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Open Access Publications from the University of California

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

  • Author(s): Slater, Paul B
  • et al.

Extensive numerical integration results lead us to conjecture that the silver mean, that is, sigmaAg = root2 - 1 approximate to 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 sigma(Ag)/3, and 10sigma(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 x 4 density matrices has volume ("hyperarea") 55sigma(Ag)/39, and that part composed of rank-three density matrices, 43sigma(Ag)/39, so the total boundary hyperarea would be 98sigma(Ag)/39. While the Bures probability of separability (approximate to0.07334) dominates that (approximate to0.050339) based on the Wigner-Yanase metric (and all other monotone metrics) for rank-four states, the Wigner-Yanase (approximate to0.18228) strongly dominates the Bures (approximate to0.03982) for the rank-three states. (C) 2004 Elsevier B.V. All rights reserved.

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