Irreducibility of Quantum Markov Semigroups, uniqueness of invariant states and related properties

TL;DR

This paper explores the concept of irreducibility in Quantum Markov Semigroups, establishing its equivalence with primitivity and relaxation properties, and provides practical criteria for checking irreducibility in various settings.

Contribution

It offers new characterizations of irreducibility, links it with other dynamical features, and presents methods to verify irreducibility using generator operators.

Findings

Irreducibility, primitivity, and relaxation are equivalent under certain conditions.

Provides criteria for checking irreducibility in uniformly continuous QMSs.

Includes elementary proofs and covers finite and infinite dimensional cases.

Abstract

We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.