Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure

2015/04/17 by Paul B. Slater, Slater, Paul B.
Chemistry · Computer Science · Mathematics · Physics and Astronomy · #33C20 #62E17 #62E20 #65Q10 #81P45 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Probability (math.PR) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph) #math-ph #math.MP #math.PR #msc:33C20 #msc:62E17 #msc:62E20 #msc:65Q10 #msc:81P45 #quant-ph

paper · pdf · doi:10.48550/arxiv.1504.04555

20 pages, 11 figures

arxiv created 2015/04/17 · openalex publication_date 2015/04/17 · arxiv updated 2015/04/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We find equivalent hypergeometric- and difference-equation-based formulas, Q(k,α)= G₁^k(α) G₂^k(α), for k = -1, 0, 1,…,9, for that (rational-valued) portion of the total separability probability for generalized two-qubit states endowed with random induced measure, for which the determinantal inequality |ρ^PT| >|ρ| holds. Here ρ denotes a 4 × 4 density matrix and ρ^PT, its partial transpose, while α is a Dyson-index-like parameter with α= 1 for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is 4 × (4 +k). For the symmetric case of k=0, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) Q(0,(1)/(2)) = (29)/(128), (the standard two-qubit case) Q(0,1)=(4)/(33), and (the two-quater[nionic]bit case) Q(0,2)= (13)/(323). The factors G₂^k(α) can be written as the sum of weighted hypergeometric functions _pF_p-1, p ≥ 7, all with argument (27)/(64) =((3)/(4))³. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express G₂^k(α) in terms of first-order difference equations. The factors G₁^k(α) are equal to ((27)/(64))^α-1 times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable α- and k-specific invariant asymptotic properties (again, involving (27)/(64) and related quantities) of separability probability formulas emerge.

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