2014/03/07 by Paul B. Slater, Slater, Paul B.
Chemistry · Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Probability (math.PR) #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math-ph #math.MP #math.PR #quant-ph
paper · pdf · doi:10.48550/arxiv.1403.1825
9 pages, one figure
arxiv created 2014/03/07 · openalex publication_date 2014/03/07 · arxiv updated 2014/03/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Compelling evidence-though yet no formal proof--has been adduced that the probability that a generic two-qubit state (ρ) is separable is (8)/(33) (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal moment formula of C. Dunkl (Appendix), we reach the conclusion that one-half of this probability arises when the determinantal inequality |ρPT|>|ρ|, where PT denotes the partial transpose, is satisfied, and, the other half, when |ρ|>|ρPT|. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of 4 × 4 density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of (29)/(64) and (26)/(323), respectively. The computational results reported lend strong support to those obtained earlier--including the "concise formula" that yields them--most conspicuously amongst those findings being the (29)/(64), (8)/(33) and (26)/(323) probabilities noted.