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arXiv:1207.2650 · Physical Review A

Monogamy of quantum discord by multipartite correlations

Helena Braga, C. C. Rulli, Thiago Rodrigues de Oliveira, M. S. Sarandy

Published Dec 12, 2012• 50 citations
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Abstract

We introduce a monogamy inequality for quantum correlations, which implies that the sum of pairwise quantum correlations is upper limited by the amount of multipartite quantum correlations as measured by the global quantum discord. This monogamy bound holds either for pure or mixed quantum states provided that bipartite quantum discord does not increase under discard of subsystems. We illustrate the monogamy behavior for multipartite pure states with Schmidt decomposition as well as for $W$-GHZ mixed states. As a by-product, we apply the monogamy bound to investigate residual multipartite correlations.

DOI
10.1103/physreva.86.062106

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We introduce a monogamy inequality for quantum correlations, which implies that the sum of pairwise quantum correlations is upper limited by the amount of multipartite quantum correlations as measured by the global quantum discord. This monogamy bound holds either for pure or mixed quantum states provided that bipartite quantum discord does not increase under discard of subsystems. We illustrate the monogamy behavior for multipartite pure states with Schmidt decomposition as well as for WWW-GHZ mixed states. As a by-product, we apply the monogamy bound to investigate residual multipartite correlations.
We consider a monogamy inequality of quantum discord in a pure tripartite state and show that it is equivalent to an inequality between quantum mutual information and entanglement of formation of two parties. Since this inequality does not hold for arbitrary bipartite states, quantum discord can generally be both monogamous and polygamous. We also carry out numerical calculations for some special states. The upper bounds of quantum discord and classical correlation are also discussed and we give physical analysis on the invalidness of a previous conjectured upper bound of quantum correlation. Our results provide new insights for further understanding of distributions of quantum correlations.
Abstract The original quantum discord (QD) is shown to be not monogamous except for the three-qubit states. Recently, a complete monogamy relation for multiparty quantum system was established for entanglement in (Guo and Zhang 2020 Phys. Rev. A 101 032301), and in addition, a new multipartite generalization of QD was proposed in (Radhakrishnan et al 2020 Phys. Rev. Lett. 124 110401). In this work, we firstly define the complete monogamy for this multipartite quantum discord (MQD) and the global quantum discord (GQD). MQD, with the same spirit as the complete monogamy of entanglement, is said to be completely monogamous (i) if it does not increase under coarsening of subsystems and (ii) if some given combination of subsystems reach the total amount of the correlation, then all other combination of subsystems that do not include all the given subsystems do not contain such a correlation any more. Here, coarsening of subsystems means discarding or combining some subsystems up to the given partition. Simultaneously, the complete monogamy of GQD is also defined with slight modification on the coarsening relation. Consequently, we explore all the coarsening relations of MQD and show that it is completely monogamous with a modicum assumption. In addition, with the same spirit, we investigate all the coarsening relations for GQD and show that GQD is not completely monogamous. That is, in the sense of the complete monogamy relation, MQD as a generalization of the original QD captures the nature of such a quantum correlation, and thus it is nicer than GQD as a generalization.
We introduce the concept of monogamy deficit for quantum correlation by combining two types of monogamy inequalities depending on different measurement sides. For tripartite pure states, we demonstrate a relation which connects two types of monogamy inequalities for quantum discord and provide the difference between them. By using this relation, we obtain a unified physical interpretation for these two monogamy deficits. In addition, we find the interesting fact that there is a general monogamy condition for several quantum correlations for tripartite pure states. We then provide a necessary and sufficient condition for the establishment of one kind of monogamy inequality for tripartite mixed states and generalize it to multipartite quantum states.
Quantum entanglement and quantum nonlocality are known to exhibit monogamy; that is, they obey strong constraints on how they can be distributed among multipartite systems. Quantum correlations that comprise and go beyond entanglement are quantified by, e.g., quantum discord. It was observed recently that for some states quantum discord is not monogamous. We prove, in general, that any measure of correlations that is monogamous for all states and satisfies reasonable basic properties must vanish for all separable states: only entanglement measures can be strictly monogamous. Monogamy of other than entanglement measures can still be satisfied for special, restricted cases: we prove that the geometric measure of discord satisfies the monogamy inequality on all pure states of three qubits.