On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schroedinger equations

TL;DR

This paper explores the connection between quantum Markov semigroups and classical diffusion processes, establishing conditions under which properties like irreducibility are equivalent or related.

Contribution

It proves that a quantum Markov semigroup's irreducibility is equivalent to the totality of associated classical diffusions and analyzes the relationship between irreducibility and other properties.

Findings

Quantum Markov semigroup irreducibility iff classical diffusions are total in the Hilbert space

Irreducibility of the semigroup relates to properties like accessibility and Lie algebra rank condition

These properties are generally weaker than semigroup irreducibility but are equivalent in key classes

Abstract

A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schr\"odinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, weaker than irreducibility of the quantum Markov semigroup, nevertheless, they are equivalent for some important classes of semigroups.