Analytic Fits to Separable Volumes and Probabilities for Qubit-Qubit and Qubit-Qutrit Systems

Abstract

We investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices \rho (math-ph/0202002) -- would be the products of 12-dimensional Haar measure \mu_{Haar} (common to each metric) and 3-dimensional measures \mu_{metric} (specific to each metric) over the 3d-simplex formed by the four eigenvalues of \rho. We attempt here to estimate/determine the 3-dimensional integrands (the products of the various [known] \mu_{metric}'s and an unknown symmetric weighting function W) remaining after the (putative) 12-fold integration of \mu_{Haar} over the twelve Euler angles. We do this by fitting W so that the conjectured HS separable volumes and hyperareas (quant-ph/0410238; cf. quant-ph/0609006) are reproduced. We further evaluate a number of possible choices of W by seeing how well they also yield the conjectured separable volumes for the Bures, Kubo-Mori, Wigner-Yanase and (arithmetic) average monotone metrics and the conjectured separable Bures hyperarea (quant-ph/0308037,Table VI). We, in fact, find two such exact (rather similar) choices that give these five conjectured (non-HS) values all within 5%. In addition to the above-mentioned Euler angle parameterization of \rho, we make extensive use of the Bloore parameterization (J. Phys. A 9 [1976], 2059) in a companion set of two-qubit separability analyses.