Generalized two-qubit whole and half Hilbert-Schmidt separability probabilities

Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state (ρ) is separable/disentangled is 833 (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal 4F3-hypergeometric moment formula (Appendix A), we reach, via density-approximation (inverse) procedures, the conclusion that one-half (433) of this probability arises when the determinantal inequality |ρPT|>|ρ|, where PT denotes the partial transpose, is satisfied, and, the other half, when |ρ|>|ρPT|. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of 4×4 density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-“re[al]bit” and two-“quater[nionic]bit” separability probabilities of 2964 and 26323, respectively. The new determinantal 4F3-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states (|ρ|=0), and its consistency manifested-also using density-approximation-with an important theorem of Szarek, Bengtsson and Å»yczkowski (arXiv:quant-ph/0509008). This theorem states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.