Dyson-Index-Like Behavior of Bures Separability Functions

2007/08/30 by Paul B. Slater, Slater, Paul B.
Chemistry · Mathematics · Physics and Astronomy · #Molecular spectroscopy and chirality #Quantum Mechanics and Applications #Random Matrices and Applications #math-ph #math.MP #msc:15A90 #msc:28A75 #msc:52A38 #msc:81P05 #quant-ph

paper · pdf · doi:10.48550/arxiv.0708.4208

13 pages, 2 figures

arxiv created 2007/08/30 · arxiv updated 2009/12/01

Abstract

We conduct a study based on the Bures (minimal monotone) metric, analogous to that recently reported for the Hilbert-Schmidt (flat or Euclidean) metric (arXiv:0704.3723v2). Among the interesting results obtained there had been proportionalities--in exact correspondence to the Dyson indices β= 1, 2, 4 of random matrix theory--between the fourth, second and first powers of the separability functions S_type(μ) for real, complex and quaternionic qubit-qubit scenarios, Here μ=√\fracρ₁₁ ρ₄₄ρ₂₂ ρ₃₃, with ρbeing a 4 x 4 density matrix. Separability functions have proved useful--in the framework of the Bloore (correlation coefficient/off-diagonal scaling) parameterization of density matrices--for the calculation of separability probabilities. We find--for certain, basic simple scenarios (in which the diagonal entries of ρare unrestricted, and one or two off-diagonal [real, complex or quaternionic] pairs of entries are nonzero) --that these proportionalities no longer strictly hold in the Bures case, but do come remarkably close to holding.

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