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

We find equivalent hypergeometric- and difference-equation-based formulas, $Q(k,\alpha )=G_1^k(\alpha )G_2^k(\alpha )$, for $k=-1,0,1,\dots ,9$, 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 $\mathrm{\mid }{\rho }^{PT}\mathrm{\mid }>\mathrm{\mid }\rho \mathrm{\mid }$ holds. Here $\rho$ denotes a $4 \times 4$densitymatrixand$\rho^{PT}$,itspartialtranspose,while$\alpha$isaDyson-index-likeparameterwith$\alpha = 1$ 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 ${}_p{F}_{p-1}$, $p\ge 7$, 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.