On the generators of quantum dynamical semigroups with invariant subalgebras

TL;DR

This paper provides a unified framework for characterizing quantum dynamical semigroups and CP-maps with invariant subalgebras, offering normal forms and connecting various results in the literature.

Contribution

It introduces a normal form for invariant GKLS-generators based on known forms for CP-maps and offers a normal form for invariant CP-maps when the algebra is atomic, unifying different approaches.

Findings

Normal form for $ ext{A}$-invariant GKLS-generators derived from CP-map forms

Normal form for $ ext{A}$-invariant CP-maps in atomic algebras

Reproduction of existing results as special cases

Abstract

The problem of characterizing GKLS-generators and CP-maps with an invariant appeared in different guises in the literature. We prove two unifying results which hold even for weakly closed *-algebras: First, we show how to construct a normal form for $\mathcal{A}$-invariant GKLS-generators, if a normal form for $\mathcal{A}$-invariant CP-maps is known - rendering the two problems essentially equivalent. Second, we provide a normal form for $\mathcal{A}$-invariant CP-maps if $\mathcal{A}$ is atomic (which includes the finite-dimensional case). As an application we reproduce several results from the literature as direct consequences of our characterizations and thereby point out connections between different fields.