Distinguishing classical correlations from quantum correlations

The aim of this paper is to distinguish classical correlations from quantum ones. First, a new characterization of a classical correlated (CC) state is established. Corresponding results for the left and right classical correlations are also obtained. Second, a sufficient and necessary condition for a convex combination of two CC states to be CC is given. It is proved that the set CC(H-A circle times H-B) of all CC states on H-A circle times H-B is a perfect, nowhere dense and compact subset of the metric space D(H-A circle times H-B) of all states (density operators) on H-A circle times H-B. Based on our new characterization of CC states, a quantity Q(rho) is associated with a state rho. It is shown that a state rho is CC if and only if Q(rho) = 0. In particular, we prove that the Werner state W-lambda in a two-qubit system with a single real parameter lambda is CC if and only if lambda = 0.25.