- 08-199 Ulrich Mutze
- Quantum Image Dynamics - an entertainment application of separated quantum dynamics
Oct 24, 08
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Abstract. A bijective mapping is established between
the set of pure qubit states and the set $[0,1]^3$.
This latter set corresponds naturally to the RGB data
which most digital image formats associate with the color
This allows to associate with a digital color image
(considered as a $[0,1]^3$-valued matrix) a rectangular
lattice of qubits, where for each pixel there is a qubit, the
state of which is determined by the pixel's color data
according to the correspondence mentioned above.
We thus associate with a digital color image an
idealized physical system.
We define a law of dynamical evolution for this system in a manner that
not only the initial state but also each evolved state
can be represented as a color image.
This will be done in two steps:
1. A Hamiltonian is specified which represents interaction of the pixel-based qubits
with a homogeneous magnetic field together with a Heisenberg spin interaction
between adjacent qubits.
2. Evolution is defined not as the exact quantum dynamics defined by the specified Hamiltonian,
but as the approximate quantum dynamics which treats this interaction
via the time-dependent Hartree equations and thus leaves each qubit in a pure state,
to which there corresponds a well-defined color.
This approximate dynamics is computationally very cheap with a computational
complexity proportional to the number of pixels, whereas the complexity of exact dynamics
is well known to grow exponentially with that number.
This method allows to evolve a digital color image, thus producing a
`movie' from it.
In such a movie the image undergoes changes which may be considered as interesting
graphical effects in a first phase.
In the course of further evolution, the larger and obvious structures of the image
fade away and what remains is a dull, grainy, uniformity of seemingly random origin.
Since, however, the evolution scheme is reversible, the initial image can be recovered
from the final image by reversed evolution. Two examples of such evolving
images are presented, one together with the reversed evolution.