dorsal/arxiv
View SchemaSilver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0308037 |
| URL | https://arxiv.org/abs/quant-ph/0308037 |
| DOI | 10.1016/j.geomphys.2004.04.011 |
| Journal | J. Geometry and Physics 53/1 (2005), 74-97 |
Abstract
Extensive numerical integration results lead us to conjecture that the silver mean, that is, s = \sqrt{2}-1 = .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of separable states consisting generically of rank-four 4 x 4 density matrices has volume (``hyperarea'') 55s/39 and that part composed of rank-three density matrices, 43s/39, so the total boundary hyperarea would be 98s/39. While the Bures probability of separability (0.07334) dominates that (0.050339) based on the Wigner-Yanase metric (and all other monotone metrics) for rank-four states, the Wigner-Yanase (0.18228) strongly dominates the Bures (0.03982) for the rank-three states.
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"abstract": "Extensive numerical integration results lead us to conjecture that the silver\nmean, that is, s = \\sqrt{2}-1 = .414214 plays a fundamental role in certain\ngeometries (those given by monotone metrics) imposable on the 15-dimensional\nconvex set of two-qubit systems. For example, we hypothesize that the volume of\nseparable two-qubit states, as measured in terms of (four times) the minimal\nmonotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori\nmonotone metric. Also, we conjecture, in terms of (four times) the Bures\nmetric, that that part of the 14-dimensional boundary of separable states\nconsisting generically of rank-four 4 x 4 density matrices has volume\n(``hyperarea\u0027\u0027) 55s/39 and that part composed of rank-three density matrices,\n43s/39, so the total boundary hyperarea would be 98s/39. While the Bures\nprobability of separability (0.07334) dominates that (0.050339) based on the\nWigner-Yanase metric (and all other monotone metrics) for rank-four states, the\nWigner-Yanase (0.18228) strongly dominates the Bures (0.03982) for the\nrank-three states.",
"arxiv_id": "quant-ph/0308037",
"authors": [
"Paul B. Slater"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/j.geomphys.2004.04.011",
"journal_ref": "J. Geometry and Physics 53/1 (2005), 74-97",
"title": "Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems",
"url": "https://arxiv.org/abs/quant-ph/0308037"
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