We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of $\frac{ 8}{ 33} = \frac{ 2^3}{ 3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{ 29}{ 64} =\frac{ 29}{ 2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{ 27}{ 1000} = (\frac{ 3}{ 10})^3 =\frac{ 3^3}{ 2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{ 860}{ 6561} =\frac{ 2^2 \cdot 5 \cdot 43}{ 3^8} \approx 0.131078$. 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 of $X$-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is $\frac{ 1}{ 3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{ 8}{ 33}$, its $k=1$ induced measure counterpart is $\frac{ 1}{ 4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{ 61}{ 143} =\frac{ 61}{ 11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".
Categories: quant-ph math-ph math.MP math.PR