Quantized recurrence time in iterated open quantum dynamics

TL;DR

This paper proves that in certain open quantum systems, the expected return time to the initial state is quantized as an integer equal to the dimension of the relevant Hilbert space, linking classical and quantum cases.

Contribution

It establishes a general condition under which the expected return time in open quantum systems is quantized, extending previous results for classical and unitary quantum systems.

Findings

Expected return time is an integer when the superoperator is unital.

The expected return time equals the dimension of the relevant Hilbert space.

Results unify classical Markov chains and quantum walks under a common framework.

Abstract

The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a general quantum channel, in each timestep followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unital in the relevant Hilbert space (the part of the Hilbert space explored by the system), then the expectation value of the return time is an integer, equal to the dimension of this relevant Hilbert space. We illustrate our results on partially coherent quantum walks on finite graphs. Our work connects the previously known quantization of the expected return time for bistochastic Markov chains and for unitary quantum walks, and shows that these are special cases of a more general statement. The expected return time is thus a quantitative measure of the size of the part of the Hilbert space available to the system when the dynamics is started from a certain state.