Irreducible Decompositions and Stationary States of Quantum Channels

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 [3]: this result is, in a probabilistic language, a decomposition of a general quantum channel into its irreducible recurrent components. More precisely, we prove that the positive recurrent subspace (i.e. the space supporting the invariant states) can be decomposed as the direct sum of supports of extremal invariant states; this decomposition is not unique, in general, but we can determine all the possible decompositions. This allows us to describe the full structure of invariant states.