Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems

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.

30 pages, 6 tables, 17 figures; nine new figures and one new table in new section VII.B pertaining to 14-dimensional hyperareas associated with various monotone metrics

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• An Explicit Computation of the Bures Metric Over the Space of $N$-Dimensional Density Matrices
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• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
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• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence
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