A priori probability that two qubits are unentangled

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 \pi^8 / 1680 = \pi^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 \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 14-dimensional 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.