{
  "id": "1404.1860",
  "version": "v4",
  "published": "2014-04-07T17:50:59.000Z",
  "updated": "2014-10-23T18:43:05.000Z",
  "title": "Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities",
  "authors": [
    "Paul B. Slater",
    "Charles F. Dunkl"
  ],
  "comment": "24 pages, minor changes/corrections. Paper expands substantially upon arXiv::1403.1825",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($\\rho$) is separable/disentangled is $\\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\\it via} density-approximation procedures, the conclusion that one-half ($\\frac{4}{33}$) of this probability arises when the determinantal inequality $|\\rho^{PT}|>|\\rho|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|\\rho|>|\\rho^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \\times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-\"re[al]bit\" and two-\"quater[nionic]bit\"separability probabilities of $\\frac{29}{64}$ and $\\frac{26}{323}$, respectively. The new determinantal $4F3$-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($|\\rho|=0$), and its consistency manifested-also using density-approximation-with a theorem of Szarek, Bengtsson and {\\.Z}yczkowski (arXiv:quant-ph/0509008). This theorem states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-30T16:03:37.000Z",
      "abstract": "Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($\\rho$) is separable/disentangled is $\\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\\it via} density-approximation procedures, the conclusion that one-half ($\\frac{4}{33}$) of this probability arises when the determinantal inequality $|\\rho^{PT}|>|\\rho|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|\\rho|>|\\rho^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \\times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-\"re[al]bit\" and two-\"quater[nionic]bit\"separability probabilities of $\\frac{29}{64}$ and $\\frac{26}{323}$, respectively. The new determinantal $4F3$-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($|\\rho|=0$), and its consistency manifested-also using density-approximation-with a theorem of Szarek, Bengtsson and {\\.Z}yczkowski (arXiv:quant-ph/0509008). It states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.",
      "comment": "23 pages, 1 figure, added Appendix D concerning Hilbert-Schmidt qubit-qutrit and Bures qubit-qubit cases. Paper expands substantially upon arXiv::1403.1825",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-10-23T18:43:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81P45",
      "33C90",
      "62G07"
    ],
    "keywords": [
      "probability",
      "half hilbert-schmidt separability probabilities",
      "generalized two-qubit",
      "hypergeometric moment formula",
      "generic minimally degenerate two-qubit states"
    ],
    "publication": {
      "doi": "10.1016/j.geomphys.2015.01.006",
      "journal": "Journal of Geometry and Physics",
      "year": 2015,
      "month": "Apr",
      "volume": 90,
      "pages": 42
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JGP....90...42S"
    }
  }
}