Density Matrix Diagonal-Block Lovas-Andai-type singular-value ratios for qubit-qudit separability/PPT probability analyses

Abstract

An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of $\frac{29}{64}$ for the 9-dimensional convex set of two-rebit density matrices, was the ratio ($\varepsilon =\frac{\sigma_2}{\sigma_1}$) of the two singular values ($\sigma_1 \geq \sigma_2 \geq 0$) of $D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$. There, $D_1$ and $D_2$ were the diagonal $2 \times 2$ blocks of a $4 \times 4$ two-rebit density matrix $\rho$. Working within the Lovas-Andai "separability function" ($\tilde{\chi}_d(\varepsilon)$) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of $\frac{8}{33}$ and $\frac{26}{323}$ for the 15-dimensional and 26-dimensional convex sets of two-qubit and two-quater[nionic]-bit density matrices. Here, we investigate the behavior of the three singular value ratios of $V=D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$, where now $D_1$ and $D_2$ are the $3 \times 3$ diagonal blocks of $6 \times 6$ rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing $8 \times 8$ density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending $\tilde{\chi}_d(\varepsilon)$ to higher dimensional systems--an issue we also approach using certain novel numeric means. Section 3.3 of the 2017 A. Lovas doctoral dissertation (written in Hungarian) appears germane to such an investigation.