dimensional Hilbert space $ {\mathcal{H}} $: \begin{definition} \label{def:naife quantum trace distance} \end{definition} QUANTUM TRACE DISTANCE ON $ \mathcal{D} ( { \mathcal{H}} ) $: \begin{equation} D_{T} ( \rho^{(A)} , \rho^{(B)} ) \; := \; \frac{1}{2} \| \rho^{(A)} - \rho^{(B)} \|_{1} \end{equation} It is remarkable that also in the quantum case an analogous of theorem\ref{th:physical meaning of the naife classical trace distance} holds \cite{Nielsen-Chuang-00}: \begin{theorem} \label{th:physical meaning of the naife quantum trace distance} \end{theorem} QUANTUM TRACE DISTANCE AS DISTANCE OF THE QUANTUM PROBABILITY OF ANTIPODAL EVENTS: \begin{equation} D_{T} ( \rho^{(A)} , \rho^{(B)} ) \; = \; \max_{P \in B({\mathbb{H}})_{+} } Tr P ( \rho^{(A)} - \rho^{(B)} ) \end{equation} Theorem\ref{th:physical meaning of the naife quantum trace distance} has an immediate operational interpretation to appreciate which we have to enter the highly insidious lands of quantum measurements. Given a $ W^{\star}$-algebra A: \begin{definition} \label{def:obsevational channel on A} \end{definition} OBSERVATIONAL CHANNEL ON A: \begin{equation*} \alpha \in CPU(C,A) \; : \; C \text{ commutative space} \end{equation*} \begin{definition} \label{def:POVM on a W-star algebra} \end{definition} POSITIVE OPERATOR VALUED MEASURE (POVM) ON A : a partial map $ E \, \in \, \stackrel{\circ}{MAP} ( X , A_{+} ) $ such that: \begin{itemize} \item $ HALTING( E ) $ is a $ \sigma $-algebra over a set X \item \begin{multline} \sum_{i} \, E ( F_{i} ) \; = \; E ( \cup_{i} F_{i} ) \\ \forall \{ F_{i} \in HALTING( E ) \} \, : \, F_{i} \cap F_{j} = \emptyset \, , \, \forall i \neq j \end{multline} \item \begin{equation} E(X) \; = \; {\mathbb{I}} \end{equation} \end{itemize} \begin{definition} \label{def:PVM on a W-star algebra} \end{definition} PROJECTION VALUED MEASURE (PVM) ON A: a POVM E on A such that: \begin{equation} E(F) \; \in \; {\mathcal{P}}(A) \; \; \forall F \in HALTING( E ) \end{equation} \begin{theorem} \label{th:Naimark's theorem} \end{theorem} NAIMARK'S THEOREM: \begin{hypothesis} \end{hypothesis} \begin{equation*} {\mathcal{H}} \text{ Hilbert space} \end{equation*} \begin{equation*} E \text{ POVM on } {\mathcal{B}} ( {\mathcal{H}} ) \end{equation*} \begin{thesis} \end{thesis} There exist an Hilbert space $ {\mathcal{K}} \, \supset \, {\mathcal{H}} $ and a PVM $ \tilde{E} $ on $ {\mathcal{B}} ( {\mathcal{H}} ) $ such that: \begin{equation*} \tilde{E} (F) \; = \; P_{{\mathcal{H}}} \, E(F) \, P_{{\mathcal{H}}} \; \; \forall F \in HALTING( E ) \end{equation*} where $ P_{{\mathcal{H}}} $ is the projector from $ {\mathcal{K}} $ to $ {\mathcal{H}} $. \smallskip \begin{definition} \label{def:operational partition of unity on a W-star algebra} \end{definition} OPERATIONAL PARTITIONS OF UNITY ON A: \begin{multline} OPU(A) \; := \; \{ {\mathcal{V} } \, := \, ( V_{1} \, , \, \cdots \, V_{n} ) \, ( n \in {\mathbb{N}}) \, : \\ V_{i} \; \in \; A_{+} \; \; i=1, \cdots , n \; and \\ \sum_{i=1}^{n} V_{i}^{\star} V_{i} \; = \; I \} \end{multline} Given a operational partition of unity $ {\mathcal{V} } \, := \, ( V_{1} \, , \, \cdots \, V_{n} ) \, \in \, OPU(A) $: \begin{definition} \label{def:channels' set of an operational partition of unity} \end{definition} CHANNELS' SET OF $ {\mathcal{V} } $: the set $ \{ \alpha_{1}( {\mathcal{V} }) \, , \, \cdots \, , \alpha_{n}( {\mathcal{V} }) \} $, where $ \alpha_{i}( {\mathcal{V} })$ is the channel (owing to theorem\ref{th:Kraus-Stinespring's theorem}) of A : \begin{equation} \alpha_{i}( {\mathcal{V}}) (a) \; := \; V_{i}^{\star} a V_{i} \; \; a \, \in A \end{equation} \begin{definition} \label{def:reduction channel of an operational partition of unity} \end{definition} REDUCTION CHANNEL OF $ {\mathcal{V} } $: the channel (owing to theorem\ref{th:Kraus-Stinespring's theorem}) $ R({\mathcal{V}}) \in CPU(A) $: \begin{equation} R({\mathcal{V}}) \; := \; \sum_{i=1}^{n} \alpha_{i}( {\mathcal{V}}) \end{equation} The interrelation among these notions is the following: \begin{itemize} \item a POVM $ E \, : \, X \, \stackrel{\circ}{\rightarrow} \, A_{+} $ whose halting set is the Borel-$\sigma$-algebra of a topology on X may be seen as an observational channel $ E \, : \, C(X) \rightarrow A $ \item an observational channel $ \alpha \in CPU( C,A) $ such that $ cardinality_{NC} ( C ) \, \in \, {\mathbb{N}} $ indivuates an operational partition of unity \end{itemize} We have spent much efforts, in section\ref{sec:Why to treat sequences of qubits one has to give up the Hilbert-Space Axiomatization of Quantum Mechanics}, to discuss a statement by Walter Thirring in which he claimed that Sch\"{o}dinger's positive answer to the separability issue was owed to the fact he didn't understand Von Neumann's paper on infinite tensor products. But we are perfectly aware that if there is someone knowing exceptionally all the mirabilities of the noncommutative approach is precisely Walter Thirring, who has given in \cite{Thirring-81} and \cite{Thirring-83} one of its most authoritative presentations. We have implicitely adopted such an approach in section\ref{sec:On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics}, though not exlicitely presenting its underlying axiomatization. We will do this here, partly moving away from the basic-assumption-2.2.32 of \cite{Thirring-81}. \begin{definition} \label{def:noncommutative axiomatization of Quantum Mechanics} \end{definition} NONCOMMUTATIVE AXIOMATIZATION OF QUANTUM MECHANICS: any axiomatization of Quantum Mechanics assuming the following two axioms: \begin{axiom} \label{ax:noncommutative axiom on observables} \end{axiom} NONCOMMUTATIVE AXIOM ON OBSERVABLES: The \textbf{observables} of a \textbf{quantum mechanical systems} are POVM's over a noncommutative space A, called its \textbf{observables' algebra} \begin{axiom} \label{ax:noncommutative axiom on states} \end{axiom} NONCOMMUTATIVE AXIOM ON STATES: The \textbf{states} of a \textbf{quantum mechanical systems} are states over its \textbf{observables' algebra} \medskip We will assume, from here and beyond, a Noncommutative Axiomatization of Quantum Mechanics endowed with the other following axioms: \begin{axiom} \label{ax:noncommutative axiom on closed dynamics} \end{axiom} NONCOMMUTATIVE AXIOM ON DYNAMICS OF A CLOSED SYSTEM: The dynamical evolution of a \textbf{closed} \textbf{quantum mechanical system} S is given by a \textbf{strongly-continuous group of inner automorphisms} of its \textbf{observables' algebra} \smallskip \begin{axiom} \label{ax:noncommutative axiom on measurement} \end{axiom} NONCOMMUTATIVE AXIOM ON MEASUREMENT: If on a \textbf{quantum mechanical system} S, prepared in the state $ \omega \in S(A) $, it is performed the measurement mathematically described by the operational partition of unity $ {\mathcal{V}} \, := \, \{ V_{1} \, , \, \cdots \, , V_{n} \} $ then: \begin{itemize} \item during the measurement S is \textbf{open} \item by definition one says that the \textbf{$ i^{th}$-experimental outcome occurs} if on a suitable \textbf{classical display} one reads the number $ i \in \{ 1 , \cdots , n \} $ \item if the \textbf{$ i^{th}$-experimental outcome occurs} then then S's observables' algebra evolves according to the the \textbf{channel} $ \alpha_{i}( {\mathcal{V}}) $ \item the \textbf{$ i^{th}$-experimental outcome occurs} with probability: \begin{equation} p_{i} \; := \; \omega( \alpha_{i}(I) ) \end{equation} \end{itemize} \medskip Axiom\ref{ax:noncommutative axiom on closed dynamics} and axiom\ref{ax:noncommutative axiom on measurement} are consistent owing to the following: \begin{theorem} \label{th:dynamics of an open quantum mechanical system} \end{theorem} DYNAMICS OF AN OPEN QUANTUM MECHANICAL SYSTEM: The dynamical evolution of an \textbf{open} \textbf{quantum mechanical system} S is given by a \textbf{one-parameter family of channels} of its \textbf{observables' algebra} \smallskip \begin{remark} \label{rem:noncommutative axiomatizations and unbounded operators} \end{remark} NONCOMMUTATIVE AXIOMATIZATIONS AND UNBOUNDED OPERATORS: A first natural reaction to definition\ref{def:noncommutative axiomatization of Quantum Mechanics} is to ask what about unbounded operators: if the \textbf{observables' algebra} of a quantum system is assumed to be a noncommutative space $ A \, \subseteq \, {\mathcal{B}} ( {\mathcal{H}}) $ isn't one arbitarily throwing away all the self-adjoints elements of $ {\mathcal{O}} ( {\mathcal{H}} ) \, - \, {\mathcal{B}} ( {\mathcal{H}} ) $? The answer to such an objection touches the original argument that led Irving Segal in 1947 to introduce the algebraic approach (cfr. the introduction of \cite{Haag-96}): given a self-adjoint unbounded operator T on an Hilbert space $ {\mathcal{H}} $ the Spectral Theorem allows us to define the operator $ f(T) $ for every Lebesgue-integrable function $ f \in {\mathcal{B}} ( {\mathbb{R}}) $, the set of linears operators obtained varying f being called the \textbf{abelian $ W^{\star}$-algebra generated by T} (cfr. the section7.2 of \cite{Reed-Simon-80}. Since one can look at the passage from T to f(T) simply as a relabeling of the possible experimental outcomes, the physically relevant notion is that of the \textbf{abelian $ W^{\star}$-algebra generated by T}, of which one can always choose a bounded element such as $ e^{T} $. \smallskip \begin{remark} \label{rem:on superselection rules} \end{remark} ON SUPERSELECTION RULES Let us observe that axiom\ref{ax:noncommutative axiom on observables} says that an observable of a quantum mechanical system is a POVM on its observables' algebra, but it doesn't say that any POVM on such an observable's algebra is an observable of the system. Similarly, axiom\ref{ax:noncommutative axiom on states} says that a physical state of a quantum mechanical system is a state on its observables' algebra, but it doesn't say that any state on such an observable's algebra is a physical state of the system. Finally, axiom\ref{ax:noncommutative axiom on measurement} tells us what happens when on a quantum mechanical system it is performed the measurement mathematically described by a partitional operation of unity, but it doesn't say that any operational partition of unity describes a possible measurement. If on a quantum mechanical systems with observables' algebra A there exist a self-adjoint operator $ a \in A_{sa} $ such that the PVM associated to it by the Spectral Theorem cannot be physical observable, one says that the system has Superselection Rules. \begin{remark} \label{rem:if God plays dices is a kinematical issue and not a dynamical one} \end{remark} IF GOD PLAYS DICES IS A KINEMATICAL ISSUE AND NOT A DYNAMICAL ONE The way axiom\ref{ax:noncommutative axiom on measurement} is formalized may be partially misleading owing to the fact that it would seem to introduce a \textbf{dynamical indeterminism} in the theory. This is not the case: we spoke about of collection $ \{ \alpha _{i} \} $ of possible dynamical evolutions, each with a classical probability $ p_{i} $ of occurring, only for simplicity, but it is exactly the same as saying the that the observable's algebra evolves according to the \textbf{reduction channel} $ R( {\mathcal{V}} ) $ of $ {\mathcal{V}} $ \medskip We can now appreciate the physical meaning of Theorem\ref{th:physical meaning of the naife quantum trace distance}: it tells us that $ D_{T} ( \rho_{1} , \rho_{2} ) $ is the maximal distance of the classical probabilities of a measurement outcome between the case in which the state before the measurement is $ \rho_{1} $ and the case in which the state before the measurement is $ \rho_{2} $. \medskip \textbf{Trace distance} is not, anyway, the only reasonable distance over $ \mathcal{D} ( M ) $ one can introduce. An other example is the following: \begin{definition} \label{def:naife classical angle distance} \end{definition} CLASSICAL ANGLE DISTANCE ON $ \mathcal{D} ( M ) $: \begin{equation} D_{A} ( \vec{p}^{(A)} , \vec{p}^{(B)} ) \; := \; \arccos F ( \vec{p}^{(A)} , \vec{p}^{(B)} ) \end{equation} where: \begin{definition} \label{def:naife classical fidelity} \end{definition} CLASSICAL FIDELITY ON $ \mathcal{D} ( M ) $: \begin{equation} F ( \vec{p}^{(A)} , \vec{p}^{(B)} ) \; := \; \sum_{i \in M} \sqrt{ p^{(A)}_{i} \, p^{(B)}_{i} } \end{equation} The name \textbf{angle distance} is justified by the following considerations: the vectors $ \vec{\xi}^{(A)} = \begin{pmatrix} \sqrt{p^{A}_{1}} \\ \vdots \\ \sqrt{p^{A}_{n}} \end{pmatrix} , \vec{\xi}^{(B)} = \begin{pmatrix} \sqrt{p^{B}_{1}} \\ \vdots \\ \sqrt{ p^{B}_{n}} \end{pmatrix} $ belong to the n-sphere of unitary radius $ S^{(n)} $. So $ D ( p_{1} , p_{2} ) $ is precisely the angle between $ \vec{\xi}_{1} $ and $ \vec{\xi}_{2} $, i.e. the geodesic distance between them on the riemannian manifold $ ( S^{(n)} , g_{S^{(n)}} ) $, $ g_{S^{(n)}} := i^{\star} \delta $ being the metric induced on $ S^{(n)} $ by its inclusion's embedding $ i : S^{(n)} \rightarrow {\mathbb{R}}^{n}$ in the euclidean space $ ( {\mathbb{R}}^{n} , \delta )$ \cite{Nakahara-95}. The quantum corrispective of definition\ref{def:naife classical angle distance} used by the Quantum Computation's community is the following: \begin{definition} \label{def:naife quantum angle distance} \end{definition} QUANTUM ANGLE DISTANCE ON $ \mathcal{D} ( { \mathcal{H}} ) $: \begin{equation} D_{A} ( \rho^{(A)} , \rho^{(B)} ) \; := \; \arccos F ( \rho^{(A)} , \rho^{(B)} ) \end{equation} where: \begin{definition} \label{def:naife quantum fidelity} \end{definition} QUANTUM FIDELITY ON $ \mathcal{D} ( { \mathcal{H}} ) $: \begin{equation} F ( \rho^{(A)} , \rho^{(B)} ) \; := \; Tr \sqrt{ \sqrt{\rho^{(A)}} \rho^{(B)} \sqrt{\rho^{(A)}}} \end{equation} A geometric interpretation of the \textbf{quantum angle distance} analogous to the classical one is furnished by the following: \begin{theorem} \label{th:first Uhlmann's theorem} \end{theorem} FIRST UHLMANN'S THEOREM: \begin{hypothesis} \end{hypothesis} \begin{equation*} \rho^{(A)} \, , \, \, \rho^{(B)} \; \in \; \mathcal{D}({\mathcal{H}}) \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} F ( \rho^{(A)} \, , \, \rho^{(B)} ) \; = \; \max_{ | \psi_{A} > \in PUR( \rho^{(A)} , {\mathcal{H}} ) \; , \; | \psi_{B} > \in PUR( \rho^{(B)} , {\mathcal{H}} )} \, | < \psi_{A} | \psi_{B} > | \end{equation*} where, given two generical Hilbert spaces $ \mathcal{H}_{A} $ and $ \mathcal{H}_{B} $ and a density matrix $ \rho \in {\mathcal{D}}( {\mathcal{H}}_{A})$ : \begin{definition} \end{definition} PURIFICATIONS OF $ \rho $ WITH RESPECT TO $ \mathcal{H}_{B} $: \begin{equation} PUR( \rho \, , \, {\mathcal{H}_{B}}) \; := \; \{ | \psi > \in {\mathcal{H}_{A}} \bigotimes {\mathcal{H}_{B}} \, : \, Tr_{{\mathcal{H}_{B}}} | \psi > \; = \; \rho \} \end{equation} So the cosin of the angle distance between two density matrices is equal to the maximum inner product between purifications of such density matrices. \smallskip The \textbf{quantum angle distance} is not, anyway, the only possible quantum corrispective of definition\ref{def:naife classical angle distance} that, as many other notions, had been extensively studied in the Mathematical-Physics' literature, many years before the Quantum Computation's community rediscovered it. Indeed the geometric interpretation of the \textbf{classical angle distance} between two distributions as the geodesic distance on the unit sphere among their square-root densities, may be seen as the first taste of \textbf{Classical Information Geometry}, namely the approach to Classical Probability Theory studying the set of all the probability measures on a given sample space from a differential-geometric viewpoint \cite{Cencov-82}, \cite{Amari-85}. Indeed, the more relevant application of Information Geometry concerns \textbf{Statistical Estimation}: given a submanifold $ N \subset {\mathbb{R}}^{m} $. \begin{definition} \label{def:classical statistical model} \end{definition} CLASSICAL STATISTICAL MODEL WITH SAMPLE SPACE M AND PARAMETER SPACE N: \begin{multline} CSM ( M , N ) \; := \; \{ \vec{\xi}(\theta) \, = \, \begin{pmatrix} \xi_{1} (\theta) \\ \vdots \\ \xi_{n} (\theta) \end{pmatrix} \: : \\ \xi_{i} (\theta) = \sqrt{p_{i} (\theta)} \, , \, \vec{p}( \vec{\theta} ) := \begin{pmatrix} p_{1} (\theta) \\ \vdots \\ p_{n} (\theta) \end{pmatrix} \in \mathcal{D} ( M ) \, \forall \theta := \begin{pmatrix} \theta_{1} \\ \vdots \\ \theta_{m} \end{pmatrix} \in N \} \end{multline} So a \textbf{classical statistical model} $ CSM ( M , N ) $ with \textbf{sample space} M and \textbf{parameter space} N is a collection of square-roots probability distributions over M parametrized through points of N. By construction $ CSM ( M , N ) $ is a submanifold of $ S^{n} $. As we will know show, definition\ref{def:naife classical angle distance} naturally induces a riemannian metric on $ CSM ( M , N ) $, called the \textbf{Fisher-Rao riemannian metric}, playing a key rule, through the Cramer-Rao Inequality, in the theory of the statistical inference of $ \vec{\theta} $ (or, more generally, a suitable function of it), from a finite set of statistical data. Given a function $ f \in C^{\infty} ( N , {\mathbb{R}}) $ and a parametrized family of classical random variables $ X(\vec{\theta} ) $ over M with distribution $ p(x | \vec{\theta} )$: \begin{definition} \label{def:unbiased estimator} \end{definition} $ X(\vec{\theta} ) $ IS AN UNBAISED ESTIMATOR OF THE FUNCTION f: \begin{equation} E[ X(\vec{\theta})] \; = \; f(\vec{\theta}) \; \; \forall \vec{\theta}\in N \end{equation} The meaning of definition\ref{def:unbiased estimator} lies in that from sampling of X we get some information about the function $ f(\vec{\theta}) $ we want to estimate. Clearly, the smaller is the variance of $ X(\vec{\theta} ) $ the higher is the classical information we gain by the estimation process. The Cramer-Rao inequality states the existence of an upper bound about such information. What is surprising in that, is the simple geometric nature underlying such a bound. Let us introduce the following: \begin{definition} \label{Fisher riemannian metric} \end{definition} FISHER-RAO RIEMANNIAN METRIC: the riemannian metric over CSM ( M , N ): \begin{equation} g_{ CSM ( M , N )} \; := \; g_{N} |_{CSM ( M , N )} \; = \; i^{\star} ( \delta ) \end{equation} where $ \delta $ is again the euclidean metric on $ {\mathbb{R}}^{n} $ while $ i : CSM ( M , N ) \rightarrow {\mathbb{R}}^{n} $ is the inclusion-embedding of $ CSM ( M , N ) $ in $ {\mathbb{R}}^{n} $. Then one has that: \begin{theorem} \label{th:Cramer Rao inequality} \end{theorem} CRAMER RAO INEQUALITY: \begin{equation} Var[ X(\vec{\theta} ) ] \; \geq \; g_{ CSM ( M , N )}^{i j} \partial_{i} f \partial_{j} f \end{equation} where we have expressed the Fisher-Rao riemannian metric through the global coordinates $ \{ \theta^{i} \} $ over N: \begin{equation} g_{ CSM ( M , N )} \; = \; ( g_{ CSM ( M , N )})_{i j} d \theta^{i} \bigotimes d \theta^{j} \end{equation} and where $ \partial_{i} f \; := \frac{\partial f}{ \partial \theta^{i}} $. \smallskip We can now appreciate how the issue of finding