Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities

Abstract

Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($\rho$) is separable/disentangled is $\frac{8}{33}$ (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, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|{\rho }^{PT}|>|\rho |$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|\rho |>|{\rho }^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4\times 4$ 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 $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The new determinantal $4F3$-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($|\rho |=0$), and its consistency manifested-also using density-approximation-with a theorem of Szarek, Bengtsson and {Ż}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.

25 pages, minor changes, to appear in the Journal of Geometry and Physics. (Paper expands substantially upon arXiv::1403.1825.)

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Cited by in corpus (7)

• Invariance of separability probability over reduced states in 4x4 bipartite systems
• Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities
• Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States
• Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures
• Formulas for Generalized Two-Qubit Separability Probabilities
• Two-Qubit Separability Probabilities as Joint Functions of the Bloch Radii of the Qubit Subsystems
• The Sums of a Double Hypergeometric Series and of the First m+1 Terms of 3F2(a,b,c;(a+b+1)/2,2c;1) when c = -m is a Negative Integer