On Conjectures of Classical and Quantum Correlations in Bipartite 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 $I\Pa{ρ^{AB}}$ 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 $C\Pa{ρ^{AB}}$ 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 $Q\Pa{ρ^{AB}}$ in the state ${\rho }^{AB}$ is defined by $Q\Pa{ρ^{AB}}= I\Pa{ρ^{AB}} - C\Pa{ρ^{AB}}$, we show also that it is upper bounded by the von Neumann entropy of subsystem $B$. It is also obtained that $Q\Pa{ρ^{AB}}$ is upper bounded by the von Neumann entropy of subsystem $A$ for a class of states.

6 pages, LaTeX, Concluding Remarks is added and some refereces are listed also. To appear in J. Phys. A

References in corpus (4)

• Gaussian quantum discord
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Maximally discordant mixed states of two qubits
• Universal bounds for the Holevo quantity, coherent information \\ and the Jensen-Shannon divergence

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• Quantum and Classical Correlations Inside the Entanglement Wedge
• Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy
• Monogamy, polygamy, and other properties of entanglement of purification
• Necessary and sufficient condition for saturating the upper bound of quantum discord
• Entanglement structures in qubit systems
• Data processing for the sandwiched Rényi divergence: a condition for equality
• Relative entropies and their use in quantum information theory
• Universal upper bound for the Holevo information induced by a quantum operation
• Remark on the coherent information saturating its upper bound