Formulas for Generalized Two-Qubit Separability Probabilities

Abstract

To begin, we find certain formulas $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,...,9$. These yield that part of the total separability probability, $P(k,\alpha)$, for generalized (real, complex, quaternionic,\ldots) two-qubit states endowed with random induced measure, for which the determinantal inequality $|\rho^{PT}| >|\rho|$ holds. Here $\rho$ denotes a $4 \times 4$ density matrix, obtained by tracing over the pure states in $4 \times (4 +k)$-dimensions, and $\rho^{PT}$, its partial transpose. Further, $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ 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,\alpha)/2$. The factors $G_2^k(\alpha)$ are sums of polynomial-weighted generalized hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, 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 $G_2^k(\alpha)$ in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms, parallel to the one obtained previously for $P(0,\alpha) =2 Q(0,\alpha)$. For nonnegative half-integer and integer values of $\alpha$, $Q(k,\alpha)$ has descending roots starting at $k=-\alpha-1$. Then, we (C. Dunkl and I) construct a remarkably compact (hypergeometric) form for $Q(k,\alpha)$ itself. The possibility of an analogous "master" formula for $P(k,\alpha)$ is, then, investigated, and a number of interesting results found.