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 29 64 for the 9-dimensional convex set of two-rebit density matrices, was the ratio (ε = σ 2 σ 1 ) of the two singular values (σ 1 ≥ σ 2 ≥ 0) of D 1 2 2 D − 1 2 1 . There, D 1 and D 2 were the diagonal 2 × 2 blocks of a 4 × 4 two-rebit density matrix ρ. Working within the Lovas-Andai "separability function" ( χd (ε)) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of 8 33 and 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 1 2 2 D − 1 2 1, where now D 1 and D 2 are the 3 × 3 diagonal blocks of 6 × 6 rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing 8 × 8 density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending χd (ε) 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.