Formulas for Generalized Two‐Qubit Separability Probabilities

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.