Bloch Radii Repulsion in Separable Two-Qubit Systems

TL;DR

This paper investigates the behavior of separability probabilities in two-qubit systems under various measures, revealing invariance properties with respect to Bloch radii and a phenomenon of Bloch radii repulsion affecting separability likelihoods.

Contribution

It demonstrates that separability probabilities are invariant along Bloch radii under Hilbert-Schmidt and induced measures, and introduces the concept of Bloch radii repulsion in two-qubit systems.

Findings

Separable two-qubit probabilities are invariant along Bloch radii with Hilbert-Schmidt measure.

Bures measure shows decreasing separability probability with increasing radii.

Separable probabilities decrease when the Bloch radii of subsystems are similar.

Abstract

Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of $\frac{8}{33}$ for two-qubit systems ($\rho$) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius ($r$) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, $r=1$). Further, they also observed (personal communication) such separability probability $r$-invariance, when using, more broadly, random induced measure ($K=3,4,5,\ldots$), with $K=4$ corresponding to the (symmetric) Hilbert-Schmidt case. Among the findings here is that this invariance is maintained even after splitting the separability probabilities into those parts arising from the determinantal inequality $|\rho^{PT}| >|\rho|$ and those from $|\rho| > |\rho^{PT}| >0$, where the partial transpose is indicated. The nine-dimensional set of generic two-re[al]bit states endowed with Hilbert-Schmidt measure is also examined, with similar $r$-invariance conclusions. Contrastingly, two-qubit separability probabilities based on the Bures (minimal monotone) measure {\it diminish} with $r$. Moreover, we study the forms that the separability probabilities take as joint (bivariate) functions of the radii ($r_A, r_B$) of the Bloch balls of {\it both} single-qubit subsystems. Here, a form of Bloch radii {\it repulsion} for separable two-qubit systems emerges in {\it all} our several analyses. Separability probabilities tend to be smaller when the lengths of the two radii are closer. In Appendix A, we report certain companion analytic results for the much-investigated, more amenable (7-dimensional) $X$-states model.