{
  "annotation_id": "5b4eac19-4bec-4781-b23a-d1cc5f61eb85",
  "date_created": "2026-03-02T18:02:13.741000Z",
  "date_modified": "2026-03-02T18:02:13.741000Z",
  "file_hash": "c98b588b3f38bf646543044b58723f4198d0574529ce8c7407867d12af0baec6",
  "private": false,
  "record": {
    "abstract": "Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for\nthe case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to\nhigh numerical accuracy, the formulas of Sommers and Zyczkowski\n(quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional\nhyperarea of the (separable and nonseparable) N x N density matrices, based on\nthe Bures (minimal monotone) metric -- and also their analogous formulas\n(quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same\nseven billion well-distributed (``low-discrepancy\u0027\u0027) sample points, we estimate\nthe unknown volumes and hyperareas based on five additional (monotone) metrics\nof interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate\nall of these seven volume and seven hyperarea (unknown) quantities when\nrestricted to the separable density matrices. The ratios of separable volumes\n(hyperareas) to separable plus nonseparable volumes (hyperareas) yield\nestimates of the separability probabilities of generically rank-six (rank-five)\ndensity matrices. The (rank-six) separability probabilities obtained based on\nthe 35-dimensional volumes appear to be -- independently of the metric (each of\nthe seven inducing Haar measure) employed -- twice as large as those (rank-five\nones) based on the 34-dimensional hyperareas. Accepting such a relationship, we\nfit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable\nvolumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of\nthe rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite\nclearly close to integral too.) The doubling relationship also appears to hold\nfor the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit\nexact formulas for the Hilbert-Schmidt separable volumes and hyperareas.",
    "arxiv_id": "quant-ph/0410238",
    "authors": [
      "Paul B. Slater"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1103/PhysRevA.71.052319",
    "journal_ref": "Phys. Rev. A 71, 052319 (2005)",
    "title": "Qubit-Qutrit Separability-Probability Ratios",
    "url": "https://arxiv.org/abs/quant-ph/0410238"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b9e09558-50c7-44f5-8fdb-bd8aeb0b39a8",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}