Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies

We first seek the rebit-retrit counterpart to the (formally proven by Lovas and Andai) two-rebit Hilbert-Schmidt separability probability of frac{29}{64} =frac{29}{2⁶} approx 0.453125 and the qubit-qutrit analogue of the (strongly supported) value of frac{8}{33} = frac{2³}{3 cdot 11} approx 0.242424. We advance the possibilities of a rebit-retrit value of frac{860}{6561} =frac{2² cdot 5 cdot 43}{3⁸} approx 0.131078 and a qubit-qutrit one of frac{27}{1000} = \(frac{3}{10}\)³ =frac{3³}{2³ cdot 5³} = 0.027. These four values for 2 times m systems $m=2,3$ suggest certain numerator/denominator sequences involving powers of m, which we further investigate for m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit X-states all equal frac{16}{3 π²} approx 0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets. Then, we extend the master Lovas-Andai formula to induced measures. For instance, the two-qubit function $k=0$ is χ_2,0\(varepsilon\)=frac{1}{3} varepsilon² \(4 -varepsilon²\), yielding frac{8}{33}, while its k=1 induced measure counterpart is χ_2,1\(varepsilon\)=frac{1}{4} varepsilon ² left\(3-varepsilon ^2right\)², yielding frac{61}{143} =frac{61}{11 cdot 13} approx 0.426573, where varepsilon is a singular-value ratio. Interpolations between Hilbert-Schmidt and operator monotone Bures, $sqrt{x}$ measures are also studied. Using a recently-developed golden-ratio-related (quasirandom sequence) approach, current (significant digits) estimates of the two-rebit and two-qubit Bures separability probabilities are 0.15709 and 0.07331, respectively.