Formulas for Generalized Two-Qubit Separability Probabilities

Abstract

To begin, we find certain formulas ( , ) = 1 ( ) 2 ( ), for = −1, 0, 1, . . . , 9. These yield that part of the total separability probability, ( , ), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality | PT | > | | holds. Here denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+ )-dimensions, and PT denotes its partial transpose. Further, is a Dyson-index-like parameter with = 1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For = 0, we obtain the previously reported Hilbert-Schmidt formulas, with (0, 1/2) = 29/128 (the real case), (0, 1) = 4/33 (the standard complex case), and (0, 2) = 13/323 (the quaternionic case), the three simply equalling (0, )/2. The factors 2 ( ) are sums of polynomial-weighted generalized hypergeometric functions −1 , ≥ 7, all with argument = 27/64 = (3/4) 3 . We find number-theoretic-based formulas for the upper ( ) and lower ( ) parameter sets of these functions and, then, equivalently express 2 ( ) in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms of (−1, ), (1, ), and (3, ), parallel to the one obtained previously (Slater 2013) for (0, ) = 2 (0, ). For nonnegative half-integer and integer values of , ( , ) (as well as ( , )) has descending roots starting at = − − 1. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for ( , ) itself. The possibility of an analogous "master" formula for ( , ) is, then, investigated, and a number of interesting results are found.

Rational-Valued, Small-Prime-Based Qubit-Qutrit and Rebit-Retrit Rank-4/Rank-6 Conjectured Hilbert-Schmidt Separability Probability Ratios

“…,counterparts [14], this seems a direction worth pursuing-especially in light of the elegant nature of the formulas so far found (not to mention also the "half-theorem" of Szarek, Bengtsson and Życzkowski [7]). Also, in terms of the Hilbert-Schmidt measure, the two-qubit separability probability is equally divided between those states for which |ρ| > |ρ P T | and those for which |ρ P T | > |ρ| [19].…”