2015/06/29 by Paul B. Slater, Slater, Paul B.
Computer Science · Mathematics · Physics and Astronomy · #15B52 #60B20 #62E17 #81P40 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Quantum optics and atomic interactions #math-ph #math.MP #math.PR #msc:15B52 #msc:60B20 #msc:62E17 #msc:81P40 #quant-ph
paper · pdf · doi:10.48550/arxiv.1506.08739
50 pages, 52 figures. Moderately revised, per referee requests. Some arguments (pertaining to doubly-stochastic measures, copulas,...) removed
openalex publication_date 2015/06/29 · arxiv created 2016/01/19 · arxiv updated 2016/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of (8)/(33) for two-qubit systems (ρ) 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,…), 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 |ρPT| >|ρ| and those from |ρ| > |ρPT| >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 (rA, rB) 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.