On conjectures of classical and quantum correlations in bipartite states

In this paper, two conjectures which were proposed in Luo et al (2010 Phys. Rev. A 82 052122) on the correlations in a bipartite state ρAB are studied. If the mutual information I(ρAB) between two quantum systems A and B before any measurement is considered as the total amount of correlations in the state ρAB, then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations C(ρAB) in the state ρAB are defined by the maximizing mutual information between two quantum systems A and B after von Neumann measurements on system B. We show that it is upper bounded by the von Neumann entropies of both subsystems A and B, which answered the conjecture on the classical correlation. If the quantum correlations Q(ρAB) in the state ρAB are defined by Q(ρAB) = I(ρAB) − C(ρAB), we also show that it is upper bounded by the von Neumann entropy of subsystem B. It is also found that Q(ρAB) is upper bounded by the von Neumann entropy of subsystem A for a class of states.