Extended studies of separability functions and probabilities and the relevance of Dyson indices

We report substantial progress in the study of separability functions and their application to the computation of separability probabilities for the real, complex and quaternionic qubit-qubit and qubit-qutrit systems. We expand our recent work [P.B. Slater, J. Phys. A 39 (2006) 913], in which the Dyson indices of random matrix theory played an essential role, to include the use of not only the volume element of the Hilbert-Schmidt (HS) metric, but also that of the Bures (minimal monotone) metric as measures over these finite-dimensional quantum systems. Further, we now employ the Euler-angle parameterization of density matrices (ρ), in addition to the Bloore parameterization. The Euler-angle separability function for the minimally degenerate complex two-qubit states is well-fitted by the sixth-power of the participation ratio, R(ρ)=1Trρ2. Additionally, replacing R(ρ) by a simple linear transformation of the Verstraete-Audenaert-De Moor function [F. Verstraete, K. Audenaert, B.D. Moor, Phys. Rev. A 64 (2001) 012316], we find close adherence to Dyson-index behaviour for the real and complex (nondegenerate) two-qubit scenarios. Several of the analyses reported help to fortify our conjectures that the HS and Bures separability probabilities of the complex two-qubit states are 833≈0.242424 and 1680(2−1)π8≈0.733389, respectively. Employing certain regularized beta functions in the role of Euler-angle separability functions, we closely reproduce-consistently with the Dyson-index ansatz-several HS two-qubit separability probability conjectures.