Trace distance ergodicity for quantum Markov semigroups

TL;DR

This paper establishes a spectral criterion for the ergodicity of quantum Markov semigroups using trace distance, and applies it to various bosonic, fermionic, and Lie algebra-based semigroups.

Contribution

It introduces a general spectral criterion for quantum ergodicity and demonstrates its application to diverse classes of quantum Markov semigroups.

Findings

Spectral criterion effectively characterizes ergodicity in quantum Markov semigroups.

Application to bosonic and fermionic Ornstein-Uhlenbeck semigroups shows practical utility.

Analysis of Lie algebra-based semigroups extends understanding of quantum ergodicity.

Abstract

We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.