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arXiv:2103.00924 · Quantum Science and Technology

Monogamy of quantum discord

Yu Jing Guo, Lizhong Huang, Yang Zhang

Published Sep 14, 2021• 14 citations

Abstract

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.

DOI: 10.1088/2058-9565/ac26b0

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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.

Multipartite entanglement includes not only the genuine entanglement but also the

k

-entanglement $(k\ensuremath{\geqslant}2)$. It is known that the most bipartite entanglement measures have been shown to be monogamous, but the monogamy relation is involved in the bipartite entanglement measures rather than the multipartite ones. So how we can explore the monogamy relation for

k

-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the

k

-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete

k

-entanglement measure and the associated complete monogamy relation according to the framework of the complete multipartite entanglement measure we established in [Phys. Rev. A 101, 032301 (2020)], which is shown to be an efficient tool for characterizing the multipartite quantum correlation as complementary to the monogamy relation associated with the bipartite measure. Consequently, the relation and the difference between monogamy, complete monogamy, and tightly complete monogamy are clearly depicted in light of the three types of coarser relation of the system partition. We then illustrate our approach with two classes of

k

-entanglement measures in detail. We find that all these

k

-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.

In quantum mechanics, quantum discord is a valuable quantum resource. As a generalization of the initial quantum discord, the multipartite quantum discord (MQD) exhibits excellent properties in the complete monogamy relation. In this paper, the dynamics of MQD are investigated for the Greenberger–Horne–Zeilinger (GHZ) state in Markovian environments. More specifically, it indicates that MQD of the GHZ state drops significantly over time in the depolarizing noise channel. Interestingly, quantum decoherence has no influence on MQD of the GHZ state in the Pauli channel [Formula: see text]. Subsequently, the analytical formulas of MQD are derived with six parameters in the Pauli channels [Formula: see text] and [Formula: see text], respectively. This is conducive to more efficient use of the GHZ state for various quantum tasks.

The distribution of quantum correlations in multipartite systems plays a significant role in several aspects of quantum information theory. While it is well known that these quantum correlations cannot be freely distributed, the way that they are shared in a multipartite system is an open problem even for a small set of qubits. Based on monogamylike relations between entanglement and discord for

n

-partite systems, we show how these correlations are distributed in general, determining distinct equalities and inequalities to the quantum discord and the entanglement of formation for arbitrary multipartite pure states.

Abstract The monogamy relation of quantum states has limited the shareability properties of quantum resources in multiparty quantum systems. It plays a vital role in information distribution and transformation over many sites in quantum communications. Here, we focus on the monogamy relations of quantum correlations in the context of three-flavor neutrino oscillations, related to the squared entanglement of formation, the squared of quantum discord and its geometric variant. The monogamy relations in terms of the squared entanglement of formation work for the electron and muon antineutrino oscillations, suggesting that the bipartite entanglement measured by squared entanglement of formation of the three-flavor neutrino system set a limitation to the sum of pairwise squared entanglement of formation. Furthermore, we found that the squared quantum discord is also monogamous in three-flavor neutrino system. As a comparison, we test the monogamy of the quantum discord in neutrino oscillations with the result that the quantum discord is not monogamous. In addition, it is found that the bipartite geometric quantum discord of three-flavor systems is equal to the sum of the pairwise geometric quantum discord, i.e., the monogamy relation for geometric quantum discord is saturated for three-flavor neutrino system. These monogamy relations of quantum correlations provide a way for studying the distribution of quantum resources in neutrino oscillations, which is of significance to explore the further applications of neutrino oscillations in quantum communications.

The monogamy of entanglement means that entanglement cannot be freely shared. In 2014, Oliveira et al. [T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini, Phys. Rev. A 89, 034303 (2014)] proposed a monogamy relation in the linear version and considered it in terms of entanglement of formation. Here we generalize the above version and consider a multilinear monogamy relation for a multiqubit system in terms of entanglement of formation and concurrence. Based on the above results, we present an entanglement criterion for genuine entangled states; also we consider the absolutely maximally entangled states and present what an absolutely maximally entangled state is for a three-qubit system. Last, we apply our results to a three-qubit pure state in terms of quantum discord.

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.