{
  "id": "2204.07734",
  "version": "v1",
  "published": "2022-04-16T07:03:54.000Z",
  "updated": "2022-04-16T07:03:54.000Z",
  "title": "The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems",
  "authors": [
    "Alexander A. Andrianov",
    "Mikhail V. Ioffe",
    "Ekaterina A. Izotova",
    "Oleg O. Novikov"
  ],
  "comment": "27 p.p",
  "journal": "Symmetry, 14 (2022) 754",
  "doi": "10.3390/sym14010754",
  "categories": [
    "quant-ph",
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension $2$. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to $1$, and find a moment of time starting from which the FGKLS equation can be used - the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-16T07:03:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "two-dimensional systems",
      "fgkls equation",
      "franke-gorini-kossakowski-lindblad-sudarshan",
      "open quantum systems",
      "density matrix"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}