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

• Published in: arXiv.org
• Main Authors: Brasil, Jader E, Knorst, Josue, Lopes, Artur O
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.08.2021
• Subjects: Center of gravity; Channels; Density; Eigenvalues; Entropy; Invariants; Linear operators; Operators (mathematics)
• ISSN: 2331-8422

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 \(\phi_L\).