{
  "id": "1901.09765",
  "version": "v1",
  "published": "2019-01-28T16:12:00.000Z",
  "updated": "2019-01-28T16:12:00.000Z",
  "title": "Thermodynamic Formalism for Quantum Channels: Entropy, Pressure and Gibbs channels",
  "authors": [
    "Jader E. Brasil",
    "Josue Knorst",
    "Artur O. Lopes"
  ],
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.PR",
    "quant-ph"
  ],
  "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 $L:M_k\\to M_k$ 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 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 $X_n$, $n\\in \\mathbb{N}$, taking values on the projective space $ P(\\C^k)$ and analyze the question of existence of invariant probabilities. We also consider an associated process $\\rho_n$, $n\\in \\mathbb{N}$, 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 $\\phi_L$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-28T16:12:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37A50",
      "37A60",
      "81Q35"
    ],
    "keywords": [
      "quantum channel",
      "gibbs channel",
      "thermodynamic formalism",
      "invariant probabilities",
      "density operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}