a quantum generalization of definition\ref{def:naife classical angle distance} fits in the more ambitious process of constructing a \textbf{Quantum Information Geometry} playing in \textbf{Quantum Estimation Theory} the same rule \textbf{Classical Information Geometry} plays in \textbf{Classical Estimation Theory}. Such a project has been pursued extensively by many authors \cite{Hasegawa-Petz-97}, \cite{Petz-Sudar-99} \cite{Brody-Hughston-98}, \cite{Streater-00b}, conceptually in the framework of Helstrom's Quantum Statistical Decision Theory for a modern presentation of which we demand to the section2.2 of \cite{Holevo-99}. All the proposed approaches, anyway, or reconduct the quantum case to the classical one (e.g. the approach by Brody and Hughston based on an application of the Fisher riemannian metric to the horizontal lift of paths in the Stiefel bundle underlying the Aharonov-Anandan geometric phase \cite{Bohm-93}) or introduce suitable riemannian geometric structures on the space of the quantum states (e.g. the riemannian geometric structure underlying Hasegawa's $ \alpha $-divergence, or Petz's monotone riemannian metrics \cite{Lesniewski-Ruskai-98} on which we will return in section\ref{sec:The problem of characterizing mathematically the notion of a quantum algorithm} on discussing the rule of the Wigner-Araki-Dyson skew information for superselection-rules). But then the considerations of remark\ref{rem:it is wrong to apply commutative geometry to noncommutative sets} apply. To sketch an idea of how \textbf{Quantum Information Geometry} should be constructed in terms of noncommutative riemannian spaces, we have to go further in the construction of the noncommutative tower. The next floor after Noncommutative Topology and Noncommutative Measure Theory is Noncommutative Differential Calculus \cite{Connes-92}, \cite{Connes-94}, \cite{Connes-98}. Commutative Differential Calculus started with Leibniz's \textbf{infinitesimals} (or equivalentely Newton's fluxions). The difficulty of furnishing a rigorous mathematical formalization of the notion of \textbf{infinitesimal} inside Commutative Analysis, led the fathers of Commutative Calculus to replace them by well-defined objects, i.e integrals, recasting the foundations of Commutative Analysis inside the boundaries of Commutative Measure Theory. In fact, though usually used as a linguistic shortcut by physicists, statements like the following: \begin{center} " Let us call $ dp(x) $ the probability that a particle is found in the interval $ [ x , x + dx ]" $ \end{center} has no rigorous meaning, as can be seen observing that the natural condition that the \textbf{commutative infinitesimal} dp(x) should satisfy, namely: \begin{equation} \label{eq:impossible constraint on a commutative infinitesimal} dp(x) \; < \; \epsilon \; \; \forall \epsilon > 0 \end{equation} obviously implies that: \begin{equation} \label{eq:catastrophic conseguence of the impossible constraint on a commutative infinitesimal} dp(x) \; = \; 0 \end{equation} The well-defined quantities are only the integrals: \begin{equation} \int dp(x) f \end{equation} of suitable functions. So the fate of the poor infinitesimal dp(x) was very sick until Robinson's Nonstandard Analysis gave it a rigorous mathematical status as a nonstandard-real, though at the price of high logico-mathematical sophistications. So it is very curious that, as we will now show the issue of defining an \textbf{infinitesimal} in a \textbf{noncommutative space} is highly simpler. Given an Hilbert space $ {\mathcal{H}} $ let us analyze if there is some natural way of defining an infinitesimal element of the noncommutative space $ {\mathcal{B}} ( \mathcal{H} ) $. Exactly as in the commutative case, then natural condition that one would require in order of considering an element $ a \in {\mathcal{B}} ( {\mathcal{H}} ) $ as an infinitesimal is that: \begin{equation} \label{eq:impossible constraint on a noncommutative infinitesimal} \| a \| \; < \; \epsilon \; \; \forall \epsilon > 0 \end{equation} But in the same way eq.\ref{eq:impossible constraint on a commutative infinitesimal} implies eq.\ref{eq:catastrophic conseguence of the impossible constraint on a commutative infinitesimal} one has that eq.\ref{eq:impossible constraint on a noncommutative infinitesimal} implies that: \begin{equation} \label{eq:catastrophic conseguence of the impossible constraint on a noncommutative infinitesimal} a \; = \; 0 \end{equation} Contrary to the commutative case, anyway, the condition of eq.\ref{eq:impossible constraint on a noncommutative infinitesimal} may be slightly modified in order of becoming meaningful, substituting the condition of eq.\ref{eq:impossible constraint on a noncommutative infinitesimal} by the condition: \begin{equation} \label{eq:constraint on a noncommutative infinitesimal} \forall \epsilon > 0 \; , \; \exists \text{ a subspace } E_{\epsilon} \subset {\mathcal{H}} \: : \: dim(E) < \infty \; and \; \| a |_{ E^{\bot} } \| \, < \, \epsilon \end{equation} Since an operator a satisfies the condition of eq.\ref{eq:constraint on a noncommutative infinitesimal} iff it is compact, it follows that the set of the infinitesimals elements of $ {\mathcal{B}} ( {\mathcal{H}} ) $ are exactly the the set $ {\mathcal{C}} ( {\mathcal{H}} ) $ of the compact operators on $ {\mathcal{H}} $: \begin{definition} \label{def:noncommutative infinitesimal} \end{definition} $ a \in {\mathcal{B}} ( {\mathcal{H}} ) $ IS INFINITESIMAL: \begin{equation} a \; \in \; {\mathcal{C}} ( {\mathcal{H}} ) \end{equation} Given a noncommutative infinitesimal $ a \in {\mathcal{C}} ( {\mathcal{H}} ) $: \begin{definition} \label{def:characteristic values of a noncommutative infinitesimal} \end{definition} $ n^{th} $ CHARACTERISTIC VALUE OF a: \begin{equation} \mu_{n} (a) \; := \; \inf \{ \| a \|_{ E^{\bot} } \, , \, dim(E) \leq n \} \end{equation} We can then classify the order of noncommutative infinitesimals in the following way: given an $ \alpha \in {\mathbb{R}}_{+} $: \begin{definition} \label{def:noncommutative infinitesimal of given order} \end{definition} INFINITESIMALS OF ORDER $ \alpha $: \begin{equation} {\mathcal{I}}_{\alpha} ({\mathcal{H}}) \; := \; \{ a \in {\mathcal{C}} ( {\mathcal{H}} ) \, : \, \mu_{n} (a) \; = \; O ( \frac{1}{ n^{\alpha}}) \; for \; n \rightarrow \infty \} \end{equation} \smallskip The next step in the construction of Noncommutative Calculus is the definition of noncommutative integration. Given a trace-class operator $ a \in {\mathcal{C}}_{1} ( {\mathcal{H}} ) $ one would be tempted to define its noncommutative integral over $ {\mathcal{B}} ( {\mathcal{H}} ) $ simply as its trace: \begin{equation} Tr(a) \; := \; \sum_{n} < \psi_{n} | a | \psi_{n} > \end{equation} that is independent from the choice of the orthonormal basis $ \{ | \psi_{n} > \} $ of $ {\mathcal{H}} $. Since the characteristic values $ \mu_{0} (a) \, \geq \, \mu_{1} (a) \, \geq \, \cdots \, \mu_{n} (a) \rightarrow 0 $ of an operator $ a \in {\mathcal{B}} ( {\mathcal{H}} ) $ are nothing but the eigenvalues of $ | a | $ one has that: \begin{equation} Tr(a) \; = \; \sum_{n=0}^{\infty} \mu_{n} (a) \; \; \forall a \in ({\mathcal{B}} ( {\mathcal{H}}) )_{+} \end{equation} But let us now observe that a reasonable notion of noncommutative integration has to satisfy the following constraints: \begin{enumerate} \item \textbf{the integral of infinitesimals of order one must converge} \begin{equation} \label{eq:the integral of infinitesimals of order one must converge} \int_{NC} a \; < \; + \infty \; \; \forall a \in {\mathcal{I}}_{1} ( {\mathcal{H}}) \end{equation} \item \textbf{the integral of infinitesimals of order greater than one must vanish} \begin{equation} \label{eq:the integral of infinitesimals of order greater than one must converge} \int_{NC} a \; < \; 0 \; \; \forall a \in {\mathcal{I}}_{\alpha} ( {\mathcal{H}}) , \alpha > 1 \end{equation} \end{enumerate} Since $ {\mathcal{I}}_{1} ( {\mathcal{H}}) \, \nsubseteq \, {\mathcal{C}}_{1} ( {\mathcal{H}} ) $ the simple trace Tr(a) doesn't satisfy the constraint of eq.\ref{eq:the integral of infinitesimals of order one must converge}. Furthermore it doesn't satisfy also eq.\ref{eq:the integral of infinitesimals of order greater than one must converge}. Hence it doesn't work. A good notion of noncommutative-integral was, instead, obtained by J. Dixmier starting from the observation that: \begin{equation} \sum_{n=0}^{\infty} \mu_{n-1} (a) \; \stackrel{ \times }{\leq} \; \log n \end{equation} So he introduced a quantity, now called the Dixmier trace, that, informally speaking, extracts the coefficient of the logarithmic divergence. Though in all the more important cases it is given simply by: \begin{equation} \label{eq:naife Dixmier trace} \lim_{n \rightarrow \infty} \frac{1}{\log n } \sum_{n=0}^{n-1} \mu_{n} (a) \end{equation} in the general case such an expression presents two problems: its linearity and its convergence. Given an infinitesimal $ a \in {\mathcal{C}}( {\mathcal{H}} ) $ let us consider the argument of the limit at the r.h.s. of eq.\ref{eq:naife Dixmier trace}, namely: \begin{equation} \gamma_{n} (a) \; := \; \frac{1}{\log n } \sum_{n=0}^{n-1} \mu_{n} (a) \end{equation} Since it obeys the relation \cite{Connes-94}: \begin{equation} \label{eq:asymptotic additivity a Dixmier sequence} \gamma_{n} (a_{1} + a_{2}) \; \leq \; \gamma_{n} (a_{1}) + \gamma_{n} (a_{2}) \; \leq \; \gamma_{2 n} (a_{1} + a_{2}) ( 1 + \frac{ \log 2}{ \log n } ) \end{equation} we see that linearity would follow from convergence. Unfortunately, though always bounded, the sequence $ \{ \gamma_{n} \} $ doesn't always converge. Considered the Banach space of bounded sequences $ l^{\infty} ( {\mathbb{N}} ) $ let us introduce the space of all the linear forms $ \lim_{\omega} $ on it such that: \begin{enumerate} \item \begin{equation} \gamma_{n} \, \leq \, 0 \; \Rightarrow \; \lim_{\omega} \gamma_{n} \, \leq \, 0 \end{equation} \item \begin{equation} \exists \lim_{ n \rightarrow + \infty} \gamma_{n} \; \Rightarrow \; \lim_{\omega} \gamma_{n} \, = \, \lim_{ n \rightarrow + \infty} \gamma_{n} \end{equation} \item \begin{equation} \lim_{\omega} \{ (\gamma_{n} ) ^{n} \} \; = \; \lim_{\omega} \gamma_{n} \end{equation} \item \begin{equation} \lim_{\omega} \gamma_{2 n} \; = \; \lim_{\omega} \gamma_{2 n} \end{equation} \end{enumerate} To each of such linear forms $ \lim_{\omega} $ (they are infinite) it is associated a \textbf{Dixmier trace}, according to the following: \begin{definition} \label{def:Dixmier trace} \end{definition} DIXMIER TRACE OF $ a \, \in \, ({\mathcal{B}} ( {\mathcal{H}} ))_{+} \, \bigcap \, {\mathcal{I}}_{1} ( {\mathcal{H}} ) $: \begin{equation} tr_{\omega} (a) \; := \; \lim_{\omega} \gamma_{n} \; = \; \lim_{\omega} \frac{1}{\log n } \sum_{n=0}^{n-1} \mu_{n} (a) \end{equation} By eq.\ref{eq:asymptotic additivity a Dixmier sequence} it follows that a Dixmier trace is additive on positive infinitesimals of order one, so that, owing to eq.\ref{eq:asymptotic additivity a Dixmier sequence}, it can be extended by linearity to the whole $ {\mathcal{I}}_{1} ( {\mathcal{H}} ) $. That a Dixmier trace is indeed a trace,i.e. that: \begin{equation} tr_{\omega} ( \alpha(a) ) \; = \; tr_{\omega} (a) \; \; \forall a \in {\mathcal{I}}_{1} ( {\mathcal{H}} ) , \forall \alpha \in INN ( {\mathcal{B}}( {\mathcal{H}} )) \end{equation} follows immediately by the unitary invariance of the characteristic values of an infinitesimal a owed to the fact that they are nothing but the eigenvalues of $ | a | $. Since any linear form $ lim_{\omega} $ assumes only finite values, a Dixmier trace satisfies by construction the first constraint, namely eq.\ref{eq:the integral of infinitesimals of order one must converge}, we required for a reasonable notion of noncommutative integration. Furthermore, since the space of all infinitesimal of order higher than one is a two-sided ideal whose elements satisfy the condition: \begin{equation} \lim_{n \rightarrow \infty} \mu_{n} (a) \; = \; 0 \end{equation} and so: \begin{equation} \lim_{n \rightarrow \infty} \gamma_{n} \end{equation} it follows that a Dixmier trace satisfies also the second constraint, namely eq.\ref{eq:the integral of infinitesimals of order greater than one must converge}, we ask to a noncommutative integral. \smallskip After \textbf{noncommutative integration} let us pass to \textbf{noncommutative differentiation}. Let us observe, at this purpose, that a key rule of commutative differentiation is given by Leibniz's rule for the differential of products: \begin{equation}\label{eq:Leibniz rule} d ( f_{1} f_{2} ) \; = \; d ( f_{1} ) f_{2} \, + \, f_{1} d ( f_{2} ) \end{equation} So it appears natural to attempt to characterize the notion of noncommutative differentiation imponing that eq.\ref{eq:Leibniz rule} holds in the noncommutative case too. The resulting notion, introduced by Kaplansky in 1953, is that of a derivation \cite{Sakai-91}, namely the following: \begin{definition} \label{def:derivation on a C-star algebra} \end{definition} DERIVATION ON A $C^{\star}$-ALGEBRA A: a linear operator $ \delta \, : D( \delta ) \rightarrow A $ from a $ \star $-subalgebra $ D( \delta ) $ to A such that: \begin{equation} \delta (a b ) \; := \; \delta (a ) b \, + \, a \delta ( b ) \; \; \forall a , b \in D( \delta ) \end{equation} Given a derivation $ \delta $ on a $C^{\star}$-algebra A: \begin{definition} \label{def:involutive derivation on a C-star algebra} \end{definition} $ \delta $ IS AN INVOLUTIVE DERIVATION ( $ \star $- DERIVATION ): \begin{equation} \delta ( a^{\star} ) \; = \; \delta (a)^{\star} \; \; \forall a \in D( \delta) \end{equation} \smallskip Let us now observe that we are used, by Functional Analysis, to the fact that the assignment of a one-parameter strongly continuous group (or semigroup) of operators is equivalent to the assignment of its generator: \begin{itemize} \item by the Stone's Theorem \cite{Reed-Simon-80} we know that the assignment of a strongly continuous one-parameter group U(t) of unitary operators on an Hilbert space $ {\mathcal{H}} $ is equivalent to the assignment of its generator, namely the (unique) self-adjoint operator A such that: \begin{equation} U(t) \; = \; e^{i t A} \; \; \forall t \in {\mathbb{R}} \end{equation} \item by the Hille-Yosida's Theorem \cite{Reed-Simon-75} we know that the assignment of a strongly continuous one-parameter semigroup C(t) of contractions on an Hilbert space $ {\mathcal{H}} $ is equivalent to the assignment of its generator, namely the (unique) self-adjoint operator A such that: \begin{equation} C(t) \; = \; e^{- t A} \; \; \forall t \in {\mathbb{R}}_{+} \end{equation} \end{itemize} So we are not surprised that a similar situation occurs also for strongly continuous one-parameter subgroups of the automorphisms' group AUT(A) of a generic $ C^{\star} $ algebra. Demanding to the paragraph3.4 of \cite{Sakai-91} for details it is sufficient here to recall that exactly as Edward Nelson's notion of analytic vectors allows to construct directly the exponential $ e^{A} $ of a self-adjoint operator A as a power series, the same happens in our operator-algebraic setting allowing to define, as a series-power, the exponential $ e^{\delta} $ of an involutive derivation $ \delta $ on a $ C^{\star}$-algebra. Then one has the following: \begin{theorem} \label{th:on the generators of strongly continuous one-parameter groups of automorphisms} \end{theorem} ON THE GENERATORS OF STRONGLY CONTINUOUS ONE-PARAMETER GROUPS OF AUTOMORPHISMS \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; \; C^{\star}-algebra \end{equation*} \begin{equation*} (\alpha_{t} )_{t \in {\mathbb{R}}} \text{strongly continuous one-parameter subgroup of AUT(A)} \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} \exists ! \: \delta \text{ $\star$-derivation on A } \; : \; ( \alpha_{t} \: = \: e^{t \delta} \; \; \forall t \in {\mathbb{R}} ) \end{equation*} \smallskip Let us then consider the particular case of one-parameter groups of inner automorphisms. Theorem\ref{th:on the generators of strongly continuous one-parameter groups of automorphisms} immediately implies the following: \begin{corollary} \label{th:on the generators of strongly continuous one-parameter groups of inner automorphisms} \end{corollary} \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; \; C^{\star}-algebra \end{equation*} \begin{equation*} (\alpha_{t} )_{t \in {\mathbb{R}}} \text{ strongly continuous one-parameter subgroup of INN(A)} \end{equation*} \begin{equation*} \delta \text{ generator of the group } (\alpha_{t} )_{t \in {\mathbb{R}}} \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} \exists ! \; D \in A_{sa} \; : \; \delta( a ) \: = \: i \, [ D , a ] \; \forall a \in A \end{equation*} \smallskip Let us now introduce the following notion: \begin{definition} \label{def:spectral triple} \end{definition} SPECTRAL TRIPLE: a therne $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ such that: \begin{itemize} \item $ {\mathcal{H}} $ is an Hilbert space \item $ A \subseteq {\mathcal{B}}( {\mathcal{H}} ) $ is a $ \star $ - subalgebra of $ {\mathcal{B}}( {\mathcal{H}} ) $ \item D is a self-adjoint operator on $ {\mathcal{H}} $ such that: \begin{align*} [ & D , a ] \; \in \; {\mathcal{B}}( {\mathcal{H}} ) \; \; \forall a \in A \\ ( & D - \lambda )^{- 1} \; \in \; {\mathcal{C}}( {\mathcal{H}} ) \; \; \forall \lambda \in {\mathbb{C}} - {\mathbb{R}} \end{align*} \end{itemize} Given a spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $: \begin{definition} \label{def:even spectral triple} \end{definition} $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ IS EVEN: there is a $ {\mathbb{Z}}_{2} $ grading on $ {\mathcal{H}}$,i.e. an operator $ \Gamma $ on $ {\mathcal{H}} $ such that: \begin{align} \Gamma^{\star} & \; = \; \Gamma \\ \Gamma^{2} & \; = \; 1 \\ \{ \Gamma & , D \} \; := \; \Gamma D \, - \, D \Gamma \; = \; 0 \\ [ \Gamma & , a ] \; = \; 0 \; \; \forall a \in A \end{align} \begin{definition} \label{def:odd spectral triple} \end{definition} $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ IS ODD: it is not even \smallskip Given an $ n > 0 $: \begin{definition} \label{def:dimension of a spectral triple} \end{definition} $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ HAS DIMENSION $ n \; \; ( \, dim([ \, ( A \, , \, {\mathcal{H}} \, , \, D ) \, ]) \; = \; n ) \, $: \begin{equation} | D |^{- 1} \in {\mathcal{I}}_{ \frac{1}{n} } ( {\mathcal{H}} ) \end{equation} \smallskip We can at last formalize all the previously machinery about noncommutative differentiation and integration in the following way: given an n-dimensional spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ \begin{definition} \label{def:noncommutative integration in a spectral triple} \end{definition} NONCOMMUTATIVE INTEGRAL OF $ a \in A$: \begin{equation} \int_{NC} a \; := \frac{1}{V} tr_{\omega} a | D |^{- n} \end{equation} where V is a normalization factor such that: \begin{equation} \int_{NC} I \; := \frac{1}{V} tr_{\omega} | D |^{- n} \; = \; 1 \end{equation} By the previously discussed properties of the Dixmier trace one has that: \begin{theorem} \label{th:basic properties of the noncommutative integral in a spectral triple} \end{theorem} BASIC PROPERTIES OF THE NONCOMMUTATIVE INTEGRAL IN A SPECTRAL TRIPLE: \begin{enumerate} \item \begin{equation} \int_{NC} a b \; = \; \int_{NC} b a \; \; \forall a , b \in A \end{equation} \item \begin{equation} \int_{NC} a^{\star} a \; \geq \; 0 \; \; \forall a \in A \end{equation} \item \begin{equation} \int_{NC} a \; = \; 0 \; \; \forall a \in {\mathcal{I}}_{\alpha} ( {\mathcal{H}} ) \, , \, \alpha > 1 \end{equation} \end{enumerate} Then: \begin{definition} \label{def:noncommutative differential in a spectral triple} \end{definition} NONCOMMUTATIVE DIFFERENTIAL OF $ a \in A$: \begin{equation} d_{NC} a \; := \; [ D , a ] \end{equation} \begin{example} \end{example} QUANTIZED CALCULUS ON THE CIRCLE AND MANDELBROT'S SET Let us consider the spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $, where: \begin{itemize} \item \begin{equation} {\mathcal{H}} \; := \; L^{2} ( S^{(1)} , d \vec{x}_{Lebesgue} ) \end{equation} \item \begin{equation} A \; := \; L^{\infty} ( S^{(1)} , d \vec{x}_{Lebesgue} ) \end{equation} where a function $ f \in A $ is seen as a multiplication operator: \begin{equation} ( f \psi ) (t) \; := \; f(t) \psi (t) \; \; f \in A , \psi \in {\mathcal{H}} \end{equation} \item D is the linear operator on $ {\mathcal{H}} $ defined by: \begin{equation} D e_{n} \; := \; sign(n) e_{n} \: , \: e_{n} ( \theta ) \; := e^{ i n \theta } \; \forall \theta \in S^{(1)} \end{equation} \end{itemize} Let us now consider again the quadratic maps on the complex plane $ p_{c} (z) \, := \, z^{2} + c $ we considered in section\ref{sec:Brudno algorithmic entropy versus the Uspensky abstract approach}. Demanding to $ 14^{th} $ chapter of \cite{Falconer-90} for details it will be sufficient to our purposes to remind that the Julia set of the application $ p_{c} (z) $ may be simply expressed as: \begin{equation} J [ p_{c} (z) ] \; = \; \partial \, \{ z \in {\mathbb{C}} : \sup_{n \in {\mathbb{N}}} | p_{c}^{(n)} (z)| < \infty \} \end{equation} and that it may be proved that there exist an homeomorphism $ Z : S^{(1)} \, \rightarrow J [ p_{c} (z) ] $. Denoted by D the Hausdorff dimension of $ J [ p_{c} (z) ] $ Alain Connes was able to prove that: \begin{enumerate} \item $ | d_{NC} Z | $ is an infinitesimal of order $ \frac{1}{D} $ \item \begin{equation}\label{eq:integration of continuous functions on Julia's set} \exists \lambda > 0 \; : \; ( \int_{J [ p_{c} (z) ]} f d \Lambda_{D} ) \, = \; \lambda \int_{NC} f(Z) | D |^{ - 1} | d_{NC} Z |^{D} \; \; \forall f \in C( J [ p_{c} (z) ] ) \end{equation} where $ d \Lambda_{D} $ is the Hausdorff measure on $ J [ p_{c} (z) ] $. \end{enumerate} The eq.