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arXiv:1109.1696 · Physical Review A

Monogamy properties of quantum and classical correlations

Gian Luca Giorgi

Published Nov 2, 2011• 122 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.84.054301

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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 WWW 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.
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.
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.
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.
It is well known that a particle cannot freely share entanglement with two or more particles. This restriction is generally called monogamy. However the formal quantification of such restriction is only known for some measures of entanglement and for two-level systems. The first and broadly known monogamy relation was established by Coffman, Kundu, and Wootters for the square of the concurrence. Since then, it is usually said that the entanglement of formation is not monogamous, as it does not obey the same relation. We show here that despite that, the entanglement of formation cannot be freely shared and therefore should be said to be monogamous. Furthermore, the square of the entanglement of formation does obey the same relation of the squared concurrence, a fact recently noted for three particles and extended here for N particles. Therefore the entanglement of formation is as monogamous as the concurrence. We also numerically study how the entanglement is distributed in pure states of three qubits and the relation between the sum of the bipartite entanglement and the classical correlation.
Quantum discord, an information-theoretic quantum correlation measure, can satisfy as well as violate monogamy for three-party quantum states. We quantify the feature using the concept of discord monogamy score. We find a necessary condition of a vanishing discord monogamy score for arbitrary three-party states. A necessary and sufficient condition is obtained for pure states. We prove that the class of states having a vanishing discord monogamy score cannot have arbitrarily high genuine multipartite entanglement, as quantified by generalized geometric measure. In the special case of three-qubit pure states, their classification with respect to the discord monogamy score reveals a rich structure that is different from that which had been obtained by using the monogamy score corresponding to the entanglement measure called concurrence. We investigate properties like genuine multipartite entanglement and violation of the multipartite Bell inequality for these states.