On Conjectures of Classical and Quantum Correlations in Bipartite States

In this paper, two conjectures which were proposed in [Phys. Rev. A \textbf{82}, 052122(2010)] on the correlations in a bipartite state $\rho^{AB}$ are studied. If the mutual information $I\Pa{\rho^{AB}}$ between two quantum systems $A$ and $B$ before any measurement is considered as the total amount of correlations in the state $\rho^{AB}$, then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations $C\Pa{\rho^{AB}}$ in the state $\rho^{AB}$, 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 $B$, this answered the conjecture on the classical correlation. If the quantum correlations $Q\Pa{\rho^{AB}}$ in the state $\rho^{AB}$ is defined by $Q\Pa{\rho^{AB}}= I\Pa{\rho^{AB}} - C\Pa{\rho^{AB}}$, we show also that it is upper bounded by the von Neumann entropy of subsystem $B$. It is also obtained that $Q\Pa{\rho^{AB}}$ is upper bounded by the von Neumann entropy of subsystem $A$ for a class of states.