\ref{eq:integration of continuous functions on Julia's set} tells us that the integral w.r.t. the Hausdorff measure of continuous functions over the Julia set $ J [ p_{c} (z) ] $ may be computed as a noncommutative integral in the spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $. Since the Mandelbrot's set $ {\mathcal{M}} $ we introduced by definition\ref{def:Mandelbrot set} is linked to the family of Julia sets $ J [ p_{c} (z) ] $ by the condition: \begin{equation} {\mathcal{M}} \; = \; \{ c \in {\mathbb{C}} \, : \, J [ p_{c} (z) ] \text{ is connected } \} \end{equation} eq.\ref{eq:integration of continuous functions on Julia's set} could be useful to investigate some of the still unknown properties of $ {\mathcal{M}} $ \medskip So, up to this point, we have seen how the notion of a spectral triple implements Noncommutative Calculus. We shall now see that it, indeed, makes much more: it implements Noncommutative Riemannian Geometry. Demanding to \cite{Nakahara-95}, \cite{Gilkey-95}, \cite{Landi-97}, \cite{Esposito-98}, \cite{Gracia-Bondia-Varilly-Figueroa-01} for details, let us recall that a \textbf{spin structure} on an n-dimensional riemannian manifold $ ( M , g ) $ is a lifting of its orthonormal frame bundle $ O(M) \stackrel{\pi}{\rightarrow} M $ to a bundle $ S(M) \stackrel{\pi}{\rightarrow} M $, said a \textbf{spin bundle} over M, in which the structure group $ O(n) $ is replaced by its universal covering group (that is by definition the spin group SPIN(n)) through the substitution of the transition functions $ t_{ij} $ by new transition functions $ \tilde{t}_{ij} $ such that: \begin{equation} \phi ( \tilde{t}_{ij} ) \; = \; t_{ij} \end{equation} where $ \phi : SPIN(n) \mapsto SO(n) $ is the double covering. \begin{definition} \label{def:spin manifold} \end{definition} SPIN MANIFOLD: a riemannian manifold $ ( M , g ) $ which admit a spin structure A well known theorem of riemannian geometry states that $ ( M ,g ) $ is a spin manifold iff its first two Stiefel-Whitney classes $ w_{1}(M) $ and $ w_{2}(M) $ vanish (the vanishing of $ w_{1}(M) $ being equivalent to the orientability of M) In this case it may, of course, admit different spin structures, corresponding to different choices of the transition functions $ \tilde{t}_{ij} $. \smallskip Given an n-dimensional spin manifold $ ( M , g ) $ let us consider a section $ \{ e_{a} , a = 1, \cdots , n \} $ of its orthonormal frame bundle and let us relate it to the natural basis $ \{ \partial_{\mu} \} $ by the n-beins, with components $ e_{a}^{\mu} $, so that the components $ \{ g^{\mu \nu} \} $ of the metric g and the components $ \{ \eta^{a b} \} $ of the flat metric over M may be related by the equations: \begin{align} g^{\mu \nu} & \; = \; e_{a}^{\mu} e_{b}^{\nu} \eta^{a b} \\ \eta^{a b} & \; = \; e_{a}^{\mu} e_{b}^{\nu} g_{\mu \nu} \end{align} We will assume, from here and beyond, that the curve indices $ \{ \mu \} $ are raised and lowered by the curved metric g while the flat indices $ \{ a \} $ are raised and lowered by the flat metric $ \eta $. Denoted by $ \nabla $ the Levi-Civita connection of $ ( M , g ) $ (i.e. the unique torsion free affine connection on M that is compatible with g) let us introduce its connection coefficients $ \omega_{ \mu a }^{b} $ defined by the condition \begin{equation} \omega_{ \mu a }^{b} e^{b} \; := \; \nabla_{\mu} e_{a} \; \; \mu , \nu = 1 , \cdots , n \end{equation} Let us now introduce the Clifford bundle $ C(M) \stackrel{\pi}{\rightarrow} $ over M, whose fiber at $ x \in M $ is the complexified Clifford algebra $ Cliff_{{\mathbb{C}}} $, and the space $ \Gamma ( M , C(M) ) $ of sections on it. Called $ {\mathcal{H}} \, := \, L^{2} ( M , S ) $ the Hilbert space of the irreducible spin bundle over M, with the inner product given by: \begin{equation} < \psi_{1} | \psi_{2} > \; := \; \int_{M} d \mu (g) \bar{\psi}_{1}( x ) \psi_{2} (x) \end{equation} let us now introduce that the map $ \gamma \, : \, \Gamma ( M , C(M) ) \; \rightarrow \; {\mathcal{B}} ( {\mathcal{H}} ) $ defined by the condition: \begin{equation} \label{eq:curved Dirac matrices} \gamma^{\mu} (x) \; := \;\gamma ( dx^{\mu} ) \; := \; \gamma^{a} e_{a}^{\mu} \end{equation} and extended as an algebra map requiring its linearity under linear combinations with coefficients taking values in the algebra $ A \; := \; {\mathcal{F}} (M) $ of complex valued smooth functions over M. $ \Gamma ( M , C(M) ) $ is as $ \star $-algebra and $ \gamma $ is an involutive morphism. By the definition of a Clifford algebra and by eq.\ref{eq:curved Dirac matrices} the curved Dirac matrices $ \{ \gamma^{\mu}(x) \} $ and the flat Dirac matrices $ \{ \gamma^{a} \} $ obey the relations: \begin{align}\label{eq:Clifford structure on Dirac matrices} \gamma^{\mu}(x) & \gamma^{\mu}(x) \, + \, \gamma^{\nu}(x) \gamma^{\mu}(x) \; = \; -2 g( dx^{\mu} , dx^{\nu} ) \; = \; - 2 g^{\mu \nu} \; \; \mu , \nu = 1 , \cdots , n \\ \gamma^{a} & \gamma^{b} \, + \, \gamma^{b} \gamma^{a} \; = \; - 2 \eta^{a b} \; \; a , b = 1 , \cdots , n \end{align} The lift of the Levi-Civita connection to the bundle of spinors is then: \begin{equation} \nabla^{S}_{\mu} \; = \; \partial_{\mu} \, + \, \omega^{S}_{\mu} \; = \; \partial_{\mu} + \frac{1}{4} \omega_{\mu a b} \gamma^{a} \gamma^{b} \end{equation} We can then introduce the following: \begin{definition} \label{def:Dirac operator} \end{definition} DIRAC OPERATOR: the linear operator D on the Hilbert space $ {\mathcal{H}} $ given by: \begin{equation} D \; := \; \gamma \circ \nabla^{S} \end{equation} The properties of the Dirac operator justify the following: \begin{definition} \label{def:canonical spectral triple of a spin manifold} \end{definition} CANONICAL SPECTRAL TRIPLE OF $ ( M , g )$: the n-dimensional spectral triple $ ( A , {\mathcal{H}} , D ) $, where (we recall that): \begin{enumerate} \item $ A \; := \; {\mathcal{F}} (M) $ is the algebra of all complex valued smooth functions on M \item $ {\mathcal{H}} $ is the Hilbert space of square integral sections (w.r.t. the the metric measure $ d \mu (g) $ ) of the irreducible spinor bundle over M \item D is the Dirac operator of $ ( M , g ) $ \end{enumerate} If n is even the canonical spectral triple of $ ( M , g ) $ is even, the $ {\mathbb{Z}}_{2} $-grading being given by: \begin{equation} \Gamma \; := \; \gamma^{n+1} \; := \; i^{\frac{n}{2}} \gamma^{1} \cdots \gamma^{n} \end{equation} \smallskip And now comes the first astonishing fact: \begin{theorem} \label{th:the canonical spectral triple of a spin manifold encodes all its riemannian structure} \end{theorem} THE CANONICAL SPECTRAL TRIPLE OF A SPIN MANIFOLD ENCODES ALL ITS RIEMANNIAN STRUCTURE \begin{enumerate} \item the geodesic distance d( p , q ) of two points of $ ( M , g ) $ is given by: \begin{equation}\label{def:geodesic distance in algebraic terms} d(p,q) \; = \; \sup_{f \in A} \{ | f(p) - f(q) | \, : \, \| [ D , f ] \| \leq 1 \} \; \; \forall p ,q \in M \end{equation} \item the integration of a function $ a \in A $ w.r.t. the metric measure of $( M , g) $ is substantially given by its noncommutative integral: \begin{equation} \int_{M} d \mu ( g ) a \; = \; c(n) \int_{NC} a \; \; \forall a \in A \end{equation} where: \begin{equation} c(n) \; := \; 2^{ n - \lfloor \frac{n}{2} \rfloor - 1} \, \pi^{\frac{n}{2}} \, \Gamma ( \frac{n}{2} ) \end{equation} \end{enumerate} By a suitable definition of an equivalence relation among spectral triples, also the converse holds, i.e., given a commutative spectral triple $ ( \, A \, , \, {\mathcal{H}} \, , \, D ) $ there exist a closed finite-dimensional riemannian spin manifold $ ( M , g ) $ whose canonical spectral triple is equivalent to $ ( \, A \, , \, {\mathcal{H}} \, , \, D ) $. Furthermore: \begin{itemize} \item given a diffeomorphism $ \phi \in Diff(M) $ of a closed finite-dimensional riemannian spin manifold $ ( M , g ) $ we may associate to it the automorphism $ \alpha_{\phi} $ of the corrispondent involutive algebra $ A \, := \, {\mathcal{F}}(M) $, defined as: \begin{equation} \label{eq:automorphism associated to a diffeomorphism} \alpha_{\phi} (f) (x) \; := \; f( \phi^{- 1}(x)) \; \; \forall f \in {\mathcal{F}}(M) , \forall x \in M \end{equation} \item given an automorphism $ \alpha \in AUT(A) $ of a commutative spectral triple $ ( \, A \, , \, {\mathcal{H}} \, , \, D ) $ there exist a diffeomorphism $ \phi \in Diff(M) $ of the associated closed finite-dimensional riemannian spin manifold $ ( M , g ) $ such that $ \alpha \; = \; \alpha_{\phi} $ \end{itemize} \smallskip All these results may then be enunciated in the abstract language of Categories' Theory as the following: \begin{conjecture} \label{con:the category isomorphism at the basis of Noncommutative Geometry} \end{conjecture} THE CATEGORY EQUIVALENCE AT THE BASIS OF NONCOMMUTATIVE GEOMETRY The \textbf{category} having as \textbf{objects} the \textbf{closed finite-dimensional riemannian spin manifolds} and as \textbf{morphisms} the \textbf{diffeomorphisms of such manifolds} is equivalent to the \textbf{category} having as \textbf{objects} the \textbf{abelian spectral triples} and as \textbf{morphisms} the \textbf{automorphisms of the involved involutive algebras}. \smallskip We can now, at last, see how Noncommutative Geometry allows to afford Quantum Information Geometry. Given a spectral triple $ ( A , {\mathcal{H}} , D ) $ over the $ W^{\star}-algebra $ A by eq.\ref{def:geodesic distance in algebraic terms} and conjecture\ref{con:the category isomorphism at the basis of Noncommutative Geometry} it results natural to define the following noncommutative generalization of the geodesic distance among probability distributions: \begin{definition} \label{def:noncommutative geodesic distance among states} \end{definition} NONCOMMUTATIVE GEODESIC DISTANCE AMONG $ \omega_{1} \in S(A) $ AND $ \omega_{2} \in S(A) $ : \begin{equation} d ( \omega_{1} \, , \, \omega_{2} ) \; := \; \sup_{ a \in A } \{ | \omega_{1} (a) - \omega_{2} (a) | \, : \, \| [ D , a ] \| \leq 1 \} \end{equation} So, given a Von Neumann algebra $ A \, \subseteq \, {\mathcal{B}} ({\mathcal{H}}) $, our genuinely noncommutative approach to Quantum Information Geometry is completely specified if we succeed in individuating a \emph{"natural"} \textbf{noncommutative Dirac operator} to use in order of defining the correct spectral triple $ ( A \, , \, {\mathcal{H}} \, , \, D ) $ to use for computing the distance among noncommutative probability measures by eq.\ref{def:noncommutative geodesic distance among states}. Let us consider, at this purpose, the commutative case: given a spin manifold $ ( M , g) $ the correct operator to use in order to obtain the correct expression for the geodesic distance, namely the Dirac operator D of (M ,g ), is that obtained minimizing the action : \begin{equation} \label{def:commutative spectral action} S ( x , \Lambda ) \; := \; Tr_{ L^{2}(M,S) } ( \chi ( \frac{ x^{2}}{ \Lambda^{2}} ) ) \end{equation} where $ \Lambda $ is a cut-off with the dimensions of the inverse of a length, $ \chi $ is a proper cut-off function throwing away the contribution of the $ x ^{2} $'s eigenvalues greater than $ \Lambda ^{2} $ that we will assume to be Heaviside's step function and x denotes the unknown operatorial quantity. The action of eq.\ref{def:commutative spectral action} is defined on the set $ OP[ {\mathcal{F}}(M) \, , \, L^{2}(M,S) ] $ of all the self-adjoint operators x on $ L^{2}(M,S) $ such that $ ( {\mathcal{F}}(M) \, , \, L^{2}(M,S) \, , \, x ) $ is a spectral triple. The meaning of the cut-off $ \Lambda $ for the variational problem of eq.\ref{def:commutative spectral action} is the following: \begin{enumerate} \item one impose the variational condition: \begin{equation} \frac{ \delta S ( x , , \Lambda )}{ \delta x } \; = \; 0 \end{equation} obtaining an equation of the form: \begin{equation} \label{eq:cut-offed equation for the operator} F_{1} [ x , \Lambda ] \; = \; 0 \end{equation} where $ F_{1} $ is a certain functional of the unknown quantity x and the cut-off $ \Lambda $ \item one takes the limit $ \Lambda \rightarrow \infty $ in eq.\ref{eq:cut-offed equation for the operator} obtaining a new equation of the form: \begin{equation} \label{eq:equation for the operator} F_{2} [ x ] \; = \; 0 \end{equation} where $ F_{2} $ is another functional of the only unknown operator x \end{enumerate} By conjecture\ref{con:the category isomorphism at the basis of Noncommutative Geometry} is appears then natural to generalize noncommutatively such a variational procedure, i.e. to choose the operator x by which to compute, via eq.\ref{def:noncommutative geodesic distance among states}, the noncommutative geodesic distance between two states as the solution of the variational problem for the following: \begin{definition} \label{def:noncommutative spectral action} \end{definition} NONCOMMUTATIVE SPECTRAL ACTION FOR $ ( A , {\mathcal{H}}) $: the map $ S \, : \, OP[ {\mathcal{H}} \, , \, A ] \: \rightarrow \: {\mathbb{R}} $ given by: \begin{equation} S [ x \, , \, \Lambda ] \; := \; Tr_{{\mathcal{H}}} ( \chi ( \frac{ x^{2}}{ \Lambda^{2}} ) ) \end{equation} where: \begin{equation} OP[ {\mathcal{H}} \, , \, A ] \; := \; \{ x \, : \, ( {\mathcal{H}} \, , \, A \, , \, x ) \text{ is a spectral triple } \} \end{equation} \smallskip \begin{example} \end{example} DISTANCE OF A NONCOMMUTATIVE PROBABILITY DISTRIBUTION ON THE NONCOMMUTATIVE SPACE OF QUBITS' SEQUENCES BY THE UNBIASED ONE. Let us apply our noncommutative-geometric approach to Quantum Information Geometry to answer the following question: \smallskip how much a given noncommutative probability distribution $ \omega \in S( \Sigma_{NC}^{\infty}) $ differs from the unbaised one $ \tau_{unbaised} $? \smallskip According to our strategy such a distance is given by: \begin{equation} d ( \omega \, , \, \tau_{unbiased} ) \; := \; \sup_{ \bar{x}_{NC} \in \Sigma_{NC}^{\infty} } \{ | \omega (\bar{x}_{NC}) - \tau_{unbiased}( \bar{x}_{NC} ) | \, : \, \| [ D , \bar{x}_{NC} ] \| \leq 1 \} \end{equation} where D is the element of $ OP[ {\mathcal{H}}_{ \tau_{{\mathbb{Z}}}} \, , \, \Sigma_{NC}^{\infty} ] $ minimizing the spectral action of definition\ref{def:noncommutative spectral action}. \medskip Leaving, anyway, aside Quantum Information Geometry, let us investigate a more direct strategy of Quantum Statistical Inference, namely Quantum Bayesianism. The issue of formalizing Quantum Bayesian Theory has been recentely analyzed by various authors \cite{Ozawa-97}, \cite{Schack-Brun-Caves-00}, \cite{Caves-Fuchs-Schack-01}. Unfortunately no one of them adopts the general language of Quantum Probability Theory, on which the mathematical foundations of the whole matter, from a less empirical point of view, lies. An exeption to this attitude is the authorithative exposition of Quantum Statistical Decision Theory made by Alexander S. Holevo in the section2.2 of \cite{Holevo-99}, from which, as already happened as to Quantum Information Geometry, we will implicitly move away. Though well-knowing that, as we will see, Bayes' Formula is nothing but a matter of conditional expectations, Holevo keeps away from its natural noncommutative generalization, namely definition\ref{def:Bayes formula}. Indeed, according to him: \begin{center} \textit{"Conditional expectations play a less important part in quantum than in classical probability, since in general the conditional expectation into a given subalgebra $ {\mathcal{B}} $ with respect to a given state S exists only if $ {\mathcal{B}} $ and S are related in a very special way which in a sense reduces the situation to the classical one; for more see n. 1.3 in Chapter 3." (extracted from section1.3.2 of \cite{Holevo-99})} \end{center} Holevo's argument, clarified in section3.1.3, is based on Takesaki's theorem, namely our theorem\ref{th:Takesaki's theorem}; while the conseguences we will infer from this theorem we be essentially: \begin{enumerate} \item the impossibility of quantum-bayesian-subjectivism (as first remarked by Miklos Redei: cfr. the seection 8.2 of \cite{Redei-98}) \item the existence of the constraint\ref{def:constraint for the feasibility of the bayesian statistical inference} \end{enumerate} Holevo's claim that Takesaki's Theorem implies a reduction to the classical case is wrong, being based on the observation that, in our terminology, the constraint\ref{def:constraint for the feasibility of the bayesian statistical inference}, is certainly satisfied if the modular operator of S belongs to the commutant of $ {\mathcal{B}} $; such a condition, though sufficient, is indeed far from being necessary. \smallskip Let us so start to analyze one of the deepest conseguences of \textbf{Quantum Probability Theory}: the \textbf{Bayesian Statistical Inference Theory}: \bigskip Let us consider a statistician having access only to the information concerning the all algebraic random variables belonging to a $W^{\star}$-sub-algebra $ A_{accessible} $ of a quantum probability space $ ( A \, , \, \omega ) $. This means that he doesn't know the state $ \omega \in S(A) $ but only its restriction to the algebra $ A_{accessible} $ he can test, i.e.: \begin{equation} \omega_{accessible} \; := \; \omega \, |_{ A_{accessible} } \; \in S ( A_{accessible} ) \end{equation} \medskip Let us suppose that, at the beginning, he hasn't used even this partial available information: according to the Bayesian Theory the best estimation of the true state $ \omega $ that he can make in this situation is to assume as estimation the uniform algebraic probability distribution: \begin{equation} \omega_{A \; PRIORI} \; := \; \tau_{unbiased} \end{equation} \medskip Here it does arises the first problem, common both to the classical case and to the quantum case: the \textbf{canonical trace} $ \tau_{unbiased} $ exists if and only if the Von Neumann algebra A is finite. Supposed, anyway, that this is the case let us consider the \textbf{statistical-inference's problem }: \textbf{which is the optimal way by which the statistician can improve his estimation of the true state $ \omega $ using the information that is available to him, i.e. using $ \omega_{accessible} $ ?