Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties
英文摘要：
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 $L:M_k\to M_k$ and the measure $μ$ on $M_k$ we define the linear operator $ϕ_L:M_k \to M_k$, via the expression $ρ\,\to\,ϕ_L(ρ) = \int_{M_k} L(v) ρ{L(v)}^\dagger \, \dm(v)$. 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 $ϕ_L$ we analyze the eigenvalue property for $ϕ_L$ and we define entropy for such channel. For a fixed $μ$ (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 $μ$ (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 $X_n$, $n\in \mathbb{N}$, 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 $ρ_n$, $n\in \mathbb{N}$, 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 $ϕ_L$.
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