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We report substantial progress in the study of separability functions andtheir application to the computation of separability probabilities for thereal, complex and quaternionic qubit-qubit and qubit-qutrit systems. We expandour recent work (arXiv:0704.3723), in which the Dyson indices of random matrixtheory played an essential role, to include the use of not only the volumeelement of the Hilbert-Schmidt (HS) metric, but also that of the Bures (minimalmonotone) metric as measures over these finite-dimensional quantum systems.Further, we now employ the Euler-angle parameterization of density matrices(rho), in addition to the Bloore parameterization. The Euler-angle separabilityfunction for the minimally degenerate complex two-qubit states is well-fittedby the sixth-power of the participation ratio, R(rho)=1/Tr(rho)^2.Additionally, replacing R(rho) by a simple linear transformation of theVerstraete-Audenaert-De Moor function (arXiv:quant-oh/00), we find closeadherence to Dyson-index behavior for the real and complex (nondegenerate)two-qubit scenarios. Several of the analyses reported help to fortify ourconjectures that the HS and Bures separability probabilities of the complextwo-qubit states are 8/33 = 0.242424 and 1680 (sqrt{2}-1)/pi^8 = 0.733389,respectively. Employing certain regularized beta functions in the role ofEuler-angle separability functions, we closely reproduce--consistently with theDyson-index ansatz--several HS two-qubit separability probability conjectures.

Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices