Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States

Abstract: Previously, a formula, incorporating a $5F4$ hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments $\left\langle \left\vert \rho^{PT}\right\vert ^{n}\left\vert \rho\right\vert ^{k}\right\rangle /\left\langle \left\vert \rho\right\vert ^{k}\right\rangle$ of $4 \times 4$ density-matrices ($\rho$), and their partial transposes ($\rho^{PT}|$) was applied with $k=0$ to the generalized two-qubit separability-probability question. The formula can, further, be viewed we note here, as an averaging over "induced measures in the space of mixed quantum states". The associated induced-measure separability probabilities ($k =1, 2,\ldots$) are found--{\it via} a high-precision density approximation procedure--to assume interesting, relatively simple rational values in the two-re[al]bit ($\alpha = \frac{1}{2}$), (standard) two-qubit ($\alpha = 1$) and two-quater[nionic]bit ($\alpha =2$) cases. We deduce rather simple companion (rebit, qubit, quaterbit,\ldots) formulas that successfully reproduce the rational values assumed for {\it general} $k$. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.