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

Previously, a formula, incorporating a 5F4 hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments leftlangle leftvert ρ^PTrightvert ⁿleftvert ρrightvert ^krightrangle /leftlangle leftvert ρrightvert ^krightrangle of 4 times 4 density-matrices (ρ), and their partial transposes $ρ^PT|$ 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 $α= frac{1}{2}$, (standard) two-qubit $α= 1$ and two-quater[nionic]bit $α=2$ 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.