Absorption and Fixed Points for Semigroups of Quantum Channels

TL;DR

This paper investigates fixed points of semigroups of quantum channels, showing they form a W*-algebra under certain conditions, and introduces absorption operators to analyze their structure, including a novel example of a recurrent semigroup.

Contribution

It refines the understanding of fixed points of quantum channel semigroups using noncommutative potential theory and absorption operators, providing new structural insights and examples.

Findings

Fixed points form a W*-algebra under recurrence.

Absorption operators characterize fixed point sets.

An example of a recurrent semigroup without orthogonal minimal invariant domains.

Abstract

In the present work we review and refine some results about fixed points of semigroups of quantum channels. Noncommutative potential theory enables us to show that the set of fixed points of a recurrent semigroup is a W*-algebra; aside from the intrinsic interest of this result, it brings an improvement in the study of fixed points by means of absorption operators (a noncommutative generalization of absorption probabilities): under the assumption of absorbing recurrent space (hence allowing non-trivial transient space) we can provide a description of the fixed points set and a probabilistic characterization of when it is a W*-algebra in terms of absorption operators. Moreover we are able to exhibit an example of a recurrent semigroup which does not admit a decomposition of the Hilbert space into orthogonal minimal invariant domains (contrarily to the case of classical Markov chains and positive recurrent semigroups of quantum channels).