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.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.

arXiv:1111.5163Quantum Information and ComputationMay 1, 201314 citations

Monogamy properties of quantum and classical correlations

Gian Luca Giorgi

In contrast with entanglement, as measured by concurrence, in general, quantum discord does not possess the property of monogamy; that is, there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party. Here, we show that, as far as monogamy is considered, quantum discord of pure states is equivalent to the entanglement of formation. This result allows one to analytically prove that none of the pure three-qubit states belonging to the subclass of $W$ states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.In contrast with entanglement, as measured by concurrence, in general, quantum discord does not possess the property of monogamy; that is, there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party. Here, we show that, as far as monogamy is considered, quantum discord of pure states is equivalent to the entanglement of formation. This result allows one to analytically prove that none of the pure three-qubit states belonging to the subclass of $W$ states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.

arXiv:1109.1696Physical Review ANov 2, 2011122 citations

Monogamy of quantum discord by multipartite correlations

Helena Braga, C. C. Rulli, Thiago Rodrigues de Oliveira +1 more

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.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.

arXiv:1207.2650Physical Review ADec 12, 201250 citations

Monogamy deficit for quantum correlations in a multipartite quantum system

Siyuan Liu, Bo Li, Wen‐Li Yang +1 more

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.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.

arXiv:1301.7600Physical Review AJun 24, 201317 citations

Multipartite Generalization of Quantum Discord

Chandrashekar Radhakrishnan, Mathieu Laurière, Tim Byrnes

A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information. The former gives a measure of the bipartite nonclassical correlations and is a non-negative quantity, while the latter is related to the monogamy of the nonclassical correlations.A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information. The former gives a measure of the bipartite nonclassical correlations and is a non-negative quantity, while the latter is related to the monogamy of the nonclassical correlations.

arXiv:1909.08194Physical Review LettersMar 16, 202060 citations

Are General Quantum Correlations Monogamous?

Alexander Streltsov, Gerardo Adesso, Marco Piani +1 more

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.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.

arXiv:1112.3967Physical Review LettersAug 1, 2012171 citations

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states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.

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-GHZ mixed states. As a by-product, we apply the monogamy bound to investigate residual multipartite correlations.