Quantized recurrence time in unital iterated open quantum dynamics

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 superoperator (quantum channel) in each time step 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 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 shows that the expected return time is 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.