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.) are 8/33 and 8/17, respectively. Central to our reasoning are the modifications of two ansatze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions B.(a, b), where. v = rho(11)rho(44)/rho(22)rho(33). We, now, set the separability function S-real(v) proportional to Bv(v, 1/2, 2) = 2/3(3 -v)root v. Then, in the complex case - conforming to a pattern we find, manifesting the Dyson indices (beta = 1, 2, 4) of random matrix theory - we take S-complex(v) proportional to S-real(2)(v). We also investigate the real and complex qubit-qutrit cases. Now, there are two variables,v(1) =rho(11)rho(55)/rho(22)rho(44),v(2) =rho(22)rho(66)/rho(33)rho(55), but they appear to remarkably coalesce into the product. eta = v(1)v(2) = rho(11)rho(66)/rho(33)rho(44), so that the real and complex separability functions are again univariate in nature.