Hilbert-Schmidt Separability Probabilities from Bures Ensembles and vice versa: Applications to Quantum Steering Ellipsoids and Monotone Metrics

We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the $\Pi_{j<k}^{1...4} (\lambda_j-\lambda_k)^2$ term of the HS volume element (the $\lambda$'s being the four eigenvalues of the associated $4 \times 4$ density matrix $\rho$). This measure is applied to the members of a HS ensemble of random two-qubit states, yielding a QES separability probability estimate of 0.105458. Alternatively, weighting members of a Bures ensemble by the QES volume divided by the eigenvalue part $\frac{1}{\sqrt{\mbox{det} \rho}} \Pi_{j<k}^{1...4} \frac{(\lambda_j-\lambda_k)^2}{\lambda_j+\lambda_k}$ of the Bures volume element, gives a close estimate of 0.100223. We also weight members of a HS ensemble by the QES volume divided not only by the indicated HS eigenvalue term, but also by the unitary component $|\Pi_{j<k}^{1...4} \mbox{Re} (U^{-1}) \mbox{Im} (U^{-1})|$ of the volume element. For one hundred thirty (rather variable) independent separability probability estimates, we, then, obtain median and mean estimates of 0.0447729 and 0.117485 with variance 0.0381468.