Optimality and Complexity in
Measured Quantum-State Stochastic Processes

ABSTRACT: 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.

Selected animations of measurement-induced complexity and structure:

See Supplemental Material at [URL will be provided by publisher] for a suite of animations that showcase the effect of measurement angles on observed randomness and complexity. Linked in below for convenience.

Animation 1: Nemo0psweep

(Top) Mixed-state presentation states for the stochastic processes generated by measuring the cCQS shown in Fig. 10—the Nonorthogonal Nemo Process that outputs qubits in states |+> and |0>. The measurement angle φ is fixed to zero and the measurement angle θ is varied from 0 to π as the animation progresses. Half-way through the angle sweep, the value θ = π/2 coincides with the measurement basis {|+>, |->}, which is the one shown in Fig. 12. Notably, the MSP invariant measure varies smoothly with respect to the measurement.

(Bottom) Entropy rate h_μ of the measured process as a function of θ: The vertical bar tracks the value of θ at each animation frame.

Animation 2: Rip0asweep

Entropy rate h_μ for the MSPs of the measured cCQS in Fig. 13—the Random Insertion Process—as a function of the measurement angle θ along the horizontal axis. Each animation frame represents a different value of the other measurement angle φ, which is varied from 0 to 2 π. As expected, the change in entropy rate is smooth with respect to both measurement angles. Interestingly, in some φ ranges the entropy rate is greater than the entropy rate h_μ^g of the underlying cCQS for all values of θ. In other regions, certain combinations of the angles achieve observed entropy rates that smaller than that of the underlying cCQS.