Separability analyses of two-qubit density matrices

Abstract

We pursue a number of analytical directions, motivated to some extent initially by the possibility of developing a methodology for formally proving or disproving a certain conjecture of quantum-theoretical relevance (quant-ph/0308037). It asserts that the 15-dimensional volume occupied by the separable two-qubit density matrices is (\sqrt{2}-1)/3, as measured in terms of the statistical distinguishability metric (four times the Bures or minimal monotone metric). Somewhat disappointingly, however, the several various analyses that we report, though we hope of independent/autonomous interest, appear to provide small indication of how to definitively resolve the conjecture. Among our studies here are ones of: (1) the Bures volumes of the two-dimensional sections of Bloch vectors for a number of the Jakobczyk-Siennicki two-qubit scenarios; (2) the structure of certain convex polytopes of separable density matrices; and (3) the diagonalization of 15 x 15 Bures metric tensors.