Minimized State-Complexity of Quantum-Encoded Cryptic Processes

ABSTRACT: The predictive information required for proper trajectory sampling of a stochastic process can be more efficiently transmitted via a quantum channel than a classical one. This recent discovery allows quantum information processing to drastically reduce the memory necessary to simulate complex classical stochastic processes. It also points to a new perspective on the intrinsic complexity that nature must employ in generating the processes we observe. The quantum advantage increases with codeword length—the length of process sequences used in constructing the quantum communication scheme. In analogy with the classical complexity measure, statistical complexity, we use this reduced communication cost as a measure of state-complexity in the quantum representation. Previously difficult to compute, the quantum advantage is expressed here in closed form using spectral decomposition. This allows for efficient numerical computation of the quantum-reduced state-complexity at all encoding lengths, including infinite. Additionally, it makes clear how finite-codeword reduction in state-complexity is controlled by the classical process' cryptic order. And, it allows asymptotic analysis of infinite-cryptic order processes.