Formulas for Generalized Two-Qubit Separability Probabilities

To begin, we find certain formulas , for . These yield that part of the total separability probability, , for generalized (real, complex, , etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality holds. Here denotes a density matrix, obtained by tracing over the pure states in -dimensions, and denotes its partial transpose. Further, is a Dyson-index-like parameter with for the standard (15-dimensional) convex set of (complex) two-qubit states. For , we obtain the previously reported Hilbert-Schmidt formulas, with (the real case), (the standard complex case), and (the quaternionic case), the three simply equalling . The factors are sums of polynomial-weighted generalized hypergeometric functions , , all with argument . We find number-theoretic-based formulas for the upper ( ) and lower ( ) parameter sets of these functions and, then, equivalently express in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise†forms of , , and , parallel to the one obtained previously (Slater 2013) for . For nonnegative half-integer and integer values of , (as well as ) has descending roots starting at . 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.