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

Extensive numerical integration results lead us to conjecture that the silver mean, that is, σAg=2−1≈0.414214 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 σAg/3 , and 10σAg in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that part of the 14-dimensional boundary of separable states consisting generically of rank-four4×4 density matrices has volume (“hyperarea”) 55σAg/39 , and that part composed of rank-three density matrices, 43σAg/39 , so the total boundary hyperarea would be 98σAg/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.