Reversibility in Quantum Models of Stochastic Processes

Natural phenomena such as time series of neural firing, orientation of layers in crystal stacking and successive measurements in spin-systems are inherently probabilistic. The provably minimal classical models of such stochastic processes are $\varepsilon $-machines, which consist of internal states, transition probabilities between states and output values. The topological properties of the $\varepsilon $-machine for a given process characterize the structure, memory and patterns of that process. However $\varepsilon $-machines are often not ideal because their statistical complexity (C$_{\mathrm{\mu }})$ is demonstrably greater than the excess entropy (\textbf{E}) of the processes they represent. Quantum models (q-machines) of the same processes can do better in that their statistical complexity (C$_{\mathrm{q}})$ obeys the relation C$_{\mathrm{\mu }}\ge $C$_{\mathrm{q}}\ge $\textbf{E}. q-machines can be constructed to consider longer lengths of strings, resulting in greater compression. With code-words of sufficiently long length, the statistical complexity becomes time-symmetric -- a feature apparently novel to this quantum representation. This result has ramifications for compression of classical information in quantum computing and quantum communication technology.