Formulas for Generalized Two-Qubit Separability Probabilities

Abstract

To begin, we find certain formulas Q(k,α)=G1k(α)G2k(α), for k=-1,0,1,…,9. These yield that part of the total separability probability, P(k,α), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k)-dimensions, and ρPT denotes its partial transpose. Further, α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For k=0, we obtain the previously reported Hilbert-Schmidt formulas, with Q(0,1/2)=29/128 (the real case), Q(0,1)=4/33 (the standard complex case), and Q(0,2)=13/323 (the quaternionic case), the three simply equalling P(0,α)/2. The factors G2k(α) are sums of polynomial-weighted generalized hypergeometric functions pFp-1, p≥7, all with argument z=27/64=(3/4)3. We find number-theoretic-based formulas for the upper (uik) and lower (bik) parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of Q(-1,α), Q(1,α), and Q(3,α), parallel to the one obtained previously (Slater 2013) for P(0,α)=2Q(0,α). For nonnegative half-integer and integer values of α, Q(k,α) (as well as P(k,α)) has descending roots starting at k=-α-1. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for Q(k,α) itself. The possibility of an analogous “master” formula for P(k,α) is, then, investigated, and a number of interesting results are found.