  Firstly, we reduce the long-standing problem of ascertaining the
Hilbert-Schmidt probability that a generic pair of qubits is separable to that
of determining the specific nature of a one-dimensional (separability) function
of the radial coordinate (r) of the unit ball in 15-dimensional Euclidean
space, and similarly for a generic pair of rebits, using the 9-dimensional unit
ball. Separability probabilities, could, then, be directly obtained by
integrating the products of these functions (which we numerically estimate and
plot) with jacobian factors of r^m over r in [0,1], with m=17 for the two-rebit
case, and m=29 in the two-qubit instance. Secondly, we repeat the analyses, but
for the replacement of r as the free variable, by the azimuthal angle phi \in
[0,2 pi]--with the associated jacobian factors now being, trivially, unity. So,
the separability probabilities, then, become simply the areas under the curves
generated. For our analyses, we employ an interesting Cholesky-decomposition
parameterization of the 4 x 4 density matrices. Thirdly, in an exploratory
investigation, we examine the Hilbert-Schmidt separability probability question
using the three-dimensional visualization (cube/tetrahedron/octahedron) of two
qubits associated with Avron, Bisker and Kenneth, and Leinaas, Myrheim and
Ovrum, among others.
