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A priori probability that two qubits are unentangled

Paul B. Slater
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Abstract

In a previous study (quant-ph/9911058), several remarkably simple exact results were found, in certain specialized m-dimensional 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 one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 x 4 density matrices (math-ph/0202002), we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits -- the total SD volume of which is known to be π^8 / 1680 = π^8 / (2^4 3 5 7) = 5.64794. Using advanced quasi-Monte Carlo procedures (scrambled Halton sequences) for numerical integration in this high-dimensional 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 π^6 /2310 = π^6 / (2 3 5 7 11) = .416186. The ratio of the two volumes, 8 / (11 π^2) = .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14-dimensional boundary of C is 142 π^7 / 12285 = 142 π^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.

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Paul B. Slater. 2003-01-22. A priori probability that two qubits are unentangled. https://arxiv.org/abs/quant-ph/0207181

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