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

We denote by M k the set of k by k matrices with complex entries. We consider quantum channels φ L of the form: given a measurable function L : M k → M k and a measure µ on M k we define the linear operator φ L :On a previous work the authors show that for a fixed measure µ it is generic on the function L the Φ-Erg property (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic process: one takes values on the projective space P (C k ) and the other on matrices in M k . The Φ-Erg property and the purification condition are good hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically onOn the previous work it was presented the concepts of entropy of a channel and of Gibbs channel; and also an example (associated to a stationary Markov chain) where this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the above mentioned example and we show that it is equal to − 1 2 h, where h is the entropy of the associated Markov probability.

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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 .

M n (C) denotes the set of n by n complex matrices. Consider continuous time quantum semigroups P t = e t L , t ≥ 0, where L : M n (C) → M n (C) is the infinitesimal generator. If we assume that L(I) = 0, we will call e t L , t ≥ 0 a quantum Markov semigroup. Given a stationary density matrix ρ = ρ L , for the quantum Markov semigroup P t , t ≥ 0, we can define a continuous time stationary quantum Markov process, denoted by X t , t ≥ 0. Given an a priori Laplacian operator L 0 : M n (C) → M n (C), we will present a natural concept of entropy for a class of density matrices on M n (C). Given an Hermitian operator A : C n → C n (which plays the role of an Hamiltonian), we will study a version of the variational principle of pressure for A. A density matrix ρ A maximizing pressure will be called an equilibrium density matrix. From ρ A we will derive a new infinitesimal generator L A . Finally, the continuous time quantum Markov process defined by the semigroup P t = e t L A , t ≥ 0, and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian A. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian A.