Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities

Abstract

A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 x 4 density matrices rho) are 8/33 and 8/17, respectively. Central to our reasoning are the modifications of two ansatze, recently advanced (quant-ph/0609006), involving incomplete beta functions B_{nu}(a,b), where nu= (rho_{11} rho_{44})/(rho_{22} rho_{33}). We, now, set the separability function S_{real}(nu) propto B_{nu}(nu,1/2},2) =(2/3) (3-nu) sqrt{nu}. Then, in the complex case -- conforming to a pattern we find, manifesting the Dyson indices (1, 2, 4) of random matrix theory-- we take S_{complex}(nu) propto S_{real}^{2} (nu). We also investigate the real and complex qubit-qutrit cases. Now, there are two Bloore ratio variables, nu_{1}= (rho_{11} rho_{55})(rho_{22} rho_{44}), nu_{2}= (rho_{22} rho_{66})(rho_{33} rho_{55}), but they appear to remarkably coalesce into the product, eta = nu_1 nu_2 = rho_{11} ρ_{66}}{ρ_{33} ρ_{44}}, so that the real and complex separability functions are again univariate in nature.

44 pages, 6 figures, modified title, trimmed abstract, to appear in J. Phys. A

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• Entanglement Classification via Witness Operators generated by Support Vector Machine
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities
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• Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
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• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures
• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence
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• One application of Duistermaat-Heckman measure in quantum information theory
• Entanglement in bipartite X-states: Analytical results for the volume of states with positive partial transpose