Two-qubit separability probabilities and Beta functions

Due to recent important work of Zyczkowski and Sommers [J. Phys. A 36, 10115 (2003); 36, 10083 (2003)], exact formulas are available (in terms of both the Hilbert-Schmidt and the Bures metrics) for the (n(2)-1)-dimensional and [n(n-1)/2-1]-dimensional volumes of the complex and real nxn density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, two-qubit systems. Making use of the density matrix (rho) parametrization of Bloore [J. Phys. A 9, 2059 (1976)], we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, nu=rho(11)rho(44)/rho(22)rho(33). The associated integrand in each case is the product of a known Jacobian (highly oscillatory near nu=1) and a certain unknown univariate function, which our extensive numerical (quasi Monte Carlo) computations indicate is very closely proportional to an (incomplete) Beta function B-nu(a,b), with a=1/2, b= root 3 in the real case, and a = 2 root 6/5,b=3 root 2 in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.