Generalized Kac's Lemma for Recurrence Time in Iterated Open Quantum Systems

TL;DR

This paper generalizes Kac's lemma to quantum channels, showing that the expected recurrence time relates to the initial state's weight in the steady state, extending previous results for unital channels.

Contribution

It introduces a generalized Kac's lemma for a broader class of quantum channels, linking recurrence time to the steady state properties.

Findings

Expected return time equals the inverse of the initial state's weight in the steady state.

Generalization applies to non-unital quantum channels.

Recurrence measurement protocol is analyzed for quantum channels.

Abstract

We consider recurrence to the initial state after repeated actions of a quantum channel. After each iteration a projective measurement is applied to check recurrence. The corresponding return time is known to be an integer for the special case of unital channels, including unitary channels. We prove that for a more general class of quantum channels the expected return time can be given as the inverse of the weight of the initial state in the steady state. This statement is a generalization of the Kac lemma for classical Markov chains.