Measured Quantum-State Stochastic Processes

Ariadna E. Venegas-Li and James P. Crutchfield |

**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.

Ariadna E. Venegas-Li and James P. Crutchfield, “Optimality and Complexity in Measured Quantum-State Stochastic Processes”, Journal of Statistical Physics

doi:10.1007/s10955-023-03112-8.

[pdf].

arxiv.org:2205.03958 [quant-ph].

**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.

(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.

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.