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- W2912558383 abstract "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 $L:M_kto M_k$ 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 assumptions on the quantum channel $phi_L$ we analyze the eigenvalue property for $phi_L$ and we also define entropy for the channel $phi_L$. For a fixed $mu$ (the {it a priori} measure) given an Hamiltonian $H: M_k to M_k$ (which defines $phi_H$) we present a variational principle of pressure and relate it to the eigenvalue problem. We introduce the concept of Gibbs channel. We describe a related process $X_n$, $nin mathbb{N}$, taking values on the projective space $ P(C^k)$ and analyze the question of existence of invariant probabilities. We also consider an associated process $rho_n$, $nin mathbb{N}$, 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 $phi_L$." @default.
- W2912558383 created "2019-02-21" @default.
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- W2912558383 date "2019-01-28" @default.
- W2912558383 modified "2023-09-27" @default.
- W2912558383 title "Thermodynamic Formalism for Quantum Channels: Entropy, Pressure and Gibbs channels" @default.
- W2912558383 hasPublicationYear "2019" @default.
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