Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities

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= rho₁₁ rho₄₄/rho₂₂ rho₃₃. 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}= rho₁₁ rho₅₅rho₂₂ rho₄₄, nu_{2}= rho₂₂ rho₆₆rho₃₃ rho₅₅, 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.