{
  "id": "math-ph/0507045",
  "version": "v3",
  "published": "2005-07-18T22:45:42.000Z",
  "updated": "2005-11-17T15:43:07.000Z",
  "title": "Geometry of quantum systems: density states and entanglement",
  "authors": [
    "Janusz Grabowski",
    "Marek Kuś",
    "Giuseppe Marmo"
  ],
  "comment": "Latex, 26 pages, minor corrections, published version",
  "journal": "J.Phys. A38 (2005) 10217-10244",
  "doi": "10.1088/0305-4470/38/47/011",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "Various problems concerning the geometry of the space $u^*(\\cH)$ of Hermitian operators on a Hilbert space $\\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into K\\\"ahler submanifolds. It is also shown that the space $\\cD(\\cH)$ of density states on an $n$-dimensional Hilbert space $\\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\\cD^k(\\cH)$ of rank-$k$ states, $k=1,...,n$, is a smooth manifold of (real) dimension $2nk-k^2-1$ and this stratification is maximal in the sense that every smooth curve in $\\cD(\\cH)$, viewed as a subset of the dual $u^*(\\cH)$ to the Lie algebra of the unitary group $U(\\cH)$, at every point must be tangent to the strata $\\cD^k(\\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\\cH=\\cH^1\\ot\\cH^2$, an abstract criterion of entanglement is proved.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-11-17T15:43:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "02.40.Ft",
      "03.65.Ud",
      "03.65.Fd"
    ],
    "keywords": [
      "density states",
      "quantum systems",
      "entanglement",
      "quantum composite system",
      "hilbert space decomposition"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2005,
      "month": "Dec",
      "volume": 38,
      "number": 47,
      "pages": 10217
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 688170,
      "adsabs": "2005JPhA...3810217G"
    }
  }
}