2008/02/01 by Paul B. Slater
Chemistry · Computer Science · Mathematics · Physics and Astronomy · #Ansatz #Curvature #Euler characteristic #Euler's formula #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Molecular spectroscopy and chirality #Monotone polygon #Physics #Pure mathematics #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Qubit #Qutrit #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.geomphys.2008.03.014
published as J. Geom. Phys. 58 (2008), 1101-1123 · 45 pages, 23 figures
arxiv created 2008/02/01 · openalex publication_date 2008/03/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 (sqrt2-1)/pi8 = 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.