Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure

Abstract

We find equivalent hypergeometric- and difference-equation-based formulas, $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,\ldots,9$, for that (rational-valued) portion of the total separability probability for generalized 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 and $\rho^{PT}$, its partial transpose, while $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is $4 \times (4 +k)$. For the symmetric case of $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard two-qubit case) $Q(0,1)=\frac{4}{33}$, and (the two-quater[nionic]bit case) $Q(0,2)= \frac{13}{323}$. The factors $G_2^k(\alpha)$ can be written as the sum of weighted hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $\frac{27}{64} =(\frac{3}{4})^3$. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express $G_2^k(\alpha)$ in terms of first-order difference equations. The factors $G_1^k(\alpha)$ are equal to $(\frac{27}{64})^{\alpha-1}$ times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable $\alpha-$ and $k$-specific invariant asymptotic properties (again, involving $\frac{27}{64}$ and related quantities) of separability probability formulas emerge.