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 with the (classical) Bayesian role of the volume elements (“Jeffreys' priors”) of Fisher information metrics. Continuing this work, we obtain exact Bures prior probabilities that the members of certain low-dimensional subsets of the fifteen-dimensional convex set of density matrices are separable or classically correlated. The main analytical tools employed are symbolic integration and a formula of Dittmann (J. Phys. A 32, 2663 (1999)) for Bures metric tensors. This study complements an earlier one (J. Phys. A 32, 5261 (1999)) in which numerical (randomization) - but not integration - methods were used to estimate Bures separability probabilities for unrestricted and density matrices. The exact values adduced here for pairs of quantum bits (qubits), typically, well exceed the estimate () there, but this disparity may be attributable to our focus on special low-dimensional subsets. Quite remarkably, for the q= 1 and states inferred using the principle of maximum nonadditive (Tsallis) entropy, the Bures probabilities of separability are both equal to . For the Werner qubit-qutrit and qutrit-qutrit states, the probabilities are vanishingly small, while in the qubit-qubit case it is .