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

TL;DR

This paper investigates quantum state separability probabilities using various measures related to quantum steering ellipsoids and explores their relation to Hilbert-Schmidt and Bures ensembles, providing new estimations and insights into quantum correlations.

Contribution

It introduces alternative measures based on QES volumes and eigenvalue terms to estimate separability probabilities, connecting geometric and ensemble-based approaches.

Findings

QES volume-based measure yields a separability probability of ~0.1055 for HS ensemble.

Bures ensemble weighted by QES volume estimates a probability of ~0.1002.

Multiple estimates show median and mean probabilities around 0.045 to 0.117.

Abstract

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.