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Abstract. We study the analogues of irreducibility, period, and communicating classes for open quantum random walks, as defined in [3]. We recover results similar to the standard ones for Markov chains, in terms of ergodic behavior, decomposition into irreducible subsystems, and characterization of stationary states.

We study open quantum random walks (OQRWs) for which the underlying graph is a lattice, and the generators of the walk are homogeneous in space. Using the results recently obtained in Carbone and Pautrat (Ann Henri Poincaré, 2015), we study the quantum trajectory associated with the OQRW, which is described by a position process and a state process. We obtain a central limit theorem and a large deviation principle for the position process. We study in detail the case of homogeneous OQRWs on the lattice Z d , with internal space h = C 2 .

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 ([3]). 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.

We prove hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup on the entire algebra B(h) of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.