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 ($\epsilon =\frac{{\sigma }_2}{{\sigma }_1}$) of the two singular values (${\sigma }_1\ge {\sigma }_2\ge 0$) 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" (${\tilde{\chi }}_d(\epsilon )$) 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 ${\tilde{\chi }}_d(\epsilon )$ 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.

21 pages, 16 figures