} The answer of the Bayesian Theory is inclosed in the following: \begin{definition} \label{def:Bayes formula} \end{definition} BAYES FORMULA: \begin{equation} \omega_{A \; PRIORI} ( \cdot ) \, = \, \tau_{unbiased} ( \cdot ) \; \rightarrow \omega_{A \; POSTERIORI} ( \cdot ) \; := \; \omega_{accessible} ( E_{unbiased} \cdot ) \end{equation} where $ E_{unbiased} \, : \, A \mapsto A_{accessible} $ is the \textbf{conditional expectation w.r.t. $ A_{accessible} \; \; \tau_{unbiased}$-invariant} whose definition and properties we are going to introduce. Given a $ W^{\star}$-algebra A: \begin{definition} \label{def:conditional expectation on a Von Neumann algebra} \end{definition} CONDITIONAL EXPECTATION ON A W.R.T. $ A_{accessible} $: a linear map $ E \, : \, A \; \rightarrow A_{accessible} $ such that: \begin{enumerate} \item \begin{equation} E(a) \; \geq \; 0 \; \; \forall a \in A_{+} \end{equation} \item \begin{equation} E(a) \; = \; a \; \; \forall a \in A_{accessible} \end{equation} \item \begin{equation} E ( a b ) \; = \; E(a) b \; \; \forall a \in A , \forall b \in A_{accessible} \end{equation} \end{enumerate} Let us suppose that the Von Neumann algebra A and its subalgebra $ A_{accessible} $ act on the Hilbert space $ {\mathcal{H}} $, i.e. $ A_{accessible} \; \subset \; A \; \subseteq \; {\mathcal{B}} ({\mathcal{H}}) $. We will say that: \begin{definition} \label{def:injective Von Neumann} \end{definition} A IS INJECTIVE: \begin{equation} \exists \, E : {\mathcal{B}} ({\mathcal{H}}) \rightarrow A \text{ conditional expectation on } {\mathcal{B}} ({\mathcal{H}}) \; w.r.t. \; A \end{equation} In an epoch making result of 1976 Alain Connes proved that: \begin{theorem} \label{th:equivalence of injectivity and hyperfiniteness} \end{theorem} \begin{equation} injectivity \; \Leftrightarrow \; hyperfiniteness \end{equation} Given a conditional expectation on A w.r.t. to $ A_{accessible} $ and a state $ \omega \in S(A) $: \begin{definition} \label{def:state preserving conditional expectation on a Von Neumann algebra} \end{definition} E IS $ \omega $-PRESERVING: \begin{equation} \omega \circ E \; = \; E \end{equation} The issue about the existence of state-preserving conditional expectations is rather subtle involving the Tomita-Takesaki Modular Theory \cite{Araki-97}. Given a Von Neumann algebra A acting on a separable Hilbert space $ {\mathcal{H}} $ and a vector $ | \psi > \in {\mathcal{H}} $: \begin{definition} \label{def:cyclic vector for a Von Neumann algebra} \end{definition} $ | \psi > $ IS CYCLIC FOR A: \begin{equation} A \, | \psi > \; \text{ is dense in } {\mathcal{H}} \end{equation} \begin{definition} \label{def:separating vector for a Von Neumann algebra} \end{definition} $ | \psi > $ IS SEPARATING FOR A: \begin{equation} ( a | \psi > \: = \: 0 \; and \; a \in A ) \; \Rightarrow \; a \, = \, 0 \end{equation} Supposing the vector $ | \psi > $ to be cyclic and separating for A let us consider the linear operator $ S_{| \psi >} $ on $ {\mathcal{H}} $ defined by the condition: \begin{equation} S_{| \psi > } a | \psi > \; := a^{\star} | \psi > \; \; a \in A \end{equation} The operator $ S_{| \psi >} $ has a closure $ \bar{S}_{| \psi > } $ that can be used to introduce the following: \begin{definition} \label{def:modular operator} \end{definition} MODULAR OPERATOR W.R.T. A AND $ | \psi > $: \begin{equation} \Delta_{ | \psi >} \; := \; S_{| \psi > } \, \bar{S}_{| \psi > } \end{equation} Let us then introduce the following: \begin{definition} \label{def:conjugate modular operator} \end{definition} MODULAR CONJUGATION W.R.T. A AND $ | \psi > $: the operator $ J_{ | \psi >} $ occurring in the polar decomposition: \begin{equation} S_{| \psi > } \; = \; J_{| \psi > } \, \Delta_{ | \psi >}^{\frac{1}{2}} \end{equation} \smallskip The corner stone of the Modular Theory is the following: \begin{theorem} \label{th:Tomita-Takesaki's theorem} \end{theorem} TOMITA-TAKESAKI'S THEOREM \begin{enumerate} \item \begin{equation} \Delta_{ | \psi >}^{it} \, A \, \Delta_{ | \psi >}^{- it} \; = \; A \; \; \forall t \in {\mathbb{R}} \end{equation} \item \begin{equation} J \, A \, J \; = \; A' \end{equation} \end{enumerate} that, in particular, justifies the following: \begin{definition} \label{def:group of modular automorphisms w.r.t. a pure state} \end{definition} GROUP OF MODULAR AUTOMORPHISMS OF A W.R.T. $ | \psi > $: the one-parameter subgroup of AUT(A): \begin{equation} \sigma^{| \psi >}_{t}(a) \; := \; \Delta_{ | \psi >}^{it} \, a \, \Delta_{ | \psi >}^{- it} \end{equation} The group of modular automorphisms $ \sigma^{| \psi >}_{t} $ depends on the cyclic and separating vector $ | \psi > \in {\mathcal{H}} $, i.e. from the normal state $ \omega_{| \psi > } \in S(A) $ with associated density operator $ \rho_{ | \psi > < \psi |}$. If one looks at outer automorphisms, anyway, such a dependence disappears: \begin{theorem} \label{th:independence from the state of the group of outer modular automorphisms} \end{theorem} INDEPENDENCE FROM THE STATE OF THE GROUP OF OUTER MODULAR AUTOMORPHIMS: \begin{hypothesis} \end{hypothesis} \begin{equation*} | \psi_{1} > \, , \, | \psi_{2} > \in {\mathcal{H}} \text{ cyclic and separating for A } \end{equation*} \begin{thesis} \end{thesis} \begin{equation} [ \sigma^{| \psi_{1} >}_{t} ]_{OUT(A)} \; = \; [ \sigma^{| \psi_{2} >}_{t} ]_{OUT(A)} \end{equation} The proof of theorem\ref{th:independence from the state of the group of outer modular automorphisms} allowed Connes to introduce the following two $\star$-isomorphisms invariants of Von Neumann algebras: \begin{definition} \label{def:first Connes' invariant of a Von Neumann algebra} \end{definition} FIRST CONNES' INVARIANT OF A: \begin{equation} Inv_{Connes}^{(1)}(A) \; := \; \bigcap_{ | \psi > } Spectrum( \Delta_{ | \psi >} ) \end{equation} \begin{definition} \label{def:second Connes' invariant of a Von Neumann algebra} \end{definition} SECOND CONNES' INVARIANT OF A: \begin{equation} Inv_{Connes}^{(2)}(A) \; := \; \{ t \in {\mathbb{R}} \: : \: \sigma^{| \psi >}_{t} \in INN(A) \} \end{equation} by which he classified type-III factors: \begin{definition} \end{definition} $ Type(A) \; := \; III_{0} $: \begin{equation} cardinality_{NC}(A) = \aleph_{2} \: and \: Inv_{Connes}^{(1)}(A) = \{ 0 , 1 \} \end{equation} \begin{definition} \end{definition} $ Type(A) \; := \; III_{1} $: \begin{equation} cardinality_{NC}(A) = \aleph_{2} \: and \: Inv_{Connes}^{(1)}(A) = {\mathbb{R}}_{+} \end{equation} \begin{definition} \end{definition} $ Type(A) \; := \; III_{\lambda} \; ( 0 < \lambda < 1 )$: \begin{equation} cardinality_{NC}(A) = \aleph_{2} \: and \: Inv_{Connes}^{(1)}(A) = \{ \lambda^{n} \, , \, n \, \in {\mathbb{Z}} \} \end{equation} Furthermore Connes proved that: \begin{theorem} \label{th:unicity of injective factors} \end{theorem} SINGLENESS OF INJECTIVE FACTORS OF EACH TYPE EXCEPT $ III_{0} $: there exist a unique injective factor of each type $ I _{n} \, , \, n \in {\mathbb{N}} $, $ I_{\infty} $, $ II_{1} $, $ II_{\infty} $, $ III_{\lambda} \, , \, \lambda \in ( 0 , 1 ] $ \begin{example} \label{ex:Powers factors} \end{example} THE HYPERFINITE $ III_{\lambda} \; ( \, 0 \, < \, \lambda \, < \, 1 \, ) $ FACTOR \smallskip To get an intuitive insight into the structure of the noncommutative space of qubits' sequences $ \Sigma_{NC}^{\infty} = R $ there is nothing better than analyzing its differences with a class of purely infinite factors, usually called the Powers factors, defined in a way very similar to that we followed in example\ref{ex:the hyperfinite finite continuous factor} to define R. It is useful, at this purpose to introduce a suitable, compact notation concerning infinite tensor products of an algebraic probability space $ ( A \, , \, \omega ) $: \begin{definition} \end{definition} \begin{equation} \bigotimes_{n=1}^{\infty} ( A \, , \, \omega ) \; := \pi_{\bigotimes_{n=1}^{\infty} \omega } ( \bigotimes_{n=1}^{\infty} A ) '' \end{equation} With this notation our space of qubits' sequences may be compactly expressed as: \begin{equation}\label{eq:compact expression of the space of qubits' sequences} \Sigma_{NC}^{\infty} \; = \; R \; = \; \bigotimes_{n=1}^{\infty} ( M_{2} ({\mathbb{C}}) \, , \tau_{unbiased} ) \end{equation} where: \begin{equation} \tau_{unbiased} ( \cdot) \; = \; Tr [ \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \ \end{pmatrix} \; \cdot ] \end{equation} A physical realization of the one-qubit unbiased quantum probability space $ ( M_{2} ({\mathbb{C}}) \, , \tau_{unbiased} ) $ is given by a $ spin 1/2 $ system in thermal equilibrium at temperature $ T \, = \; + \infty $, as can be seen observing that the canonical-ensemble's state: \begin{equation} \omega_{CAN} ( H , \beta )( \cdot ) \; := \; Tr[ \frac{e^{- \beta H} }{Tre^{- \beta H}} \; \cdot ] \end{equation} collapses to $ \tau_{unbiased} $ when $ k T \, = \, \beta^{ - 1} \, \rightarrow \, \infty $ for any self-adjoint, bounded from below hamiltonian operator H. Let us now introduce the following: \begin{definition} \label{def:Powers factors} \end{definition} POWERS FACTORS: \begin{equation} R_{ \lambda } \; := \; \bigotimes_{n=1}^{\infty} ( M_{2} ({\mathbb{C}}) \, , \omega_{\lambda} ) \; \; \lambda \in ( 0 , 1 ) \end{equation} where: \begin{equation} \omega_{\lambda} ( \cdot ) \; := \; Tr[ \begin{pmatrix} \frac{1}{1 + \lambda} & 0 \\ 0 & \frac{\lambda}{1 + \lambda} \end{pmatrix} \, \cdot ] \end{equation} It may be proved that: \begin{equation} Type( R_{ \lambda }) \; = \; \lambda \; \; \forall \lambda \in ( 0 , 1 ) \end{equation} By the same considerations made in the example\ref{ex:the hyperfinite finite continuous factor} we may infer that each $ R_{ \lambda } \, , \, \lambda \in ( 0 , 1 ) $ is hyperfinite and, hence, by theorem\ref{th:equivalence of injectivity and hyperfiniteness} and theorem\ref{th:unicity of injective factors}, it is the only hyperfinite, type $ III_{\lambda}$ factor. Clearly, for $ \lambda \in [ 0 , 1 ) $, the state $ \omega_{\lambda} $ is not unbiased: for $ \lambda = 0 $ it is simply the pure state of density matrix $ | 0 > < 0 | $. Then, when $ \lambda $ monotonically increases from 0 to 1, it becomes a mixture of $ | 0 > < 0 | $ and $ | 1 > < 1 | $ with the bias bestowing a privilege on $ | 0 > < 0 | $ decreasing so that it vanishes in the limit $ \lim_{\lambda \rightarrow 1} R_{\lambda} \, = \, R $. \smallskip Let us now observe that definition\ref{def:group of modular automorphisms w.r.t. a pure state} defines the group of modular automorphisms of the Von Neumann algebra $ A \subseteq {\mathcal{B}} ( {\mathcal{H}} ) $ w.r.t. a pure, normal state $ \omega \, \in \, \Xi(A) \bigcap S(A)_{n} $. In order of generalizing it to non-pure normal states we have to introduce a generalization of the modular operator, the \textbf{spatial derivative operator} and the associated \textbf{noncommutative Radon-Nikodym derivative} \cite{Ohya-Petz-93}, \cite{Connes-94}. Given an arbitrary state $ \psi \, \in \, S(A) $ let us introduce the following: \begin{definition} \end{definition} LINEAL OF $ \psi $: \begin{equation} D( {\mathcal{H}} , \psi ) \; := \; \{ | \xi > \in {\mathcal{H}} \, : \, \| a | \xi > \| \, \leq \, C_{| \xi >} \psi ( a a^{\star} ) \; \; \forall a \in A \} \end{equation} Considered the GNS-triplet $ ( {\mathcal{H}}_{\psi} \, , \, \pi_{\psi} \, , \, | \Psi >_{\psi} $ corresponding to the state $ psi $ and taken any $ | \xi > \in D( {\mathcal{H}} , \psi ) $, let us introduce the following operators: \begin{definition} \end{definition} $ R^{\psi}( | \xi > ) \, : \, {\mathcal{H}}_{\psi} \rightarrow {\mathcal{H}} $: \begin{equation} R^{\psi}( | \xi > ) \, \pi_{\psi}(a) \, | \Psi >_{\psi} \; := \; a | \xi > \; \; a \in A \end{equation} \begin{definition} \end{definition} \begin{equation} \Theta^{\psi} ( ( | \xi > ) ) \; := \; R^{\psi}( | \xi > ) ( R^{\psi}( | \xi > ))^{\star} \end{equation} Fixed a $ \varphi ' \, \in \, A ' $: \begin{definition} \label{def:spatial derivative operator} \end{definition} SPATIAL DERIVATIVE OPERATOR W.R.T. $ \varphi ' $ AND $ \psi $: the positive self-adjoint operator $ \Delta ( \varphi ', \psi ) $ associated by the Form Representation Theorem to the closure of the quadratic form q: \begin{equation} q ( | \xi > + | \eta > ) \; := \; \varphi ' ( \Theta^{\psi} ( ( | \xi > ) )) \end{equation} such that: \begin{enumerate} \item \begin{equation} \| \Delta ( \varphi ' \, , \, | \xi > ) ^{\frac{1}{2}} \, | \zeta > \| ^{2} \; = \; q( | \zeta > \| ) \; \; \forall | \zeta > \in Dom(q) \end{equation} \item \begin{equation} Dom(q) \text{ is the core of } \Delta ( \varphi ' \, , \, | \xi > ) ^{\frac{1}{2}} \end{equation} \end{enumerate} Given now two states $ \omega_{1} \, , \, \omega_{2} \, \in \, S(A) $: \begin{definition} \label{def:noncommutative Radon-Nikodym derivative} \end{definition} NONCOMMUTATIVE RADON NIKODYM DERIVATIVE OF $ \omega_{1} $ W.R.T. $ \omega_{2} $: \begin{equation} ( D \omega_{1} \, : \, D \omega_{2} )_{t} \; := \; \Delta ( \omega_{1} / \varphi ' )^{it} \, \Delta ( \omega_{2} / \varphi ' )^{- it} \; \; t \in {\mathbb{R}} \end{equation} where the name remarks the independence from the state $ \varphi ' $. One can now generalize definition\ref{def:group of modular automorphisms w.r.t. a pure state} in the following way: given a state $ \omega \in S(A)_{norm} $: \begin{definition} \label{def:group of modular automorphisms w.r.t. a normal state} \end{definition} GROUP OF MODULAR AUTOMORPHISMS OF A W.R.T. $ \omega $: the one-parameter subgroup of AUT(A): \begin{equation} \sigma_{t}^{\omega} (a) \; := \; \Delta( \omega , \varphi ' ) ^{i t } a \Delta( \omega , \varphi ' ) ^{- i t } \; \; a \in A \, , \, t \in {\mathbb{R}} \end{equation} We have seen how theorem\ref{th:equivalence of injectivity and hyperfiniteness} poses some constraint on the existence of conditional expectations. As to state-preserving conditional expectation, we have at last all the required ingredients to state Takesaki's theorem ruling the whole business: \begin{theorem} \label{th:Takesaki's theorem} \end{theorem} TAKESAKI'S THEOREM \bigskip \begin{hypothesis} \end{hypothesis} \bigskip \begin{equation*} ( A \, , \, \omega ) \; \; \text{ algebraic probability space} \end{equation*} \begin{equation*} A_{accessible} \; \; \text{ $W^{\star} $-subalgebra of A} \end{equation*} \bigskip \begin{thesis} \end{thesis} \bigskip \begin{enumerate} \item a \textbf{conditional expectation }$ E_{\omega} \, : \, A \rightarrow A_{accessible} $ \textbf{ w.r.t. $ A_{accessible} $} \; $ \omega $ - \textbf{invariant} exists \textbf{if and only if $ A_{accessible} $ is invariant under the modular group of $ \omega $, namely}: \begin{equation} \sigma^{\omega}_{t} ( a ) \; \in \; A_{accessible} \; \; \forall a \in A_{accessible} \, \forall t \in {\mathbb{R}} \end{equation} \item if it exists , the \textbf{conditional expectation }$ E_{\omega} \, : \, A_{accessible} \rightarrow A $ \textbf{ w.r.t. $ A_{accessible} $} \; $ \omega $ - \textbf{invariant} is unique \end{enumerate} We have at last all the necessary technical machinery to analyze how and when the Bayesian Strategy can be applied to our problem of Statistical Inference. It is important, first of all, to underline that the state on the complete algebra A involved in the \textbf{Bayes formula} is not the state $ \omega $, that the statistician doesn't know, but the \textbf{a priori estimation of it} $ \omega_{A \; PRIORI} $. Conseguentially, for the theorem\ref{th:Takesaki's theorem}, the involved conditional expectation $ E_{\omega_{A \; PRIORI}} \, : \, A_{accessible} \rightarrow A $ exist (and in this case is unique) under the following: \begin{definition} \label{def:constraint for the feasibility of the bayesian statistical inference} \end{definition} NECESSARY AND SUFFICIENT CONDITION FOR THE FEASIBILITY OF THE BAYESIAN STATISTICAL INFERENCE: \begin{equation} \sigma^{\omega_{A \; PRIORI}}_{t} ( a ) \; \in \; A_{accessible} \; \; \forall a \in A_{accessible} \, \forall t \in {\mathbb{R}} \end{equation} \medskip In the classical case such a condition is always satisfied, guaranteeing that bayesian statistical inference on finite classical probability spaces is always feasible. \medskip This doesn't happen, instead, in the quantum case with the following fundamental consequence lucidly discovered by Miklos Redei (cfr. the cap.8 of \cite{Redei-98}) and confuting the point of view exposed in \cite{Caves-Fuchs-Schack-01}. \begin{theorem} \label{th:impossibility of a subjectivistic Bayesian foundation of Quantum Probability Theory} \end{theorem} IMPOSSIBILITY OF A SUBJECTIVISTIC BAYESIAN FOUNDATION OF QUANTUM PROBABILITY THEORY as far as \textbf{Foundations of Probability Theory} is concerned Quantum Bayesian Theory can't be used to give a \textbf{subjectivist foundation} of Quantum Probability Theory as it happens in the classical case \cite{Bernardo-Smith-00}. \begin{example} \label{ex:Bayesian statistical inference for an EPR-pair} \end{example} BAYESIAN STATISTICAL INFERENCE FOR AN EINSTEIN-PODOLSKI-ROSEN PAIR Let us consider the Einstein-Podolsky-Rosen's setting (in its reformulation in terms of spin 1/2 given by David B\"{o}hm \cite{Bohm-79}, \cite{Bell-93}): Each among Alice and Bob receive from a proper source one of the two spin 1/2 particles on an \textbf{EPR pair}. The quantum