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

Abstract

Extensive numerical integration results lead us to conjecture that the silver mean, that is, σ Ag = 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 σ Ag / 3 , and 10 σ Ag 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- four 4 × 4 density matrices has volume (“hyperarea”) 55 σ Ag / 39 , and that part composed of rank- three density matrices, 43 σ Ag / 39 , so the total boundary hyperarea would be 98 σ Ag / 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.