On Conjectures of Classical and Quantum Correlations in Bipartite States

TL;DR

This paper investigates two conjectures about classical and quantum correlations in bipartite quantum states, establishing upper bounds for these correlations based on von Neumann entropies, thus advancing understanding of quantum information theory.

Contribution

It proves upper bounds for classical and quantum correlations in bipartite states, confirming conjectures and clarifying the structure of correlations in quantum systems.

Findings

Classical correlations are bounded by the von Neumann entropies of both subsystems.

Quantum correlations are bounded by the von Neumann entropy of subsystem B.

Quantum correlations are also bounded by the von Neumann entropy of subsystem A for certain states.

Abstract

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.