Irreducible decompositions and stationary states of quantum channels

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.