On entropy production of repeated quantum measurements II. Examples

TL;DR

This paper explores entropy production in repeated quantum measurements through examples, highlighting phenomena like non-Gibbsian measures and large deviation behaviors, and connecting quantum information with topics like Markov models and number theory.

Contribution

It provides concrete examples illustrating complex entropy production phenomena in quantum measurements, emphasizing the role of thermodynamic formalism and non-Gibbsian measures.

Findings

Some quantum instruments produce non-Gibbsian probability measures.

Entropy production rate can satisfy large deviation principles without obeying CLT.

Connections are drawn between quantum measurement entropy and topics like Markov chains and number theory.

Abstract

We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation-dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.