Irreducible decompositions and stationary states of quantum channels

TL;DR

This paper generalizes a decomposition of quantum channels into irreducible components to infinite dimensions, providing insights into the structure of invariant states and their supports in quantum information theory.

Contribution

It extends the Baumgartner and Narnhofer result to infinite-dimensional quantum channels, linking irreducible decompositions with invariant state structures.

Findings

Decomposition of quantum channels into irreducible components in infinite dimensions

Characterization of invariant states and their supports

Relation between decomposition and communication structure

Abstract

For a quantum channel (completely positive, trace-preserving map), we prove a generalization to the infinite dimensional case of a result by Baumgartner and Narnhofer. This result is, in a probabilistic language, a decomposition of a general quantum channel into its irreducible positive recurrent components. This decomposition is related with a communication relation on the reference Hilbert space. This allows us to describe the full structure of invariant states of a quantum channel, and of their supports.