Optimality and Complexity in
Measured Quantum-State Stochastic Processes

Ariadna E. Venegas-Li and James P. Crutchfield

Complexity Sciences Center
Physics Department
University of California at Davis
Davis, CA 95616

ABSTRACT: Temporal sequences of quantum states are essential to quantum computation protocols, as used in quantum key distribution, and to quantum computing implementations, as witnessed by substantial efforts to develop on-demand single-photon sources. To date, though, these sources emit qubit sequences in which the experimenter has little or no control over the outgoing quantum states. The photon stream emitted by a color center is a familiar example. As a diagnostic aid, one desires appropriate metrics of randomness and correlation in such quantum processes.

If an experimentalist observes a sequence of emitted quantum states via either projective or positive-operator-valued measurements, the outcomes form a time series. Individual time series are realizations of a stochastic process over the measurements' classical outcomes. We recently showed that, in general, the resulting stochastic process is highly complex in two specific senses: (i) it is inherently unpredictable to varying degrees that depend on measurement choice and (ii) optimal prediction requires using an infinite number of temporal features. Here, we identify the mechanism underlying this complicatedness as generator nonunifilarity—the degeneracy between sequences of generator states and sequences of measurement outcomes. This makes it possible to quantitatively explore the influence that measurement choice has on a quantum process' degrees of randomness and structural complexity using recently introduced methods from ergodic theory. Progress in this, though, requires quantitative measures of structure and memory in observed time series. And, success requires accurate and efficient estimation algorithms that overcome the requirement to explicitly represent an infinite set of predictive features. We provide these metrics and associated algorithms, using them to design informationally-optimal measurements of open quantum dynamical systems.

Ariadna E. Venegas-Li and James P. Crutchfield, “Optimality and Complexity in Measured Quantum-State Stochastic Processes”, (2022).
arxiv.org:2205.03958 [quant-ph].