08-199 Ulrich Mutze

Quantum Image Dynamics - an entertainment application of separated quantum dynamics (3592K, pdf) 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 of pixels. 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.

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