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

Denote $M_k$ the set of complex $k$ by $k$ matrices. We will analyze here quantum channels $\phi_L$ of the following kind: given a measurable function $L:M_k\to M_k$ and the measure $\mu$ on $M_k$ we define the linear operator $\phi_L:M_k \to M_k$, via the expression $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho {L(v)}^\dagger \, \dm(v)$. 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 $\phi_L$ we analyze the eigenvalue property for $\phi_L$ and we define entropy for such channel. For a fixed $\mu$ (the \textit{a priori} measure) and for a given a Hamiltonian $H: M_k \to M_k$ 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 $\mu$ (with more than one point in the support) the set of $L$ such that it is $\phi$-Erg (also irreducible) for $\mu$ is a generic set. We describe a related process $X_n$, $n\in \mathbb{N}$, taking values on the projective space $P(\C^k)$ and analyze the question of the existence of invariant probabilities. We also consider an associated process $\rho_n$, $n\in \mathbb{N}$, with values on $\mathcal{D}_k$ ($\mathcal{D}_k$ is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator which is fixed for $\phi_L$.