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Firstly, we reduce the long-standing problem of ascertaining theHilbert-Schmidt probability that a generic pair of qubits is separable to thatof determining the specific nature of a one-dimensional (separability) functionof the radial coordinate (r) of the unit ball in 15-dimensional Euclideanspace, and similarly for a generic pair of rebits, using the 9-dimensional unitball. Separability probabilities, could, then, be directly obtained byintegrating the products of these functions (which we numerically estimate andplot) with jacobian factors of r^m over r in [0,1], with m=17 for the two-rebitcase, and m=29 in the two-qubit instance. Secondly, we repeat the analyses, butfor 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 curvesgenerated. For our analyses, we employ an interesting Cholesky-decompositionparameterization of the 4 x 4 density matrices. Thirdly, in an exploratoryinvestigation, we examine the Hilbert-Schmidt separability probability questionusing the three-dimensional visualization (cube/tetrahedron/octahedron) of twoqubits associated with Avron, Bisker and Kenneth, and Leinaas, Myrheim andOvrum, among others.

Radial and Azimuthal Profiles of Two-Qubit/Rebit Hilbert-Schmidt Separability Probabilities and Related 3-D Visualization Analyses