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A priori probability that two qubits are unentangled  Paul B. Slater
;  Date: 
31 Jul 2002  Journal:  Quantum Info. Proc. vol. 1, no. 5, Oct. 2002, pp. 397408  Subject:  quantph  Affiliation:  University of California  Abstract:  In a previous study (quantph/9911058), several remarkably simple exact results were found, in certain specialized mdimensional scenarios (m<5), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically onefourth the statistical distinguishability [SD] metric). Here, making use of a newlydeveloped (Euler angle) parameterization of the 4 x 4 density matrices (mathph/0202002), we extend the analysis to the complete 15dimensional convex set (C) of arbitrarily paired qubits  the total SD volume of which is known to be pi^8 / 1680 = pi^8 / (2^4 3 5 7) = 5.64794. Using advanced quasiMonte Carlo procedures (scrambled Halton sequences) for numerical integration in this highdimensional space, we approximately (5.64851) reproduce that value, while obtaining an estimate of .416302 for the SD volume of separable states. We conjecture that this is but an approximation to pi^6 /2310 = pi^6 / (2 3 5 7 11) = .416186. The ratio of the two volumes, 8 / (11 pi^2) = .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14dimensional boundary of C is 142 pi^7 / 12285 = 142 pi^7 /(3^3 5 7 13) = 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of the boundary of separable states.  Source:  arXiv, quantph/0207181  Services:  Forum  Review  PDF  Favorites 


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