Formulas for Generalized Two-Qubit Separability Probabilities

To begin, we find certain formulas Q(k,α)= G₁^k(α) G₂^k(α), for k = -1, 0, 1,...,9. These yield that part of the total separability probability, P(k,α), for generalized real, complex, quaternionic,ldots two-qubit states endowed with random induced measure, for which the determinantal inequality |ρ^PT| >|ρ| holds. Here ρ denotes a 4 times 4 density matrix, obtained by tracing over the pure states in 4 times (4 +k)-dimensions, and ρ^PT, 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 (the real case) Q\(0,frac{1}{2}\) = frac{29}{128}, (the standard complex case) Q(0,1)=frac{4}{33}, and (the quaternionic case) Q(0,2)= frac{13}{323}---the three simply equalling P(0,α)/2. The factors G₂^k(α) are sums of polynomial-weighted generalized hypergeometric functions _pF_p-1, p geq 7, all with argument z=frac{27}{64} =\(frac{3}{4}\)³. We find number-theoretic-based formulas for the upper $u_ik$ and lower $b_ik$ parameter sets of these functions and, then, equivalently express G₂^k(α) in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms, parallel to the one obtained previously for P(0,α) =2 Q(0,α). For nonnegative half-integer and integer values of α, Q(k,α) has descending roots starting at k=-α-1. Then, we (C. 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 found.