Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

Abstract

Denote M k the set of complex k by k matrices. We will analyze here quantum channels φ L of the following kind: given a measurable function L : M k → M k and the measure µ on M k we define the linear operatorA 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 φ L we analyze the eigenvalue property for φ L and we define entropy for such channel. For a fixed µ (the a priori measure) and for a given an Hamiltonian H : M k → M k we present a variational principle of pressure (associated to such H) and relate it to the eigenvalue problem. 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 X n , n ∈ N, taking values on the projective space P (C k ) and analyze the question of existence of invariant probabilities.We also consider an associated process ρ n , n ∈ N, with values on D k (D k is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator fixed for φ L .