Thermodynamic formalism for quantum channels: Entropy, pressure, Gibbs channels and generic properties

Denote Mk the set of complex k by k matrices. We will analyze here quantum channels φL of the following kind: given a measurable function L: Mk → Mk and the measure μ on Mk we define the linear operator φL:Mk → Mk, via the expression ρ → φL(ρ) =fMkL(v)ρL(v)†dμ(v). A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where L was the identity. Under some mild assumptions on the quantum channel φL we analyze the eigenvalue property for φL and we define entropy for such channel. For a fixed μ (the a priori measure) and for a given a Hamiltonian H:Mk → Mk we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such H) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed μ (with more than one point in the support) the set of L such that it is φ-Erg (also irreducible) for μ is a generic set. We describe a related process Xn, n, taking values on the projective space P(k) and analyze the question of the existence of invariant probabilities. We also consider an associated process ρn, n, with values on k (k is the set of density operators). Via the barycenter, we associate the invariant probability mentioned above with the density operator fixed for φL.