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arXiv:1105.2993 · Journal of Physics A Mathematical and Theoretical

On conjectures of classical and quantum correlations in bipartite states

Lin Zhang, Junde Wu

Published Dec 8, 2011• 24 citations
arXivHTMLPDF

Abstract

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 ρ AB are studied. If the mutual information I (ρ AB ) between two quantum systems A and B before any measurement is considered as the total amount of correlations in the state ρ AB , then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations C (ρ AB ) in the state ρ AB 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 (ρ AB ) in the state ρ AB are defined by Q (ρ AB ) = I (ρ AB ) − C (ρ AB ), we also show that it is upper bounded by the von Neumann entropy of subsystem B . It is also found that Q (ρ AB ) is upper bounded by the von Neumann entropy of subsystem A for a class of states.

DOI
10.1088/1751-8113/45/2/025301

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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 ρ AB are studied. If the mutual information I (ρ AB ) between two quantum systems A and B before any measurement is considered as the total amount of correlations in the state ρ AB , then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations C (ρ AB ) in the state ρ AB 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 (ρ AB ) in the state ρ AB are defined by Q (ρ AB ) = I (ρ AB ) − C (ρ AB ), we also show that it is upper bounded by the von Neumann entropy of subsystem B . It is also found that Q (ρ AB ) is upper bounded by the von Neumann entropy of subsystem A for a class of states.
In this paper, two conjectures which were proposed in [Phys. Rev. A \textbf{82}, 052122(2010)] on the correlations in a bipartite state ρAB\rho^{AB}ρAB are studied. If the mutual information $I\Pa{\rho^{AB}}$ between two quantum systems AAA and BBB before any measurement is considered as the total amount of correlations in the state ρAB\rho^{AB}ρAB, 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 ρAB\rho^{AB}ρAB, defined by the maximizing mutual information between two quantum systems AAA and BBB after von Neumann measurements on system BBB, we show that it is upper bounded by the von Neumann entropies of both subsystems AAA and BBB, this answered the conjecture on the classical correlation. If the quantum correlations $Q\Pa{\rho^{AB}}$ in the state ρAB\rho^{AB}ρAB 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 BBB. It is also obtained that $Q\Pa{\rho^{AB}}$ is upper bounded by the von Neumann entropy of subsystem AAA for a class of states.
Abstract In Li and Luo (2007 Phys. Rev. A 76 032327), the inequality ( 1 / 2 )  ≥  was identified as a fundamental postulate for a consistent theory of quantum versus classical correlations for arbitrary measures of total  and quantum  correlations in bipartite quantum states. Besides, Hayden et al (2006 Commun. Math. Phys. 265 95) have conjectured that, in some conditions within systems endowed with infinite-dimensional Hilbert spaces, quantum correlations may dominate not only half of total correlations but total correlations itself. Here, in a two-mode Gaussian state, quantifying  and  respectively by the quantum mutual information  G and the entanglement of formation (EoF)  F G , we verify that  F , R G is always less than ( 1 / 2 )  R G when  G and  F G are defined via the Rényi-2 entropy. While via the von Neumann entropy,  F , V G may even dominate 
How strongly can one system be correlated with another? In the classical world, this basic question concerning correlative capacity has a very satisfying answer: The ``effective size'' of the marginal system, as quantified by the Shannon entropy, sets a tight upper bound to the correlations, as quantified by the mutual information. Although in the quantum world bipartite correlations, like their classical counterparts, are also well quantified by mutual information, the similarity ends here: The correlations in a bipartite quantum system can be twice as large as the marginal entropy. In the paradigm of quantum discord, the correlations are split into classical and quantum components, and it was conjectured that both the classical and quantum correlations are (like the classical mutual information) bounded above by each subsystem's entropy. In this work, by exploiting the interplay between entanglement of formation, mutual information, and quantum discord, we disprove that conjecture. We further indicate a scheme to restore harmony between quantum and classical correlative capacities. The results illustrate dramatically the asymmetric nature of quantum discord and highlight some subtle and unusual features of quantum correlations.
School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom(Dated: August 22, 2010)We investigate quantum correlations versus global von Neumann entropy, focusing explicitly on two-qubitstates. We establish a hierarchy of non-classicality indicators, showing that the measurement-induced distur-bance (MID) defined in [S. Luo, Phys. Rev. A, 77, 022301 (2008)] can be non-null and even maximal on“classical” states. We thus consider an ameliorated versio n of MID, optimized over all local measurements, andderive its expression for general two-qubit states. We thenstudy the analytical relation between amelioratedMID and quantum discord, characterizing the maximally non-classical two-qubit mixed states that simultane-ously extremize both such non-classicality quantifiers at given von Neumann entropy.