Exact Bures Probabilities that Two Quantum Bits are Classically Correlated

Abstract

In previous studies, we have explored the ansatz that the volume elements of the Bures metrics over quantum systems might serve as prior distributions, in analogy to the (classical) Bayesian role of the volume elements ("Jeffreys' priors") of Fisher information metrics. Continuing this work, we obtain exact Bures probabilities that the members of certain low-dimensional subsets of the fifteen-dimensional convex set of 4 x 4 density matrices are separable or classically correlated. The main analytical tools employed are symbolic integration and a formula of Dittmann (quant-ph/9908044) for Bures metric tensors. This study complements an earlier one (quant-ph/9810026) in which numerical (randomization) --- but not integration --- methods were used to estimate Bures separability probabilities for unrestricted 4 x 4 or 6 x 6 density matrices. The exact values adduced here for pairs of quantum bits (qubits), typically, well exceed the estimate (.1) there, but this disparity may be attributable to our focus on special low-dimensional subsets. Quite remarkably, for the q = 1 and q = 1/2 states inferred using the principle of maximum nonadditive (Tsallis) entropy, the separability probabilities are both equal to 2^{1/2} - 1. For the Werner qubit-qutrit and qutrit-qutrit states, the probabilities are vanishingly small, while in the qubit-qubit case it is 1/4.Comment: Seventeen pages, LaTeX, eleven postscript figures. In this version, subsequent (!) to publication in European Physical Journal B, we correct the (1,1)-entries of the 4 x 4 matrices given in formulas (6) and (7), that is, the numerators should both read v^2 - x^2 - y^2 - z^2, rather than v^2 - x^2 + y^2 + z^