Two-Qubit Rational-Valued Entanglement-Boundary Probability Densities and a Fisher Information Equality Conjecture

2013/03/05 by Paul B. Slater, Slater, Paul B.
Computer Science · Mathematics · Physics and Astronomy · #34A55 #81P40 #94A17 #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math-ph #math.MP #msc:34A55 #msc:81P40 #msc:94A17 #quant-ph #stat.CO

paper · pdf · doi:10.48550/arxiv.1303.1125

15 pages, 14 figures, 2 tables. Further analyses included. Substantial expansion

openalex publication_date 2013/03/05 · arxiv created 2013/05/01 · arxiv updated 2013/05/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider a pair of one-parameter (alpha) families of generalized two-qubit determinantal Hilbert-Schmidt probability distributions, p_alpha(|rho^PT|) and q_alpha(|rho|), where rho is a 4 x 4 density matrix, rho^PT, its partial transpose, with |rho^PT| ∈ [-1/16,1/256] and |rho| ∈ [0, 1/256]. The Dyson-index-like (random matrix) parameter alpha is 1/2 for the 9-dimensional generic two-rebit systems, 1 for the 15-dimensional generic two-qubit systems,... Numerical (moment-based probability-distribution-reconstruction) analyses suggest the conjecture that the Fisher information--a measure over alpha--is identical for the two distinct families. Further, we study the values of p_alpha(0), the probability densities at the separability-entanglement boundary, with evidence strongly indicating that p₂(0) =7425/34 and p₃(0)= 7696/69. Despite extensive results of such a nature, we have not yet succeeded--in contrast with the corresponding rational-valued separability probabilities (arXiv.org:1301.6617)--in generating an underlying, explanatory ("concise") formula for p_α(0), even though the corresponding denominators of both sets of rational-valued results are closely related, having almost identical (small) prime factors. The first derivatives p_alpha^'(0) are positive for alpha = 1/2 and 1, but negative for alpha > 1.

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