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 \(\frac{29}{64} =\frac{29}{2^6} \approx 0.453125\) and the qubit-qutrit analogue of the (strongly supported) value of \(\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424\). We advance the possibilities of a rebit-retrit value of \(\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078\) and a qubit-qutrit one of \(\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027\). 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 \(\frac{16}{3 \pi^2} \approx 0.54038\), 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 \(\tilde{\chi}_{2,0}(\varepsilon)=\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)\), yielding \(\frac{8}{33}\), while its \(k=1\) induced measure counterpart is \(\tilde{\chi}_{2,1}(\varepsilon)=\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2\), 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.