Dyson indices and Hilbert–Schmidt separability functions and probabilities
Paul B. Slater · Journal of Physics A Mathematical and Theoretical · 2007
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 × 4 density matrices ρ) are and , respectively. Central to our reasoning are the modifications of two ansätze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions B ν ( a , b ), where . We, now, set the separability function proportional to . Then, in the complex case—conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory—we take proportional to . We also investigate the real and complex qubit– qutrit cases. Now, there are two variables, , but they appear to remarkably coalesce into the product , so that the real and complex separability functions are again univariate in nature.