probability space of the system is $ ( \, A := M_{4} ({\mathbb{C}}) \, , \, \omega ) $, where: \begin{equation} \rho_{\omega}\; : = \; | \psi > < \psi | \end{equation} \begin{equation} | \psi > \; := \; \begin{pmatrix} 0 \\ \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} \\ 0 \ \end{pmatrix} \end{equation} \begin{equation} \rho_{\omega} \; = \; \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \ \end{pmatrix} \end{equation} Let us now consider Alice. She has at her disposal only the information relative to the subalgebra $ A _{accessible} \; := \; M_{2} ({\mathbb{C}} )$ given by the state: \begin{equation} \omega_{accessible} \; = \; \omega | _{A_{accessible} } \; = \; \tau_{2} \end{equation} The $ \tau_{unbaised} $ - conditional expectation $ E_{unbaised} \, : \, A \, \mapsto A _{accessible} $ is, simply, the orthogonal projection on$ A _{accessible} $ with respect to the following: \begin{definition} \end{definition} HILBERT-SCHIMDT SCALAR PRODUCT ON $ M_{n} ({\mathbb{C}}) $: \begin{equation} < a_{1} | a_{2} > \; := \; \tau_{n} ( a_{1}^{\star} a_{2} ) \end{equation} \medskip Considered the basis $ {\mathbb{E}}_{2} \; := \; \{ e_{1} := \sigma_{1} \, , \, e_{2} := \sigma_{2} \, , \, e_{3} := \sigma_{3} \, , \, e_{4} := {\mathbb{I}} \} $ of $ M_{2} ({\mathbb{C}}) $ and the basis $ {\mathbb{E}}_{4} \; := \; \{ e_{i,j} := e_{i} \bigotimes e_{j} \, , \, i,j = 1 , \cdots , 4 \} $ of $ M_{4} ({\mathbb{C}}) $, we have clearly that: \begin{equation} E_{unbaised} ( \sum_{i=1}^{4} \sum_{j=1}^{4} c_{i,j} e_{i,j} ) \; = \; \sum_{i=1}^{4} c_{i,4} e_{i} \end{equation} Conseguentially: \begin{equation} \omega_{A \, POSTERIORI} ( \sum_{i=1}^{4} \sum_{j=1}^{4} c_{i,j} e_{i,j} ) \; = \; \omega_{accessible} ( E_{unbaised} ( \sum_{i=1}^{4} \sum_{j=1}^{4} c_{i,j} e_{i,j} )) \end{equation} namely: \begin{equation} \omega_{A \, POSTERIORI} ( \sum_{i=1}^{4} \sum_{j=1}^{4} c_{i,j} e_{i,j} ) \; = \; \omega_{accessible} ( \sum_{i=1}^{4} c_{i,4} e_{i} ) \end{equation} and so: \begin{multline} \omega_{A \, POSTERIORI} ( \sum_{i=1}^{4} \sum_{j=1}^{4} c_{i,j} e_{i,j} ) \; = \; \tau_{2} ( \sum_{i=1}^{4} c_{i,4} e_{i}) \; = \\ \sum_{i=1}^{4} c_{i,4} \tau_{2} ( e_{i} ) \; = \; c_{i,4} \end{multline} \smallskip \begin{remark} \label{rem:noncommutative axiomatizations of Quantum Mechanics and Relativity Theory} \end{remark} NONCOMMUTATIVE AXIOMATIZATIONS OF QUANTUM MECHANICS AND RELATIVITY THEORY: Looking at definition\ref{def:noncommutative axiomatization of Quantum Mechanics} one could ask what about Relativity Theory: are we in the framework of Nonrelativistic Quantum Mechanics, of Special-relativistic Quantum Mechanics or of General Relativistic Quantum Mechanics? The answer is that it has been formulated in order of holding in any case, adding suitable further axioms: \begin{itemize} \item assuming conjecture\ref{con:the category isomorphism at the basis of Noncommutative Geometry} it appears natural \cite{Connes-98} to suppose that General Relativistic Quantum Mechanics is based on a quantum spacetime described by a spectral triple $ ( A_{\hbar} \, , \, {\mathcal{H}}_{\hbar} \, , \, D_{\hbar} ) $, where the observables' algebra of quantum space-time $ A_{\hbar} \, \subseteq \, {\mathcal{B}} ( {\mathcal{H}}_{\hbar} ) $ is a Von Neumann algebra acting on the Hilbert space $ {\mathcal{H}}_{\hbar} $. Let us observe that, in the classical limit $ \hbar \, \rightarrow \, 0 $, $ A_{\hbar} $ becomes commutative, so that by Conjecture\ref{con:the category isomorphism at the basis of Noncommutative Geometry}, the spectral triple $ ( A_{\hbar} \, , \, {\mathcal{H}}_{\hbar} \, , \, D_{\hbar} ) $ tends to a riemannian manifold $ ( M \, , \, g_{Riemannian} ) $ The lorentzian manifold constituing the classical space-time $ ( M \, , \, g_{Lorentzian} ) $ is then recovered by a suitable non-euclidean generalization of Wick's rotation \cite{Deligne-Etingof-Freed-Jeffrey-Kazhdan-Morgan-Morrison-Witten-99a}, based on the analytic continuation to the complex plane, and a suitable rotation, of a \textbf{global time function} (i.e. of a function $ t \in \Omega^{0}(M) $ such that $ \nabla_{a} t $ is a past-directed time-like vector field, whose existence is assured by the assumption of axiom\ref{ax:axiom of strong cosmic censorship} since \textbf{global-hyperbolicity} implies \textbf{stable-causality}) having the property that its level's surfaces $ \Sigma_{t} $ are Cauchy surfaces leading to the foliation $ M \; = \; \bigcup_{t} \Sigma_{t} $ of M (cfr. the $ 8^{th} $ chapter of \cite{Wald-84}). Denoted by $ n^{a} $ the unit normal vector field to the hypersurface $ \Sigma_{t} $ and called $ h_{a b } $ the riemannian metric induced on it by $ g_{a b } $ one can choose a vector field $ t^{a} $ on M such that $ t^{a} \nabla_{a} t \, = \, 1 $ such the the \textbf{lapse function}: \begin{equation} \label{eq:lapse function} N \; := \; - t^{a} n_{a} \; = \; ( n^{a} \, \nabla_{a} t )^{ - 1 } \end{equation} and the \textbf{shift vector}: \begin{equation}\label{eq:shift vector} N_{a} \; := \; h_{a b} t^{b} \end{equation} are those for coherent flows of classical test particles adapted to the chosen \textbf{foliation}, i.e. (cfr. the sections 5.4 and 11.1 of \cite{Prugovecki-92}): \begin{align} N & \; = \; 1 \\ N_{a} & \; = \; 0 \end{align} so that $ g _{a b } $ may be expressed in terms of the corresponding \textbf{syncronous (Gaussian normal) coordinates} $ ( x^{0} \, = \, t \, , \, x^{1} \, , \, x^{2} \, , \, x^{3} ) $ as: \begin{equation} g_{Lorentzian} \; = \; d t \bigotimes d t \, - \, h_{i j} d x^{i} \bigotimes d x^{j} \end{equation} Prolonging the coordinate t to the complex plane and evaluating it on the imaginary axis one results in the required riemannian manifold $ ( M \, , \, g_{riemannian} ) $. As the Wick's rotation's operation is always named as the passage from the \textbf{minkowskian} to the \textbf{euclidean} we will refer to the introduced not-flat generalization as to the passage from the \textbf{lorentzian} to the \textbf{riemannian}. \smallskip Since the Universe is closed by definition,it follows by axiom\ref{ax:noncommutative axiom on closed dynamics} that it is described by a strongly-continuous one-parameter group of inner automorphisms. Conjecture\ref{con:the category isomorphism at the basis of Noncommutative Geometry} suggests that OUT(A) plays the rule of the \textbf{quantum diffeomorphims' group} of the quantum spacetime A, while inner fluctuations of the quantum spacetime, i.e. elements of INN(A) corresponds to gauge transformations, as it is supported by the INN(A)-invariance of the minimally-coupled version of definition\ref{def:noncommutative spectral action}. Since the dynamics is made only by gauge transformations, one may conclude that such a picture respects Rovelli's suggestion of forgetting time \cite{Rovelli-88}, i.e. though not following Canonical Quantum Gravity but the: \begin{center} \textit{" $ \cdots $ gnostic subculture of workers in quantum gravity who feel that that the structure of space and time may undergo radical changes at scales of the Planck length"; from \cite{Isham-93} } \end{center} it may be catalogued in the category \emph{"Tempus Nihil Est"} of Chris Isham's classification of different approaches to the Problem of Time in Quantum Gravity \cite{Isham-93}. \item an approximation to the complete quantum theory of fields coupled with gravity is that in which one considers quantum fields on a fixed, classical space-time $ ( M \, , \, g_{a b } ) $ we will suppose to be globally-hyperbolic. A quantum field theory on $ ( M \, , \, g_{a b } ) $ may be defined in terms of the so called Weyl algebra A of $ ( M , g_{a b} ) $ and the Hadamard's states on it (for whose definition we demand to the $ 4^{th} $ chapter of \cite{Wald-94}), i.e. by the collection $ \{ A_{O} \} $ of $ C^{\star}$-sub-algebras of A, one for every open subset $ O \, \subseteq \, M $, with $ A_{O}$ representing the local observables localized on O, satisfying suitable natural conditions: \begin{equation} O_{1} \, \subseteq \, O_{2} \; \Rightarrow \; A_{O_{1}} \, \subseteq \, A_{O_{2}} \end{equation} \begin{equation} O_{1} , O_{2} \text{ causally disconnected } \; \Rightarrow \; [ A_{O_{1}} \, , \, A_{O_{2}} ] \, = \, 0 \end{equation} \begin{multline} \exists \, \{ \alpha_{g} \} \in GR-INN[Is[( M \, , \, g_{a b } )] \, , \, A] \, : \\ \alpha_{g} ( A_{O} ) \, = \, A_{g \, O} \forall g \in Is[( M \, , \, g_{a b } )] \, , \, \forall O \subseteq M \; open \end{multline} where $ Is[( M \, , \, g_{a b } )] $ is the isometries'-group of $ ( M \, , \, g_{a b } )$. In the particular case in which $ ( M = {\mathbb{R}}^{4} \, , \, \eta = \eta_{\mu \nu } d x^{\mu} \bigotimes d x^{\nu} ) $: \begin{equation} \eta_{\mu \nu} \; := \; \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \ \end{pmatrix} \end{equation} is the Minkowski space-time, the above conditions reduce to the Haag-Kastler axioms \cite{Haag-96}. \item Non-relativistic Quantum Mechanics can then be recovered taking the Inonu-Wigner's contraction $ c \, \rightarrow \, + \infty $ of the isometries'-group of the minkowskian space-time, namely of the Poincar\'{e} group, thus obtaining the Galilei group (cfr. e.g. the $ 3^{th} $ chapter of \cite{De-Azcarrega-Izquierdo-95}) \end{itemize} \smallskip \begin{remark} \end{remark} QUBITS ENTERING AND EXITING BLACK-HOLES Since, as a matter of principle, the theory of quantum-black holes is nothing but a matter of bayesian statistical inference w.r.t. a sub-$W^{\star}$-algebra $ A_{accessible} $ of the Weyl's algebra of a space-time with an event-horizon, it is a matter of Quantum Information Theory. Our knowledges in this field are infimous, but there is a thing that have often aroused our's curiosity: the specialists of these matters (apart from some very timid allusion in \cite{Bekenstein-01}), speak always of \textbf{classical information} attached to certain geometrical entities, \textbf{classical information} finishing lost inside or exiting from the event-horizon though they are treating quantum systems. It seems to us that, with this regard, the high energy physics' community have not catched the great conceptual revolution of modern Quantum Information Theory: \begin{center} \textbf{the irreducibility of quantum information to classical information, of the qubit to the cbit} \end{center} When we will at last hear about qubits entering and exiting black-holes ? \newpage \section{The problem of hidden points of a noncommutative space} \label{sec:The problem of hidden points of a noncommutative space} Let us now analyze the concept of a point in a noncommutative space. Given an abelian $ W^{\star}$-algebra A we know by theorem\ref{th:Gelfand isomorphism at C-star algebraic level} that it may be seen as (i.e. it is $ \star$-isomorphic to) the $ C^{\star}$-algebra $ C(X(A)) $ of all the continuous (w.r.t the $ w^{\star}-topology$) functions on the set X(A) of its \textbf{characters} that may thus be seen as the \textbf{points} of the \textbf{commutative space} A. We want to characterize this concept more precisely. Given a generic \textbf{algebraic space} A: \begin{definition} \label{def:points of an algebraic space} \end{definition} POINTS OF A: \begin{equation} POINTS(A) \; := \; \{ \omega \in S(A) \: : \: Var(a) \; = \; 0 \text{ in } (A , \omega ) \; \; \forall a \in A \} \end{equation} The previous considerations may then be formalized as: \begin{theorem} \label{th:on the points of a commutative space} \end{theorem} ON THE POINTS OF A COMMUTATIVE SPACE: \begin{equation} A \; commutative \; \Rightarrow \; POINTS(A) \; = \; \Xi(A) \; = \; X(A) \end{equation} \begin{remark} \label{rem:points of a commutative space as Dirac-delta measures} \end{remark} POINTS OF A COMMUTATIVE SPACE AS DIRAC-DELTA MEASURES Given the commutative $ C^{\star}$-algebra C(X) of continuous functions over the compact, Hausdorff topological space X the points of C(X) are nothing but the Dirac delta measures over X, i.e. the states of the form: \begin{equation} \delta_{x} [ f(y) ] \; := \; f(x) \; \; f \in C(X) \, , \, x \in X \end{equation} Considered a probability measure $ \mu $ on X and represented the classical probability space $ ( X \, , \, \mu ) $ as the commutative probability space $ ( L^{\infty} ( X , \mu ) \, , \, \omega_{\mu} ) $ one has that the projections of $ L^{\infty} ( X , \mu ) $ are nothing but the characteristic functions of $ \mu$-measurable subsets of X forming a classical logic. Obviously the values of $ L^{\infty} ( X , \mu ) $'s points on the projections is given by: \begin{equation} \delta_{x} [ \chi_{A} ] \; = \; \begin{cases} 1 & \text{if $ x \in A$}, \\ 0 & \text{otherwise}. \end{cases} \end{equation} \smallskip Given an algebraic probability space $ ( A , \omega ) $: \begin{definition} \label{def:determinism} \end{definition} $ ( A , \omega ) $ IS DETERMINISTIC: \begin{equation*} \omega \; \in \; POINTS(A) \end{equation*} \begin{definition} \label{classical-nondeterminism} \end{definition} $ ( A , \omega ) $ IS CLASSICALLY-NONDETERMINISTIC: $ ( A , \omega ) $ is nondeterministic and A is commutative \smallskip \begin{definition} \label{quantistically-nondeterminism} \end{definition} $ ( A , \omega ) $ IS QUANTISTICALLY-NONDETERMINISTIC: $ ( A , \omega ) $ is nondeterministic and A is noncommutative \smallskip where, obviously, the form of definition\ref{classical-nondeterminism} and definition\ref{quantistically-nondeterminism} is owed to theorem\ref{th:category isomorphism at the basis of Noncommutative Probability}. \medskip The existence of points on noncommutative spaces is inficiated by the following: \begin{theorem} \label{th:indetermination's theorem} \end{theorem} INDETERMINATION'S THEOREM: \begin{equation} | E( \frac{ [ a , b ] }{2 i } ) | \; \leq \; \sqrt{Var(a)} \sqrt{Var(b)} \; \; \forall a , b \in A \end{equation} \begin{proof} Introduced the quantity: \begin{equation} O(a,b) \; := \; \frac{a-E(a)}{\sqrt{Var(a)}} \, + \, i \frac{b-E(b)}{\sqrt{Var(b)}} \; \; a , b \in A \end{equation} we have clearly that: \begin{equation} O(a , b) \, O(a , b)^{\star} \; \in \; A_{+} \; \; \forall a , b \in A \end{equation} from which the thesis immediately follows \end{proof} Theorem\ref{th:indetermination's theorem} implies that: \begin{corollary} \label{cor:corollary of the indeterminations' theorem} \end{corollary} \begin{equation} ( A \, , \, \omega ) \; deterministic \; \Rightarrow \; ( E( [ a \, , \, b ] ) \; = \; 0 \; \; \forall a , b \in A ) \end{equation} from which it follows that: \begin{theorem} \label{th:first Von Neumann's theorem} \end{theorem} FIRST VON NEUMANN'S THEOREM: \begin{hypothesis} \end{hypothesis} \begin{equation*} A \text{ noncommutative space } \end{equation*} \begin{equation*} cardinality_{NC}(A) \, = \, \aleph_{0} \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} POINTS(A) \; = \; \emptyset \end{equation*} \begin{proof} By hypothesis A is ( $ \star$-isomorphic to) the algebra $ {\mathcal{B}} ( {\mathcal{H}}) $ of all bounded operator on a infinite-dimensional Hilbert-space $ {\mathcal{H}}$. Let us assume, for simplicity, that $ {\mathcal{H}}$ is separable. Fixed a complete orthonormal basis $ {\mathbb{E}} \, := \, \{ | n > \} $ of $ {\mathcal{H}}$ let us consider the sequence $ \{ P_{n} \} $ of projectors defined by: \begin{equation} P_{n} \; := \; \sum_{k=1}^{n} | k > < k | \end{equation} We have clearly that: \begin{equation} P_{0} \; \preceq \; P_{1} \; \preceq \; P_{2} \; \preceq \; \cdots \; \preceq \; I \end{equation} Furthermore there exist hermitian operators $ \{ a_{n} \} $ and $ \{ b_{n} \} $ such that: \begin{equation} P_{n} \; = \; [ a_{n} \, , \, b_{n} ] \; \; \forall n \in {\mathbb{N}} \end{equation} Let us then suppose ad absurdum that there exist a dispersion-free state $ \omega \in POINTS(A) $. By the corollary\ref{cor:corollary of the indeterminations' theorem} one has that: \begin{equation} \omega( P_{n} ) \; = \; \omega( [ a_{n} \, , \, b_{n} ] ) \; = \; 0 \; \; \forall n \in {\mathbb{N}} \end{equation} from which it follows that $ \omega \, = \, 0 $. \end{proof} As we will now show theorem\ref{th:first Von Neumann's theorem} can be generalized to higher noncommutative cardinality. \smallskip Given an algebra A and a subalgebra $ B \; \subset \; A $: \begin{definition} \end{definition} B IS A LEFT IDEAL OF A: \begin{equation} a \in A \; , \; b \in B \; \Rightarrow \; a b \, \in \, B \end{equation} \begin{definition} \end{definition} B IS A RIGHT IDEAL OF A: \begin{equation} a \in A \; , \; b \in B \; \Rightarrow \; b a \, \in \, B \end{equation} \begin{definition} \end{definition} B IS A TWO-SIDED IDEAL OF A: B is both a left ideal and a right ideal of A \medskip Given a $ C^{\star}$-algebra A: \begin{definition} \label{simple C-star algebra} \end{definition} A IS SIMPLE: \begin{equation*} \nexists B \subset A \: : \: \text{not-trivial two-sided ideal} \end{equation*} Given an algebraic probability space: \begin{definition} \label{simple algebraic probability space} \end{definition} $ ( A \, , \, \omega ) $ IS SIMPLE: A is simple \smallskip \begin{example} \label{ex:example of ideals} \end{example} The space $ {\mathcal{C}}_{1} ({\mathcal{H}}) $ of \textbf{trace-class operators}, the space $ {\mathcal{C}} ({\mathcal{H}}) $ of \textbf{infinitesimals operators}, the space $ {\mathcal{I}}_{\alpha} ({\mathcal{H}}) $ of \textbf{order-$\alpha$ infinitesimals operators} are all two-sided ideals of the $W^{\star}$-algebra $ {\mathcal{B}} ({\mathcal{H}}) $ of all bounded operators on an Hilbert space $ {\mathcal{H}} $. The rule of ideals for hidden variables issues is owed to the following: \begin{lemma} \label{lem:on the not-trivial ideals} \end{lemma} ON THE NOT-TRIVIAL IDEALS \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; not-trivial \; C^{\star}-algebra \; : \; \end{equation*} \begin{thesis} \end{thesis} \begin{multline} POINTS(A) \, \neq \, \emptyset \; \Leftrightarrow \\ \exists \, J \, \text{not-trivial two-sided ideal in A} \, : \frac{A}{J} \text{ is abelian} \end{multline} \begin{proof} \begin{enumerate} \item Given $ \phi \, \in \, POINTS(A) $ we will prove that: \begin{equation} J \; := \; \{ a \in A \, : \, \phi(a) \, = \, 0 \} \end{equation} is a not-trivial two-sided ideal such that $ \frac{A}{J} $ is commutative. Obviously J is a linear subspace of A and is a subset of $ A_{sa} $. Furthermore, the hypothesis of not-triviality of A implies that also A is not trivial, since the existence of an $ a \in A \, : \, a \, \neq \, I $ implies that also $ a - \phi(a) I \; \neq \; I $ so that $ a - \phi(a) I $ is a not-trivial element of J. Let us now show that J is an ideal: a generic element $ a \in A $ may be expressed as linear combination of self-adjoint elements: \begin{equation} a \; = \; \sum_{n} c_{n} x_{n} \; c_{n} \in {\mathbb{C}} , x_{n} \in A_{sa} \; \forall n \end{equation} Given $ a , b \in J $ we have, by theorem\ref{th:Cauchy-Schwarz inequality} and the fact that $ \phi $ is a point, that: \begin{multline} | \phi ( x_{n} b ) | \; \leq \; \phi ( x_{n} x_{n}^{\star} ) \phi ( b b^{\star} ) \\ = \; \phi( x_{n}^{2} ) \phi ( b^{2} ) \; = \; 