Sanov and Central Limit Theorems for output statistics of quantum Markov chains

TL;DR

This paper develops large deviations and central limit theorems for the output statistics of quantum Markov chains, enabling analysis of complex quantum dynamical behaviors beyond average measures.

Contribution

It introduces a novel framework for analyzing output statistics of quantum Markov chains using an extended quantum transition operator, establishing large deviations and CLT results.

Findings

Established a Sanov-type large deviations principle for quantum Markov chains.

Proved a central limit theorem for the empirical measure of quantum output statistics.

Demonstrated the potential to detect dynamical phase transitions through higher-level statistics.

Abstract

In this paper we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov's theorem for the empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction we give an example of a finite system whose level-one rate function is independent of a model parameter while the level-two rate is not.