{
  "annotation_id": "b382714d-6046-4474-97c7-869c9cfd0649",
  "date_created": "2026-03-02T18:02:34.311000Z",
  "date_modified": "2026-03-02T18:02:34.311000Z",
  "file_hash": "741540659fb6c040fb38de656b5c5e4f08d46604928b070efeef1d6dc00122f0",
  "private": false,
  "record": {
    "abstract": "We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3\npi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact\ntwo-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to\ncompute. Each probability corresponds to a specific scenario -- a class of 4 x\n4 density matrices (rho) with an m-subset (m\u003c6) of its six off-diagonal pairs\nof symmetrically located entries set to zero. Initially, we consider only\nscenarios in which the (6-m) non-nullified pairs are real, but then permit them\nto be complex (as well as quaternionic) in nature also. The general analytical\nstrategy we implement is based on the Bloore density matrix parameterization\n(J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the\ndimensionalities of required integrations. For each scenario, we identify a\ncertain univariate ``separability function\u0027\u0027 S_{scenario}(nu), where nu =\nrho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the\nproduct of this function with a scenario-specific (marginal) jacobian function\nJ_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of\nV^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the\nHS scenario-specific separability probability. Among the possible forms that we\nhave so far determined for S_{scenario}(nu) are piecewise combinations of c, c\nsqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu\nfor nu in [1,infty]. We also obtain bivariate separability functions\nS_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6\ndensity matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22}\nrho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we\ninvestigate parallel two-qutrit and three-qubit problems.",
    "arxiv_id": "quant-ph/0702134",
    "authors": [
      "Paul B. Slater"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Qubit-Qubit and Qubit-Qutrit Separability Functions and Probabilities",
    "url": "https://arxiv.org/abs/quant-ph/0702134"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "426af7f9-75d5-425e-bac0-544858ad7ea5",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}