0 \; \; \forall n \end{multline} so that: \begin{equation} \phi ( a b ) \; = \; 0 \end{equation} and hence $ a b \, \in \, J $, so that it is proved that J is a left-ideal. The proof that J is also a right-ideal is specular. To prove, finally, that $ \frac{A}{J} $ is abelian let us observe that the map $ h : \frac{A}{J} \rightarrow {\mathcal{C}} $ defined by: \begin{equation} h( [ a ]_{ \frac{A}{J} } ) \; := \; \phi ( a ) \end{equation} is a $ \star $ -isomorphism from $ \frac{A}{J} $ to $ {\mathcal{C}} $; in fact if $ \phi( a ) \, = \, \phi( b) $ then $ \phi( a - b) \, = \, 0 $ so that $ [ a ]_{\frac{A}{J}} \, = \, [ b ]_{\frac{A}{J}} $ and, consequentially, h is invertible; furthermore: \begin{equation} h ( [ a ]_{\frac{A}{J}} , [ b ]_{\frac{A}{J}} ) \; = \; h ( [ a b ]_{\frac{A}{J}}) \; = \; \phi( a b ) \; = \; \phi(a) \phi(b) \; \; \forall a, b, \in A \end{equation} and hence h preserves the product \item let suppose that there exist a two-sided ideal J such that $ {\frac{A}{J}} $ is abelian. Then, by theorem\ref{th:on the points of a commutative space}, it follows that $ POINTS(A) \, \neq \, \emptyset $. \end{enumerate} \end{proof} immediately implying the following: \begin{corollary} \label{cor:indeterminism of simple algebraic probability spaces} \end{corollary} INDETERMINISM OF SIMPLE ALGEBRAIC PROBABILITY SPACES \begin{equation*} ( A \, , \, \omega ) \text{ simple } \; \Rightarrow \; ( A \, , \, \omega ) \text{ nondeterministic } \end{equation*} Corollary\ref{cor:indeterminism of simple algebraic probability spaces}, anyway, is only the tip of an iceberg, as is stated by the following generalization of theorem\ref{th:first Von Neumann's theorem} \begin{theorem} \label{th:indeterminism of noncommutative probability spaces} \end{theorem} INDETERMINISM OF NONCOMMUTATIVE PROBABILITY SPACES: \begin{hypothesis} \end{hypothesis} \begin{equation*} ( A \, , \, \omega ) \; \; \text{algebraic probability space} \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} ( A \, , \, \omega ) \; noncommutative \; \Rightarrow \; ( A \, , \, \omega ) \; nondeterministic \end{equation*} \begin{proof} Proceeding exactly as in the proof of theorem\ref{th:first Von Neumann's theorem} let us consider a sequence $ \{ p_{n} \}_{n \in {\mathbb{N}}} $ such that: \begin{align} p_{n} & \; \in \; {\mathcal{P}}(A) \; \; \forall n \in {\mathbb{N}} \\ p_{i} & \; \preceq \; p_{j} \; \; \forall i \, < \, j \end{align} There will exist hermitian operators $ \{ a_{n} \} $ and $ \{ b_{n} \} $ such that: \begin{equation} p_{n} \; = \; [ a_{n} \, , \, b_{n} ] \; \; \forall n \in {\mathbb{N}} \end{equation} Let us then suppose ad absurdum that there exist a dispersion-free state $ \omega \in POINTS(A) $. By the corollary\ref{cor:corollary of the indeterminations' theorem} one has that: \begin{equation} \omega( p_{n} ) \; = \; \omega( [ a_{n} \, , \, b_{n} ] ) \; = \; 0 \; \; \forall n \in {\mathbb{N}} \end{equation} from which it follows that $ \omega \, = \, 0 $. \end{proof} \smallskip \begin{remark} \label{rem:usnharp localization of a noncommutative space} \end{remark} UNSHARP LOCALIZATION ON A NONCOMMUTATIVE SPACE: In the commutative case a point may be characterized through a monotonically descreasing sequence of projections. Given, for example, the unbaised probability space of cbits' sequences $ ( \Sigma^{\infty} \, , \, P_{unbiased} ) $ let us use the numeric representation map of definition\ref{def:numeric representation} to visualize it as the the classical probability space $ ( [ 0 , 1 ) \, , \, \mu_{Lebesgue} ) $. A point $ x \in ( 0 , 1) $ is completelly specified by a nested sequence of measurable $ ( 0 , 1) $'s subsets $ \{ A_{n} \} $ whose intersection is the singleton containing x : \begin{align} A_{i} & \supset A_{j} \; \; \forall i < j \\ \{ x \} & \; = \; \bigcap _{n \in {\mathbb{N}}} A_{n} \end{align} For example one can take: \begin{equation} A_{n} \; := \; ( x - \frac{1}{2^{n}} \, , \, x + \frac{1}{2^{n}}) \end{equation} Let us now look at the classical probability space $ ( [ 0 , 1 ) \, , \, \mu_{Lebesgue} ) $ as at the commutative probability space $ ( A := L^{\infty} ( [ 0 , 1 ) \, , \, \mu_{Lebesgue} ) \, , \, \omega_{\mu_{Lebesgue}})$. As we saw in the remark\ref{rem:points of a commutative space as Dirac-delta measures} the projections of $ {\mathcal{P}}(A) $ are nothing but the characteristic functions of measurable $ [ 0 , 1 ) $'s subsets constituting a classical logic. The sequence $ \{ A_{n} \} $ corresponds to the sequence of projections $ \{ \chi_{A_{n}} \} $ satisfying the condition: \begin{equation} \chi_{A_{i}} \, > \, \chi_{A_{j}} \; \; \forall i < j \end{equation} The point $ x \in A $, i.e the point $ \delta_{x} \in POINTS(A) $, is then characterized by the condition: \begin{equation} \delta_{x} ( \chi_{A_{n}} ) \; = \; 1 \; \; \forall n \in {\mathbb{N}} \end{equation} Given, now a noncommutative space X, one could think that, though $ POINTS(X) \,= \, \emptyset $, the characterization of the concept of an X's point can be recovered generalizing the above procedure, i.e finding a monotonically increasing sequence of projections $ \{ p_{n} \} $ over X: \begin{align} p_{n} & \, \in \, {\mathcal{P}}(X) \; \; \forall n \in {\mathbb{N}} \\ p_{i} & \, > \, p_{j} \; \; \forall i < j \end{align} and a state $ \omega \in \Xi(X) $ such that: \begin{equation} \omega( p_{n} ) \; = \; 1 \; \; \forall n \in {\mathbb{N}} \end{equation} One could, in fact, think that in such a situation it is possible, after all, to look at the sequence $ \{ p_{n} \} $ as a sequence of propositions stating the localization in a monotonically-increasing way, so that a state $ \omega $ giving value one to all these propositions, i.e. in quantum-logic language , stating the truth of all these propositions, can assume a geometrical meaning as an unsharped-localized region of the noncommutative space X. \smallskip \begin{remark} \end{remark} ON HIDDEN POINTS OF A NONCOMMUTATIVE SPACE Given a noncommutative space $ A_{accessible} $ one could think of completing it, i.e. of considering a larger noncommutative space A of which $ A_{accessible} $ is a sub-$W^{\star}$-algebra, such that $ POINTS(A) \; \neq \; \emptyset $. In such a situation the indeterminism of any noncommutative probability space $ ( A , \omega ) $ on A could then be simply attributed to the not accessibility of the algebraic random variables belonging to $ A \, - \, A_{accessible} $. That this in not the case is stated by the following obvious corollary of theorem\ref{th:indeterminism of noncommutative probability spaces}: \begin{corollary} \label{cor:not existence of hidden points of a noncommutative space} \end{corollary} NOT EXISTENCE OF HIDDEN POINTS OF A NONCOMMUTATIVE SPACE: \begin{equation} POINTS(A) \; = \; \emptyset \; \; \forall A \supset A_{accessible} \end{equation} \begin{proof} Since $ A_{accessible} $ is noncommutative, A is noncommutative too. The thesis follows immediately by theorem\ref{th:indeterminism of noncommutative probability spaces} \end{proof} Though not leading to a sharp localization, one could anyway think that completions can anyway improve the unsharp localization. If this is possible or not depends sensibly from the definition of completion one assume, as we will now show. Given a state $ \beta \in S(B) $: \begin{definition} \end{definition} CLASSICAL PROBABILITY MEASURES WITH BARYCENTER $ \beta $: the set $ M_{\beta}[ S(B)] $: \begin{multline} M_{\beta}[ S(B)] \; := \; \{ \mu \in M[S(B)] \, : \\ \beta (b) \, = \, \int_{S(B)} \omega(b) \, d \mu(\omega) \; \; \forall b \in B \} \end{multline} Given a channel $ \beta \in CPU(B,A) $: \begin{definition} \label{def:completion-channel} \end{definition} C IS A COMPLETION-CHANNEL: $ \forall \alpha \in S(A) \; , \; \exists \mu \in M_{C^{\star} \alpha}[ S(B)] $ such that: \begin{itemize} \item \textbf{in the completion the uncertainty descreases}, i.e.: \begin{multline} \sqrt{Var_{\alpha} ( C b )} \; > \; \sqrt{Var_{\omega} ( b )} \; \; \forall \omega \in supp( \mu ) \, , \\ \forall b \in B \, : \, ( C b \, \in A_{sa} \: and \: \sqrt{Var_{\alpha} ( C b )} \, > \, 0 ) \end{multline} \item \textbf{in the completion the certainty remains certain}, i.e.: \begin{multline} \sqrt{Var_{\alpha} ( C b )} \; = \; \sqrt{Var_{\omega} ( b )} \; = \; 0 \; \; \forall \omega \in supp( \mu ) \, , \\ \forall b \in B \, : \, ( C b \, \in A_{sa} \: and \: \sqrt{Var_{\alpha} ( C b )} \, = \, 0 ) \end{multline} \end{itemize} \begin{definition} \label{def:deterministic-completion-channel} \end{definition} C IS A DETERMINISTIC-COMPLETION-CHANNEL: \begin{itemize} \item C is a completion-channel \item \begin{multline} \sqrt{Var_{\omega} ( b )} \; = 0 \; \; \forall \omega \in supp( \mu ) \, , \\ \forall b \in B \, : \, ( C b \, \in A_{sa} \: and \: \sqrt{Var_{\alpha} ( C b )} \, > \, 0 ) \end{multline} \end{itemize} Von Neumann himself was the first to investigate the possibility of deterministic completion channels, though only of the following particular kind: \begin{definition} \label{def:Von Neumann's completion} \end{definition} VON NEUMANN'S COMPLETION: a deterministic-completion-channel of the form $ {\mathbb{I}} \in CPU(A) $ such that: \begin{equation} cardinality_{NC} ( A ) \; \leq \; \aleph_{0} \end{equation} formulating the first no-go theorem on hidden variables: \begin{theorem} \label{th:Von Neumann's no-go theorem} \end{theorem} VON NEUMANN'S NO-GO THEOREM: \begin{equation} \{ C \; : \; \text{ Von Neumann's completion } \} \; = \; \emptyset \end{equation} \begin{proof} It immediately follows by theorem\ref{th:first Von Neumann's theorem} \end{proof} The generalization involved in the passage from theorem\ref{th:first Von Neumann's theorem} to theorem\ref{th:indeterminism of noncommutative probability spaces} induces the following generalization of theorem\ref{th:first Von Neumann's theorem}: \begin{theorem} \label{th:first algebraic no-go theorem} \end{theorem} FIRST ALGEBRAIC NO-GO THEOREM: \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; \; \text{ algebraic space} \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} {\mathbb{I}}_{A} \text{ is a deterministic-channel completion } \; \Leftrightarrow \; \text{A is commutative} \end{equation*} \begin{proof} \begin{itemize} \item A commutative $ \; \Rightarrow \; {\mathbb{I}}_{A} \text{ is a deterministic-channel completion } $ If A is commutative we know by theorem\ref{th:Gelfand isomorphism at C-star algebraic level} that it may be seen (i.e. it is $ \star$-isomorphic to) the space $ C(X(A)) $ of the continuous functions over the characters, i.e. the points, of A. By the Riesz-Markov theorem (cfr. the section 4.4 of \cite{Reed-Simon-80}) for every state $ \phi \in S(C(X)) $ there exist a measure $ \mu_{\phi} $ on X such that: \begin{equation} \phi (f) \; = \; \int_{X} d \mu_{\phi} f \; \; \forall f \in C(X) \end{equation} By theorem\ref{th:on the points of a commutative space} if follows that: \begin{equation} supp(\mu) \; \subseteq \; POINTS(A) \end{equation} for which the thesis follows \item $ {\mathbb{I}}_{A} \text{ is a deterministic-channel completion } \; \Rightarrow \; $ A commutative Let us assume that $ {\mathbb{I}}_{A} $ is a deterministic-channel completion. Given $ x , y \, \in \, A \; : \; x \, > \, y \, \leq \, 0 $ one has, by the definition\ref{def:completion-channel}, that for every state $ \phi \in S(A) $ there exists a measure $ \mu \in M[ S(A)] $ such that: \begin{align} \phi( x^{2} ) & \; = \; \int_{S(A)} d \mu ( \omega ) \omega ( x^{2}) \; = \; \int_{S(A)} d \mu ( \omega ) \omega ( x )^{2} \\ \phi( y^{2} ) & \; = \; \int_{S(A)} d \mu ( \omega ) \omega ( y^{2}) \; = \; \int_{S(A)} d \mu ( \omega ) \omega ( y )^{2} \end{align} Since $ \omega ( x ) \; \leq \; \omega ( y ) $, it follows that: \begin{equation} \phi( x^{2} ) \, - \, \phi( y^{2} ) \; = \; \int_{S(A)} d \mu ( \omega ) ( \omega ( x )^{2} \, - \, \omega ( y )^{2} ) \; \leq \; 0 \end{equation} that implies that $ x^{2} \, \geq \, y^{2} $. The thesis follows immediately from the property: \begin{equation} ( x \, \geq \, y \: \rightarrow \: x^{2} \, \geq \, y^{2} \, \, \forall x , y \in A ) \; \Rightarrow \; A \; commutative \end{equation} \end{itemize} \end{proof} \smallskip Let us now return to the Noncommutative Bayesian Statistical Inference Theory we have outlined in section\ref{sec:On the rule Noncommutative Measure Theory and Noncommutative Geometry play in Quantum Physics}: one could think that the process of statistical inference corresponds to an improvement in the localization on a noncommutative space. This is not ,anyway, the case, as it is stated by the following \cite{Redei-98}: \begin{theorem} \label{th:bayesian statistical inference doesn't noncommutatively-localize} \end{theorem} SECOND ALGEBRAIC NO-GO THEOREM (BAYESIAN STATISTICAL INFERENCE DOESN'T NONCOMMUTATIVELY-LOCALIZE): \begin{hypothesis} \end{hypothesis} \begin{equation*} A \text{ noncommutative space} \end{equation*} \begin{equation*} A_{accessible} \; \subset \; A \; \; \text{sub-$W^{\star}$-algebra of A satisfying the condition of definition\ref{def:constraint for the feasibility of the bayesian statistical inference}} \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} E_{unbaised} \, : \, A \: \rightarrow \: A_{accessible} \text{ is not a channel-completion} \end{equation*} \smallskip Let us conclude this section by an analysis of John Bell's contribution to the Hidden Variables' Issue. This involves the dicussion of an (apparentely) different kind of completion, concerning the degree oif irreducibility of noncommutative probabilities to the commutative ones. An immediate conseguence of theorem\ref{th:category isomorphism at the basis of Noncommutative Probability} is the following: \begin{theorem} \label{th:irreducibility of noncommutative probability to commutative probability to any order} \end{theorem} IRREDUCIBILITY OF NONCOMMUTATIVE PROBABILITY TO COMMUTATIVE PROBABILITY TO ANY ORDER: \begin{hypothesis} \end{hypothesis} \begin{equation*} ( A \, , \, \omega ) \text{ noncommutative probability space } \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} \exists m \in {\mathbb{N}} \; : \; \text{A is irreducible to Classical Probability Theory up to $ m^{th}$ order} \end{equation*} \begin{remark} \label{rem:impossibility of the Osterwalder-Schrader program} \end{remark} IMPOSSIBILITY OF THE OSTERWALDER-SCHRADER'S PROGRAM A consequence of theorem\ref{th:irreducibility of noncommutative probability to commutative probability to any order} is the impossibility of founding Quantum Field Theory on the Osterwalder-Schrader axiomatization (cfr. the sixth chapter of \cite{Glimm-Jaffe-87} and \cite{Haag-96}). Indeed a quantum field theory satisfying the Osterwalder-Schrader axioms obeys the Haag-Kastler axiom's too, but the conversely doesn't hold. This implies that the formal path-integral measures comparing in euclidean field theories cannot in principle be made always rigorous since they , mathematically rigorously, cannot always exist \smallskip One the conceptually more fascinating examples of irriducibility of Quantum Probability Theory to Classical Probability Theory to a low order is given by the EPR-stuff we already introduced in the example\ref{ex:Bayesian statistical inference for an EPR-pair}. Given the noncommutative probability space $ ( A , \omega )$, with: \begin{equation} A \; := \; M_{2} ({\mathbb{C}}) \bigotimes M_{2} ({\mathbb{C}}) \; = \; M_{4}( {\mathbb{C}}) \end{equation} \begin{equation} \omega ( \cdot ) \; : = \; Tr ( \rho_{| \psi > < \psi|} \cdot ) \end{equation} \begin{equation} | \psi > \; := \; \begin{pmatrix} 0 \\ \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} \\ 0 \ \end{pmatrix} \end{equation} \begin{equation} \rho_{| \psi > < \psi|} \; = \; \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \ \end{pmatrix} \end{equation} let us consider the following noncommutative random variables: \begin{equation} q^{A}_{1} \; := \; \sigma_{1} \bigotimes I \; = \; \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \ \end{pmatrix} \end{equation} \begin{equation} q^{A}_{2} \; := \; \sigma_{2} \bigotimes I \; = \; \begin{pmatrix} 0 & 0 & - i & 0 \\ 0 & 0 & 0 & - i \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \ \end{pmatrix} \end{equation} \begin{equation} q^{A}_{3} \; := \; \sigma_{3} \bigotimes I \; = \; \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \ \end{pmatrix} \end{equation} \begin{equation} q^{B}_{1} \; := \; I \bigotimes \sigma_{1} \; = \; \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \ \end{pmatrix} \end{equation} \begin{equation} q^{B}_{2} \; := \; I \bigotimes \sigma_{2} \; = \; \begin{pmatrix} 0 & -i & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \ \end{pmatrix} \end{equation} \begin{equation} q^{B}_{3} \; := \; I \bigotimes \sigma_{3} \; = \; \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \ \end{pmatrix} \end{equation} having moments: \begin{equation} M_{n}( q^{A}_{i} ) \; = \; M_{n}( q^{B}_{i} ) \; = \; \begin{cases} 0 & \text{n even}, \\ 1 & \text{n odd}. \end{cases} \; \; i= 1 , \cdots , 3 \end{equation} \medskip The first joint moments of the six noncommutative random variables $ q^{A}_{1} \, , \, q^{A}_{2} \, , \, q^{A}_{3} \, , \, q^{B}_{1} \, , \, q^{B}_{2} \, , \, q^{B}_{3} $ are given by: \begin{equation} E( q^{A}_{i} q^{A}_{j} ) \; = \; E( q^{B}_{i} q^{B}_{j} ) \; = \;\delta_{i,j} \; \; i,j = 1 , \cdots , 3 \end{equation} \begin{equation} E ( q^{A}_{i} q^{B}_{j} ) \; = \; E ( q^{B}_{i} q^{A}_{j} ) \; = \; \begin{cases} -1 & \text{$ i = j$}, \\ 0 & \text{$ i \neq j$}. \end{cases} \; \; i,j = 1 , \cdots , 3 \end{equation} The contribution by John Bell was to show that \cite{Accardi-88} \cite{Streater-95}, \cite{Streater-00a}: \begin{theorem} \label{th:Bell's theorem} \end{theorem} BELL'S THEOREM: \begin{equation*} Q \; = \; \{ q^{A}_{1} \, , \, q^{A}_{2} \, , \, q^{A}_{3} \, , \, q^{B}_{1} \, , \, q^{B}_{2} \, , \, q^{B}_{3} \} \text{ is irreducible to classical probability up to the } 2^{th} order \end{equation*} \begin{proof} The simple numerical property: \begin{equation} a b \, - \, b c \, + \, a c \; \leq \; 1 \; \; \forall a,b,c \in \, [ - 1 \, , \, 1 ] \end{equation} implies that: \begin{align} | & a b \, - \, b c | \; \leq \; 1 \, - \, a c \; \; \forall a ,b ,c \in [ 0 ,1 ] \\ | & a b \, + \, b c | \, + \, | a d \, - \, d c | \; \leq \; 1 \, + \, a c \; \; \forall a ,b ,c,d \in [ 0 ,1 ] \\ \end{align} from which it follows that it don't exist four random variables a , b , c , d defined on a classical probability space $ ( \, \Omega \, , \, P ) $ such that: \begin{align} | & E( a b ) \, - \, E( b c ) | \; \leq \; 1 \, - \, E( a c ) \\ | & E( a b ) \, - \, E( b c ) | \; \leq \; 1 \, + \, E( a c ) \\ | & E( a b ) \, - \, E( b c ) | \, + \, | E( a d ) \, + \, E( d c ) | \; \leq \; 2 \end{align} where E denotes expectation w.r.t. the P-measure:, i.e.: \begin{equation*} E( F ) \; := \; \int_{\Omega} \, F \, dP \end{equation*} The thesis easily follows \end{proof} \begin{remark} \label{rem:Bell's theorem doesn't speak of locality} \end{remark} BELL'S THEOREM DOESN'T SPEAK OF