Thermodynamic formalism for quantum channels: Entropy, pressure, Gibbs channels and generic properties
| Autor: | Jader E. Brasil, Josue Knorst, Artur O. Lopes |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Measurable function
Formalism (philosophy) Generic property General Mathematics FOS: Physical sciences Dynamical Systems (math.DS) Quantum channel Measure (mathematics) Entropy (classical thermodynamics) FOS: Mathematics Computer Science::General Literature Mathematics - Dynamical Systems Quantum Mathematical Physics ComputingMilieux_MISCELLANEOUS Mathematics Mathematical physics Quantum Physics Computer Science::Information Retrieval Applied Mathematics Probability (math.PR) Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Mathematical Physics (math-ph) TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES 37D35 37A50 37A60 81Q35 ComputingMethodologies_DOCUMENTANDTEXTPROCESSING Quantum Physics (quant-ph) Mathematics - Probability |
| Zdroj: | Communications in Contemporary Mathematics. 25 |
| ISSN: | 1793-6683 0219-1997 |
| DOI: | 10.1142/s0219199721500905 |
| Popis: | 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 $\phi_L$. Comment: We correct some mistakes in some proofs and we add a new section on genericty of ergodicity. We change the title |
| Databáze: | OpenAIRE |
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