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Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices | | Paul B. Slater
; | | Date: |
1 Feb 2008 | | Abstract: | 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 (arXiv:0704.3723), 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
(rho), 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(rho)=1/Tr(rho)^2.
Additionally, replacing R(rho) by a simple linear transformation of the
Verstraete-Audenaert-De Moor function (arXiv:quant-oh/0011111), we find close
adherence to Dyson-index behavior 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 8/33 = 0.242424 and 1680 (sqrt{2}-1)/pi^8 = 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. | | Source: | arXiv, 0802.0197 | | Services: | Forum | Review | PDF | Favorites | |
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