LOCALITY: Our way of presenting Bell's result is someway provocative, in that it is completely different both from the form and from the spirit of Bell's papers \cite{Bell-93}: Bell's theorem was intended to be and is almost always looked as \cite{Shimony-00} the proof that all local hidden variables' theories imply an inequality which is incompatible with some of the predictions of Quantum Mechanics. Such inequality, anyway, is nothing but a consequence of the fact that there does not exist a set of six classical random variables $ \{ c^{A}_{1} \, , \, c^{A}_{2} \, , \, c^{A}_{3} \, , \, c^{B}_{1} \, , \, c^{B}_{2} \, , \, c^{B}_{3} \} $ on a suitable classical probability space, such that: \begin{equation} M_{n}( c^{A}_{i} ) \; = \; M_{n}( c^{B}_{i} ) \; = \; \begin{cases} 0 & \text{ if $ n = 0 $ or $ n = 2 $}, \\ 1 & \text{if $ n = 0 $}. \end{cases} \; \; i= 1 , \cdots , 3 \end{equation} \begin{equation} E( c^{A}_{i} c^{A}_{j} ) \; = \; E( c^{B}_{i} c^{B}_{j} ) \; = \;\delta_{i,j} \; \; i,j = 1 , \cdots , 3 \end{equation} \begin{equation} E ( c^{A}_{i} c^{B}_{j} ) \; = \; E ( c^{B}_{i} c^{A}_{j} ) \; = \; \begin{cases} -1 & \text{$ i = j$}, \\ 0 & \text{$ i \neq j$}. \end{cases} \; \; i,j = 1 , \cdots , 3 \end{equation} The concept of locality appears nowhere and has nothing to do with the physical meaning of theorem\ref{th:Bell's theorem} concerning the irreducibility of entanglement to Classical Probability Theory. \begin{remark} \label{rem:Bell's theorem and functional integrals on superspaces} \end{remark} BELL'S THEOREM AND FUNCTIONAL INTEGRALS ON SUPERSPACES: There exist a natural reaction to theorem\ref{th:Bell's theorem}, that could lead to think that there must be certainly a mistake in its proof: considered a system of two uncoupled fermionic oscillators: \begin{equation} \hat{H} \; := \; \frac{1}{2} ( \hat{a}_{1}^{\dagger} \hat{a}_{1} \, + \, \hat{a}_{2}^{\dagger} \hat{a}_{2} ) \end{equation} where: \begin{align} \hat{a}_{i}^{2} & \; = \; ( \hat{a}_{i}^{\dagger} )^{2} \; = \; 0 \; \; i = 1,2 \\ \hat{a}_{i}^{\dagger} & \hat{a}_{j} \, + \, \hat{a}_{j} \hat{a}_{i}^{\dagger} \; = \; \delta_{i,j} \; \; i = 1,2 \end{align} every theoretical-physicists' textbook (cfr. e.g. the section 3.5 of \cite{Zinn-Justin-93}) tell us that we can compute all its correlation functions by functional derivatives of the partition function: \begin{multline} \label{eq:informal 2-fermionic oscillators' partition function} Z [ \bar{\eta}_{1} \, , \, \eta_{1} \, , \, \bar{\eta}_{2} \, , \, \eta_{2} ] \; := \; \int [ d c_{1}(t) d c_{2}(t) d \bar{c}_{1}(t) d \bar{c}_{2}(t) ] \exp [ - S( c_{1}, c_{2} , \bar{c}_{1} , \bar{c}_{2}) \, + \\ \int d s \sum_{i=1}^{2} \bar{\eta}_{i} (s) c_{i}(s) + \bar{c}_{i} (s) \eta_{i}(s) ] \end{multline} with euclidean action: \begin{equation} S( c_{1}, c_{2} , \bar{c}_{1} , \bar{c}_{2}) \; := \; \int dt \sum_{i=1}^{2} \bar{c}_{i} (t) \dot{c}_{i} (t) \, - \, \bar{c}_{i} (t) c_{i}(t) \end{equation} Isn't this fact an explicit confutation of theorem\ref{th:Bell's theorem}, implying the existence of the six classical random variables $ \{ c^{A}_{1} \, , \, c^{A}_{2} \, , \, c^{A}_{3} \, , \, c^{B}_{1} \, , \, c^{B}_{2} \, , \, c^{B}_{3} \} $ we spoke about in the remark\ref{rem:Bell's theorem doesn't speak of locality}? The reason why this is not the case is that the euristic measure of equation\ref{eq:informal 2-fermionic oscillators' partition function} cannot be defined in a mathematically rigorous way. Indeed, though being at the basis of many exciting mathematical results such as the proof of the Atiyah-Singer Index Theorem by the computation of the index of the Dirac operator D on a spin-manifold $ ( M \, , \, g ) $ as the path-integral: \begin{equation} \label{eq:path-integral's computation of the Dirac operator's index } Index(D) \; := \; \int_{p.b.c.} [ dx ] [ d \psi ] \exp [ - \int_{0}^{\beta} dt L ] \end{equation} where: \begin{equation} L \; := \; \frac{1}{2} g_{\mu , \nu} \dot{x}^{\mu} \dot{x}^{\nu} \, + \, \frac{1}{2} g_{\mu \nu} \psi^{\mu} \frac{D \psi^{\mu} }{D t} \end{equation} is the supersymmetric lagrangian of a spin-$\frac{1}{2}$ fermion living on $ ( M \, , \, g ) $ \cite{Alvarez-95}, a rigorous mathematical formalization of functional integration on superspaces (going beyond informal time-splitting procedures such as that of the fifth chapter of \cite{De-Witt-92}) doesn't exist yet. It may be worth mentioning the possibility that it could require an extension of the Kolmogorov's Axiomatization of Probability rather than simply an application of it, and could in this way converge to Quantum Probability Theory, as the section5.3 of \cite{Khrennikov-99} and the intellectual path of its author could suggest \newpage \section{Irreducibility of Quantum Computational Complexity Theory to Classical Computational Complexity Theory} \label{sec:Irreducibility of Quantum Computational Complexity Theory to Classical Computational Complexity Theory} \chapter{Quantum algorithmic randomness: where are we?} \label{chap:Quantum algorithmic randomness: where are we?} \section{The unpublished ideas of Sidney Coleman and Andrew Lesniewski} \label{sec:The unpublished ideas of Sidney Coleman and Andrew Lesniewski} The first people who began to investigate in a sistematic way the interrelations between Quantum Theory and the notion of algorithmic randomness was certainly Paul Benioff in a serie of 1970's papers \cite{Benioff-73}, \cite{Benioff-74}, \cite{Benioff-77}, \cite{Benioff-78} in which he extensively analyzed the algorithmic randomness status of the sequence of outcomes of quantum measurements. Benioff's intention was not, anyway, that of characterizng a notion of quantum-algorithmic-randomness, but that of extracting from Quantum Physics a new definition of classical-algorithmic randomness. Indeed, in those years, the great scientific revolution concerning the incommensurability of quantum information and classical information (we underlined in the example \ref{ex:the Hilbert space of Quantum Information Theory} and in the remark\ref{rem:the noncommutative combinatory information and the definition of the qubit}) was not happened yet. A very similar kind of investigation was then pursued by Sidney Coleman and Andrew Lesniewski who tried to extend previous considerations by Hartle, as well as by Sam Guttman \cite{Guttmann-95}, \cite{Mittelstaedt-01} Unfortunately Coleman and Lesniweski never published their thought that is accessible only from the exposition of it made by John Preskill in the section3.6 of his wonderful lecture notes \cite{Preskill-98} as well as from the electronic correspondence of Christopher Fuchs he gently gave to collectivity's disposition (cfr. pagg.24-30 as well as pagg.106-110 of \cite{Fuchs-01}). The starting point is the following analysis by Hartle \cite{Hartle-68}: the only point of the standard Copenhagen's axiomatization of Quantum Mechanics in which the term $ << probability >> $ appears is the \textbf{Postulate of Reduction}, stating that a measurement of an observable $ \hat{A} \; := \; \sum_{a} a | a > < a | $ on a quantum system prepared in the state $ | \psi > \; := \; \sum_{a} | a > < a | \psi > $ has the following effects: \begin{enumerate} \item \begin{equation} Probability [measurement's \: outcome \; = \; a ] \; = \; | < a | \psi >| ^{2} \end{equation} \item if the measuremnt's outcome a occurs, then the state's system collapses istantaneously to the state $ | a > $ \end{enumerate} where we have considered, for simplicity, the case when there is no degeneration. Hartle observed that the \textbf{Issue of the Interpretation of Probability} may be made to disappear from the axiomatization of Quantum Mechanics in the following way: \begin{enumerate} \item one replaces the Postulate of Reduction with the weaker \textbf{Postulate of Eigenstates}: \begin{center} If we prepare a quantum state $ | a > $ such that $ \hat{A} | a > \, = \, a | a > $, and then immediately measure $ \hat{A} $, the outcome of the measurement is a with certainty \end{center} \item the case of measurements performed in a state that is not an eigenstate of the measured observable is reconducted to the \textbf{Postulate of Eigenstates} by the assumption of a frequentistic interpretation of probability: suppose we want to make a statement about the probability of obtaining the result $ | \uparrow_{z} > $ when we measure $ \sigma_{z} $ in the state : \begin{equation} | \psi > \; = \; a | \uparrow_{z} > \, + \, b | \downarrow_{z} > \end{equation} Hartle imagines that one prepares an infinite number of copies, so that the state is: \begin{equation} | \psi^{( \bigotimes \infty)} > \; := \; \bigotimes_{n=1}^{\infty} | \psi > \end{equation} and imagines that one measures $ \sigma_{z} $ for each of the copies. Introduced the \textbf{average spin operator}: \begin{equation}\label{eq:average spin operator} \bar{\sigma}_{z} \; := \; \lim_{n \rightarrow + \infty} \frac{1}{n} \sum_{i=1}^{n} \bar{\sigma}_{z}^{(i)} \end{equation} Hartle claims that $ | \psi^{(\bigotimes n)} > $ becomes an eigenstate of $ \bar{\sigma}_{z} $ with eigenvalue $ |a|^{2} - |b|^{2} $ for $ n \rightarrow \infty $. Then he appeals to the \textbf{Postulate of Eigenstates} to infer that a measurement of $ \bar{\sigma}_{z} $ will yeld the result $ |a|^{2} - |b|^{2} $ with certainty, and that the fraction of all the spins that point up is, therefore, $ | a |^{2} $. In this sense $ | a |^{2} $ is the probability that the measurement of $ \sigma_{z} $ yelds the outcome $ | \uparrow_{z} > $. \end{enumerate} As an application of Hartle's strategy, let us suppose , for example, that: \begin{equation} | \uparrow_{x}^{( \bigotimes n)} > \; := \; \bigotimes_{i=1}^{n} \frac{1}{\sqrt{2}} ( | \uparrow_{z} > + | \downarrow_{z} > ) \end{equation} One has that: \begin{equation} < \uparrow_{x}^{(\bigotimes n)} | \bar{\sigma}_{z} | \uparrow_{x}^{(\bigotimes n)} > \; = \; 0 \end{equation} \begin{equation} < \uparrow_{x}^{(\bigotimes n)} | \bar{\sigma}_{z}^{2} | \uparrow_{x}^{(\bigotimes n)} > \; = \; \frac{1}{n} \end{equation} Thus, taking the limit $ n \rightarrow + \infty $, one concludes that $ \bar{\sigma}_{z} $ has vanishing dispersion about its mean value so that, at least in this sense, $ | \uparrow_{x}^{(\bigotimes \infty)} > $ is an "eigenstate" of $ \bar{\sigma}_{z} $ with eigenvalue zero. Coleman and Lesniewski has generalized Hartle's ideas observing that indeed one can require that the sequence $ \cdot_{i} \lambda_{i} $, where $ \lambda_{i} $ is the result of the measurement of the operator $ \sigma_{z}^{i} $, satisfies not only the Law of Randomness of 1-Borel normality, but all the Laws of randomess, i.e. that it is Martin L\"{o}f - Solovay - Chaitin random. So they introduce an orthogonal projection operator $ \hat{\Pi}_{random} $ that acting on a state $ | \psi > $ that is an eigenstate of each $ \sigma_{z}^{(i)} $ satisfies: \begin{equation} \label{eq:the Coleman-Lesniewski operator acts as the identity on eigenstates of random sequences} \hat{\Pi}_{random} | \psi > \; = \; | \psi > \end{equation} if the sequence of eigenvalues of $ \sigma_{z}^{(i)} $ is algorithmically-random, and: \begin{equation} \label{eq:the Coleman-Lesniewski operator vanishes on eigenstates of not random sequences} \hat{\Pi}_{random} | \psi > \; = \; 0 \end{equation} if the sequence of eigenvalues of $ \sigma_{z}^{(i)} $ is not algorithmically-random. Preskill reports that Coleman and Lesniewski discovered that eq.\ref{eq:the Coleman-Lesniewski operator acts as the identity on eigenstates of random sequences} and eq.\ref{eq:the Coleman-Lesniewski operator vanishes on eigenstates of not random sequences} properties, together with the condition that $ \hat{\Pi}_{random} $ is an orthogonal projection, are not sufficient to determine how $ \hat{\Pi}_{random} $ acts on all $ {\mathcal{H}}_{2}^{\bigotimes \infty} $, but that, with additional technical constrains, it exists, it is unique, and has the property that: \begin{equation} \hat{\Pi}_{random} | \uparrow_{x}^{ \bigotimes \infty } > \; = \; 1 \end{equation} These considerations seems to us rather strange, since, according to us, the operator $ \hat{\Pi}_{random} $ may be simply defined as: \begin{definition} \label{Coleman-Lesniewski operator} \end{definition} COLEMAN-LESNIEWSKI OPERATOR: \begin{equation} \hat{\Pi}_{random} \; := \; \int_{CHAITIN(\Sigma^{\infty})} dP_{unbaised} | \bar{x} > < \bar{x} | \end{equation} but than one has that: \begin{multline} \hat{\Pi}_{random} | \uparrow_{x} ^{\bigotimes \infty} > \; \\ = \; \hat{\Pi}_{random} \bigotimes_{i=1}^{\infty} [ \frac{1}{\sqrt{2}}( | 0 > + | 1 > ) \\ = \; ( \lim_{n \rightarrow \infty } \frac{1}{ 2 \frac{n}{2} } ) \: \hat{\Pi}_{random}( | 0^{\infty} > + | 1^{\infty} > ) \; = \; 0 \end{multline} \smallskip Introduced the following notion: \begin{definition} \label{def:Coleman random sequences of qubits} \end{definition} COLEMAN RANDOM SEQUENCES OF QUBITS: \begin{equation} COLEMAN-RANDOM( {\mathcal{H}}_{2}^{\bigotimes \infty} ) \; := \; \{ | \psi > \in {\mathcal{H}}_{2}^{\bigotimes \infty} ) \, : \, \hat{\Pi}_{random} | \psi > \; = \; | \psi > \} \end{equation} it would be clear why, according to our point of view, such a notion is completelly misleading as to the characterization of quantum algorithmic randomness: as we extensively discussed in section\ref{sec:Why to treat sequences of qubits one has to give up the Hilbert-Space Axiomatization of Quantum Mechanics}, since the right space of qubits' sequences is the noncommutative space $ \Sigma_{NC}^{\infty} $ and not the Hilbert space $ {\mathcal{H}}_{2}^{\bigotimes \infty} $, the space of algorithmically-random sequences of qubits is a set of objects of the form: \begin{equation} RANDOM( \Sigma_{NC}^{\infty} ) \; \subset \; \Sigma_{NC}^{\infty} \end{equation} and not a set of the form: \begin{equation} RANDOM( {\mathcal{H}}_{2}^{\bigotimes \infty} ) \; \subset \; {\mathcal{B}}({\mathcal{H}}_{2}^{\bigotimes \infty} ) \end{equation} as $ COLEMAN-RANDOM( {\mathcal{H}}_{2}^{\bigotimes \infty} ) $. Demanding to remark\ref{rem:difference between the raising of commutative cardinality ad the raising of noncommutative cardinality} and remark\ref{rem:the phenomenon of continuous dimension from a logical point of view} for a complete analysis, let us briefly recall that the passage from $ \Sigma_{NC}^{\star} $ to $ \Sigma_{NC}^{\infty} $ corresponds to a genuine increasing of \textbf{noncommutative cardinality} by one step, with the resulting effect of \textbf{continuous dimension} and, hence, \textbf{the lost of atomicity} of the underlying quantum logic, while the passage from $ {\mathcal{B}} ( { \mathcal{H}}_{2}^{\star} ) $ to $ {\mathcal{B}} ( { \mathcal{H}}_{2}^{\infty} ) $ corresponds to an increasing of \textbf{commutative cardinality} by one step, that is different from the correct required increasing of \textbf{noncommutative cardinality} by one step. From a logico-mathematical point of view, this can be seen introducing the following: \begin{definition} \label{def:Coleman quantum propositions} \end{definition} COLEMAN PROPOSITIONS: \begin{equation} CQP \; := \; \{ | \psi > < \psi | \; : \; | \psi > \in COLEMAN-RANDOM( {\mathcal{H}}_{2}^{\bigotimes \infty} ) \} \end{equation} Clearly any Coleman quantum proposition is an \textbf{atomic quantum propositions} of the weak quantum logic $ {\mathcal{L}} ( {\mathcal{H}}_{2}^{\bigotimes \infty} ) $. The effects of erroneously supposing that the quantum logic of qubits' sequences has atomic propositions may be appreciated by the following: \begin{remark} \label{rem:Coleman and the Entscheidungsproblem} \end{remark} THE HALTING-PROBABILITY'S COLEMAN ATOMIC PROPOSITION WOULD SOLVE THE COMMUTATIVE ENTSCHEIDUNGPROBLEM Let us consider the following Coleman quantum proposition: \begin{equation} p_{\Omega_{U}} \; := \; | \Omega_{U} > < \Omega_{U} | \end{equation} where, according to definition\ref{def:halting probability}, $ \Omega_{U}$ denotes the Halting Probability w.r.t. the Chaitin universal computer U. Let us, then, introduce the \textbf{qubits' sequence operator}: \begin{equation} \label{eq:qubits' sequence operator} \hat{q}^{\bigotimes \infty} \; := \; \bigotimes_{n \in {\mathbb{N}}} \hat{q} \end{equation} where $\hat{q}$ is the qubit operator defined in eq.\ref{eq:right definition of the qubit operator}. The measurement of $ \hat{q}^{ \bigotimes \infty } $ in the state $ | \Omega_{U} > $ results in the solution of the ($ C_{\Phi}$ - classical, i.e. commutative) Enstcheidungproblem (as David Hilbert indicated the problem of determining whether or not a given formula of the (Classical) Predicate Calculus is valid \cite{Davis-65}, \cite{Odifreddi-89}). We see, then, that the predicate $ p_{\Omega_{U}} $ encodes the solution of the Commutative Entsheidungsproblem. So we would have that the Quantum Propositional Calculus admits an\textbf{ atomic} proposition (from which other not-atomic i.e. not-elementary, propositions may be constructed logically-connecting it with other propositions through the connectives $ \bigvee , \bigwedge , \perp $), that, just alone, implies a violation of the Church-Turing Thesis. Assuming the Church-Turing Thesis, we have then to reject such a situation. \smallskip Let us explain, by the way, more precisely the meaning of the expression Commutative Enschteidungsproblem we used: the impossibility of developing all Mathematical-Logics simply by the distributive orthocomplemended lattice of Classical Predicate Calculus appears only when one wants to take into account quantifications. In this case, even restricting the analysis to First Order Theories in which one predicate cannot have other predicates or functions as arguments and quantification on predicates or functions is forbidden, one has to pass to Classical Formal Systems, their models and so on. Also the Classical Predicate Calculus may, of course, be then embedded in such a more sophisticate language: there exist many ways of axiomatize it as a classical formal system, and the general theory of formal systems may be applied to it to conclude that, as a formal system (in any way we axiomatized it), Classical Predicate Calculus is consistent though, as we have seen, undecidable. As far as Quantum Logic is concerned, many people, and we among them, tried to go beyond Quantum Predicate Calculus (i.e. the theory of orthocomplemented orthomodular lattices) to deveop a general theory of Quantum Formal Systems, the first attempts being of Von Neumann himself: \begin{center} \textit{"Dear Doctor Silsbee, It is with great regret that I am writing these lines to you, but I simply cannot help myself. In spite of very serious attempts to write the article " Logics of Quantum Mechanics" I find it it completely impossible to do it at this time. As you may know, I wrote a paper on this subject with Garrett Birkhoff in 1936 ("Annals of Mathematics", vol. 37 , pp. 823-843) and I have thought a good deal on the subject since. My work on continuous geometry , on which I gave the Amer. Math. Soc. Colloqiuim lectures of 1937, comes to a considerable extent from this source. Also a good deal concerning the relationship between strict- and probability logics (upon which I touched briefly in the Henry Joseph Lecture) and on the extension of this "Propositional calculus" work to "logic with quantifiers" (which I never discussed in public)"} (letter to Doctor Solbee ; July 2, 1945 ; cfr. \cite{von-Neumann-01}) \end{center} Personally we tried to develop: \begin{enumerate} \item a quantum corrispective of John Mc Carthy's LISP \cite{Mc-Carthy-60}, i.e. more precisely, of Chaitin's version of it in which the evaluation operator \emph{"eval"} of syntax: \begin{center} eval S-expression \end{center} is replaced by a time-constrained \cite{Chaitin-98} version of it, whose syntax: \begin{center} try time-limit S-expression \end{center} specifies the time-interval after which the computation halts furnishing as output the partial computation performed \footnote{Such a time-constraining is necessary since, otherwise, the request of evaluating a formal axiomatic system would never halt since, for all the not-trivial formal systems, the inferential chain of theorem-proving is infinite} We studied a new language, that we called the quantum-LISP, defined by the replacement of the instruction \emph{"try"} with a new instruction \emph{"quantum try"} with sintax: \begin{center} quantum-try time-limit quantum-S-expression \end{center} where a \textbf{quantum-S-expression} is a list of the form: \begin{equation}\label{eq:syntax of the quantum-try} (( S-expression_{G} \; S-expression_{H} ) \; ( a \; b )) \end{equation} with $ a , b \, \in \, {\mathbb{C}} \; | a |^{2} + | b |^{2} \, = \, 1 $. Under the command of eq.\ref{eq:syntax of the quantum-try} the computer chooses at random one value of a binary random variable h such that: \begin{align} Prob( & h = \frac{1}{2} ) \; = \; | a |^{2}\\ Prob( & h = - \frac{1}{2} ) \; = \; | b |^{2} \end{align} and then operates as follows: \begin{itemize} \item if occurs $ h = \frac{1}{2} $ then it sets the \textbf{halt-qubit-list} to $ (1 0)$ and operates as Chaitin-LISP would do under the instruction: \begin{center} try time-limit $ S-expression_{G} $ \end{center} \item if occurs $ h = - \frac{1}{2} $ then it sets the \textbf{halt-qubit-list} to $ (0 1)$ and operates as Chaitin-LISP would do under the instruction: \begin{center} try time-limit $ S-expression_{H} $ \end{center} \end{itemize} The idea underlying such a definition of the quantum-try instruction is to make it equivalent to a Deutsch's quantum Turing machine in which the periodic monitoring of the halting-qubit occurs at temporal-steps of time-interval \cite{Deutsch-85}. By the impossibility of having a fair random generator extensively discussed in section\ref{sec:Irreducibility of Quantum Computational Complexity Theory to Classical Computational Complexity Theory} Quantum-LISP is not implementable on a classical computer and, for practical purposes, must be replaced with a Virtual-quantum-LISP, i.e. a language completelly identical to Quantum-LISP, but for the fact that the fair random generator is replaced with a PRG. \item proceeding euristically, we tried to characterize the notion of a \textbf{quantum Post systems} associated to a classical Post system \cite{Odifreddi-89} $ {\mathcal{G}} \; := \, ( \Sigma , A , Q ) $ with the \textbf{axioms' set } $ A \subset \Sigma^{\star} $ and \textbf{productions' set } Q as a triple $ \hat{{\mathcal{G}}} \; := \; ( {\mathcal{H}}_{\Sigma} \, , \, {\mathcal{H}}_{A} \, , \, \hat{Q} )$, where $ {\mathcal{H}}_{\Sigma} \; = \; {\mathcal{H}}_{2}^{ \bigotimes \star} $, $ {\mathcal{H}}_{A} $ is an Hilbert sub-space of $ {\mathcal{H}}_{\Sigma} $, while $ \hat{Q} $ is the set of the \textbf{quantum productions}, i.e. operators on $ {\mathcal{H}}_{\Sigma} $ acting as the productions Q of $ {\mathcal{G}} $ on the computational basis. A theorem of a quantum Post system is then defined as an element of $ {\mathcal{H}}_{2}^{\star} $ reachable by a vector belonging to $ {\mathcal{H}}_{A} $ by a finite number of application of suitable quantum productions giving rise to a plethora of logical-mathematical notions specularizing the classical ones. \end{enumerate} We then discovered that formalizations of the theory of Quantum Formal Systems already existed in the literature: in 1996 Philiph Maymin introduced a quantum analogue of Alonzo Church's Lambda Calculus \cite{Mayimin-96}, an idea already independentely (and in a completely different way) developed by David Finkelstein in the section 14.3.7 of his monograph \cite{Finkelstein-97}. In 1997 Christopher Moore and James P. Crutchfield introduced quantum analogues of the whole Chomsky hierarchy \cite{Moore-Crutchfield-97}. A similar idea was concretelly implemented in his Masters Thesis by Bernard \"{O}emer who developed QCL: an high-level, architecture independent programming language for quantum computing whose interpreter is downloadable from the author's homepage \cite{Oemer-98}; the syntactic structure of a QCL program is described by a context-free grammar, in a way concisely explained in the section4.19.4 of \cite{Calude-Paun-01}. A step forward in the formalization of what a General Theory of Quantum Formal Systems has been done, according to us, by Paul Benioff \cite{Benioff-98} who, introducing (with the usual generalization on the computational basis) a quantum analogue of a toy-formal-system by Raymond M. Smullyian (cfr. the first chapter of \cite{Smullyan-92}) discusses not only its \textbf{syntax} but also its \textbf{semantic}. This is something new since, up to date, interpretations and models has been studied by the Quantum Logic Community only at the Quantum Propositional Calculus' level \label{Dalla-Chiara-Giuntini-01}. Now, exactly as the consideration of quantifications requires, in the classical case, to give up the simple lattice-theoretic Classical Propositional Calculus passing to the more sophisticated language of Classical Formal Systems and arriving, on this way, to axiomatize Classical Propositional Calculus itself formalizing its Entscheidungsproblem (that we call the Commutative Entscheidungsproblem) and discovering its unsolvability, we think the same must happen as to Quantum Propositional Calculus, whose Enstcheidungsproblem will be called the \textbf{Noncommutative Enstcheidungsproblem} from here and beyond. These preliminary, euristic considerations concerning \textbf{quantum formal systems} will be discussed, anyway, more explicitly in section\ref{sec:Karl Svozil's invention of Quantum Algorithmic Information Theory} where we will extensively discuss the quantum extension of the duality: \begin{center} \textbf{languages versus automata} \end{center} and the consequential characterization of the notion of \textbf{quantum formal system} obtained using such a duality at the correct level of Moore's generalization of Chomsky's hierarchy. \newpage \section{Karl Svozil's invention of Quantum Algorithmic Information Theory} \label{sec:Karl Svozil's invention of Quantum Algorithmic Information Theory} In 1995 Karl Svozil first introduced the idea that the irreducibility of Quantum Information Theory to the classical one implies the necessity of developing a quantum analogue of Classical Algorithmic Information Theory, namely Quantum Algorithmic Information Theory, irreducible to the classical theory \cite{Svozil-96}. Given a quantum computer Q, i.e. a quantum-mechanical physical system with Hilbert space $ {\mathcal{H}}_{2}^{\star} $ Svozil affords the first issue: \begin{center} \textbf{have the programs of Q to be coded in cbit or qubits?} \end{center} To obtain the quantum analogue of \textbf{prefix algorithmic entropy}, Svozil claims, their lengths must satisfy the Kraft's Inequality; but if we allowed Q's programs to be qubits' strings instead of cbits' strings, than the Kraft sum would diverge. As we will see this a key point, discussed also by Paul Vitanyi \cite{Vitanyi-99}, \cite{Vitanyi-01} in his rediscovering of Svozil's results and lying at the basis of the objections Andr\'{e} Berthiaume, Wim van Dam and Sophie Laplante moved to Vitanyi \cite{Berthiaume-van-Dam-Laplante-00} in their rediscovering of what Svozil had already discussed years before. The condition that the programs of Q are classical may be easily formalized observing that any map: \begin{equation} Q \in \stackrel{\circ}{MAP}( \Sigma^{\star} \, , \, {\mathcal{H}}_{2}^{ \bigotimes \star}) \end{equation} may be equivalentely seen as a map: \begin{equation} Q \; \in \; \stackrel{\circ}{MAP}( {\mathbb{E}}_{\star} \, , \, {\mathcal{H}}_{2}^{ \bigotimes \star}) \end{equation} identifying $ \Sigma^{\star} $ with the computational basis $ {\mathbb{E}}_{\star} $. Assuming that the quantum computer is a \textbf{closed system} Q will be clearly nothing but the restriction to $ {\mathbb{E}}_{\star} $ of an \textbf{inner automorphism} of $ {\mathcal{B}} ({\mathcal{H}}_{2}^{\star} )$. Assumed the prefix-free condition: \begin{equation} HALTING( Q ) \text{ is prefix-free} \end{equation} Svozil introduces the following: \begin{definition} \label{def:Svozil's quantum algorithmic information} \end{definition} QUANTUM ALGORITHMIC INFORMATION OF $ | \psi > $ W.R.T. Q: \begin{equation} I_{Q}( | \psi > ) \; := \; \begin{cases} \min \{ \vec{x} \in HALTING(Q) \: : \: Q ( \vec{x} ) = | \psi > \} & \text{if $ \exists \vec{x} \in HALTING(Q) \: : \: Q ( \vec{x} ) = | \psi > $ }, \\ + \infty & \text{otherwise}. \end{cases} \end{equation} Then Svozil considers the definition of a quantum analogue of Chaitin's Halting Probability. To see how Svozil implements such a notion it is necessary, first of all, to discuss his analysis of the Halting Problem for Quantum Computers, i.e. his analysis of Quantum Diagonalization. Diagonalization is a proof's technique introduced by Cantor to prove that $ cardinality( 2^{{\mathbb{N}}} ) \, > \, \aleph_{0} $. It may be formalized in the following way: \begin{theorem} \label{th:diagonalization's theorem} \end{theorem} DIAGONALIZATION'S THEOREM: \begin{hypothesis} \end{hypothesis} \begin{equation*} A \; \; set \end{equation*} \begin{equation*} R \; \subseteq \; A \times A \; \; \text{ binary relation on A} \end{equation*} \begin{equation} \label{eq:diagonal set of a binary relation} D \; := \; \{ a \in A \, : \, ( a , a) \notin R \} \; \; \text{ diagonal set for R} \end{equation} \begin{equation*} R_{a} \; := \; \{ b \in A : ( a , b ) \in R \} \; \; a \in A \end{equation*} \begin{thesis} \end{thesis} \begin{equation*} D \, \neq \, R_{a} \; \; \forall a \in A \end{equation*} \begin{proof} Suppose ad-absurdum that: \begin{equation} \exists \bar{a} \in A \; : \; D \, = \, R_{\bar{a}} \end{equation} i.e.: \begin{equation} \label{eq:the diagonal is a suitable row} \exists \bar{a} \in A \; : \; D \, = \, \{ b \in A \, : \, ( \bar{a}, b ) \in R \} \end{equation} Let us now consider the following question: \begin{equation*} \label{eq:basic question in diagonalization} \bar{a} \; \in \;D \; ? \end{equation*} \begin{itemize} \item if the answer to the question in eq.\ref{eq:basic question in diagonalization} is \textbf{yes} it follows by eq.\ref{eq:diagonal set of a binary relation} that $ ( \bar{a} \, , \, \bar{a} ) \; \notin \; R $ that, by eq.\ref{eq:the diagonal is a suitable row}, implies that $ \bar{a} \; \notin \; D $ that is asburdum \item if the answer to the question in eq.\ref{eq:basic question in diagonalization} is \textbf{no} it follows by eq.\ref{eq:diagonal set of a binary relation} that $ ( \bar{a} \, , \, \bar{a} ) \; \in \; R $ that, by eq.\ref{eq:the diagonal is a suitable row}, implies that $ \bar{a} \; \in \; D $ that is again asburdum \end{itemize} \end{proof} Cantor's argument runs than as follows: \begin{theorem} \label{th:Cantor's theorem} \end{theorem} CANTOR'S THEOREM: \begin{equation} cardinality( 2^{{\mathbb{N}}} ) \, > \, \aleph_{0} \end{equation} \begin{proof} Let us suppose ad absurdum that $ 2^{{\mathbb{N}}} $ is countable. Then there exists a a way of enumerating all members of $ 2^{{\mathbb{N}}} $ as: \begin{equation} 2^{{\mathbb{N}}} \; = \; \{ R_{0} , R_{1} , R_{2} , \cdots \} \end{equation} Introduced the relation on $ {\mathbb{N}} $ as: \begin{equation} R \; := \; \{ ( i , j ) \, \in \, {\mathbb{N}} \times {\mathbb{N}} \: : \: j \in R_{i} \} \end{equation} the thesis immediately follows applying to R the theorem\ref{th:diagonalization's theorem} \end{proof} \smallskip Let us now pass to partial recursive functions and let us introduce the following two sets: \begin{definition} \label{def:first self-referential set} \end{definition} FIRST SELF-REFERENTIAL SET: \begin{equation} SR_{1} \; := \; \{ i \in {\mathbb{N}} \, : \, i \in {\mathcal{W}}_{i} \} \end{equation} \begin{definition} \label{def:second self-referential set} \end{definition} SECOND SELF-REFERENTIAL SET: \begin{equation} SR_{2} \; := \; \{ ( i , j ) \, \in \, {\mathbb{N}} \times {\mathbb{N}} \, : \, i \in {\mathcal{W}}_{j} \} \end{equation} Cantor's diagonalization argument immediately leads to the following importan theorems: \begin{theorem} \label{th:combinatorial core of the undecidability results} \end{theorem} COMBINATORIAL CORE OF THE UNDECIDABILITY RESULTS: \begin{equation*} SR_{1} \; \text{ is r.e. but not recursive} \end{equation*} \begin{proof} We have that: \begin{equation} x \in SR_{1} \; \Leftrightarrow \; \varphi_{x}(x) \downarrow \end{equation} But theorem\ref{th:Goedel's numbering of partial recursive functions} tells us that there exist a partial recursive $ \varphi $ such that: \begin{equation} \varphi (x) \; = \; \varphi_{x} (x) \end{equation} and hence: \begin{equation} SR_{1} \; = \; HALTING( \varphi ) \end{equation} So $ SR_{1} $, being the halting set of a partial recursive function, is a r.e. set. The fact that $ SR_{1} $ is not recursive follows immediately by applying theorem\ref{th:Goedel's numbering of partial recursive functions} to the relation $ SR_{2} $. \end{proof} \begin{theorem} \label{th:unsolvability of the Halting Problem} \end{theorem} UNSOLVABILITY OF THE HALTING PROBLEM: \begin{equation*} SR_{2} \; \text{ is r.e. but not recursive} \end{equation*} \begin{proof} We have that: \begin{equation} (i,j) \in SR_{2} \; \Leftrightarrow \; \varphi_{j}(i) \downarrow \end{equation} But theorem\ref{th:Goedel's numbering of partial recursive functions} tells us that there exist a partial recursive $ \varphi $ such that: \begin{equation} \varphi (i) \; = \; \varphi_{j} (i) \end{equation} and hence: \begin{equation} SR_{2} \; = \; HALTING( \varphi ) \end{equation} So $ SR_{2 } $, being the halting set of a partial recursive function, is a r.e. set. Let us then suppose by absurdum that $ SR_{2 } $ is recursive. Since: \begin{equation} x \in SR_{1} \; \Leftrightarrow \; ( x \, , \, x ) \in x \in SR_{2} \end{equation} this implies that $ SR_{1} $ is not recursive too, contradicting theorem\ref{th:combinatorial core of the undecidability results}. \end{proof} \begin{remark} \label{rem:the first self-referential set and Russell's paradox} \end{remark} THE FIRST SELF-REFERENTIAL SET AND RUSSELL'S PARADOX Bertrand Russell's Paradox is certainly the most famous example of the many subtlities that appear in the formalization of \textbf{classes}, i.e. of \textbf{sets} whose elements are \textbf{sets} themselves. It runs as follows: considered the set: \begin{equation} A \; := \; \{ x \, : \, x \notin x \} \end{equation} one has that: \begin{equation} x \in A \; \Leftrightarrow \; x \notin x \end{equation} and thus: \begin{equation} A \in A \; \Leftrightarrow \; A \notin A \end{equation} that is nonsense. Let us now observe that the set $ {\mathbb{N}} \, - \, SR_{1} $ resembles Russell's set A: it is the set of numbers not belonging to the r.e. set they code. But the is no paradox here because Russel's argument simply shows that such a set is not r.e. itself. \smallskip \begin{remark} \label{ref:programmation and meta-programmation} \end{remark} PROGRAMMATION AND META-PROGRAMMATION The meaning of theorem\ref{th:unsolvability of the Halting Problem} may be appreciated taking into account the concrete programmation on the (classical, deterministic) computers we use every day, observing that, by Church-Turing Thesis, the specific hardware nature of the consided computer is irrilevant. We can divide the set of all programming languages for a generic computer in two classes, according to if they admit \textbf{meta-programmation} or not. By meta-programmation we mean the ability of programs to deal with that particular kind of \textbf{objects} made by program themselves. Indeed the more logico-mathematically featured programming languages deal whith only one structure of objects (e.g. \textbf{lists} in Mac Carthy's LISP \cite{Mc-Carthy-60} or \textbf{expressions} in Wolfram's Mathematica). So they automatically admit \textbf{meta-programmation} since programs and the other objects on which they operate are of the same (unique) structure. What is important to observe is that the \textbf{meta-programmation ability} realizes exactly that link between \textbf{language} and \textbf{meta-language} that we indicated in the remark\ref{Godel numbering and self-reference} as the door leading (or better allowing) self-reference. \smallskip We can now explain the diagonalization argument lying behind theorem\ref{th:unsolvability of the Halting Problem} in the following more concrete way \cite{Svozil-93}, \cite{Davis-Sigal-Weyuker-94}, \cite{Lewis-Papadimitriou-98}: Let us suppose, for example to enter a Mathematica session. We could thus think that it is possible, using the meta-programming in a clever way, to define, through a suitable Mathematica expression: \begin{equation} \label{eq:halt expression} In[1] \; := \; HALT[p_{-} , x_{-}] \: := \: \cdots \end{equation} a function HALT[p,x] that, when called, returns a cbit having the value True or False according if, respectively, Mathematica halts or doesn't halt under the input p[x] If such a Mathematica expression HALT[p,x] existed it could be used to construct the following: \begin{definition} \label{def:diagonal expression} \end{definition} DIAGONAL EXPRESSION: \begin{equation} In[2] \; := \; DIAGONAL[x_{-}] \: := \: (Label[start] \, ; \, If[Halt[x,x]==True \, , \, Goto[start] \, , \,True ) \end{equation} Notice what DIAGONAL[x] does: if the HALT program decides that the program x would halt if presented with itself as input, then DIAGONAL(x) loops forever; otherwise it gives as output True and then halts. From the function DIAGONAL[x] we could, then, construct the following Mathematica expression: \begin{definition} \label{def:paradox expression} \end{definition} \begin{equation} In[3] \; := \; PARADOX \: := \: DIAGONAL[DIAGONAL] \end{equation} Let us now give to Mathematica the following input: \begin{equation} In[4] \; := \; PARADOX \end{equation} \smallskip Will Mathematica halt giving the output Out[4] or not? \smallskip It will do it iff the input HALT[DIAGONAL,DIAGONAL] gives as output False; in other words Mathematica halts if and only if it doesn't halt. That is a contradiction. So we must conclude that the only hypothesis that star