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arXiv:1109.4318 · Physical Review A

Relating monogamy of quantum correlations and multisite entanglement

R. Prabhu, Arun Kumar Pati, Aditi Sen, Ujjwal Sen

Published Nov 30, 2012• 32 citations
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Abstract

There are two important paradigms for defining quantum correlations in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. We find an analytical relation between two measures of quantum correlations, one in each paradigm, and show that only a certain cone-like region on the two-dimensional space spanned by these measures is accessible to pure three-qubit states. The information-theoretic multiparty quantum correlation measure is related to the monogamy considerations of a bipartite information-theoretic quantum correlation measure, while the entanglement-separability multiparty measure is the generalized geometric measure, a genuine multiparty entanglement measure. We also find an analytical relation between two multiparty entanglement measures, and again obtain a cone-like accessible region in this case. One of the multisite measures in this case is related to the monogamy of a bipartite entanglement measure, while the other is again the generalized geometric measure.

DOI
10.1103/physreva.86.052337

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There are two important paradigms for defining quantum correlations in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. We find an analytical relation between two measures of quantum correlations, one in each paradigm, and show that only a certain cone-like region on the two-dimensional space spanned by these measures is accessible to pure three-qubit states. The information-theoretic multiparty quantum correlation measure is related to the monogamy considerations of a bipartite information-theoretic quantum correlation measure, while the entanglement-separability multiparty measure is the generalized geometric measure, a genuine multiparty entanglement measure. We also find an analytical relation between two multiparty entanglement measures, and again obtain a cone-like accessible region in this case. One of the multisite measures in this case is related to the monogamy of a bipartite entanglement measure, while the other is again the generalized geometric measure.
\n\nIn order to define quantum correlations, there are two important paradigms in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. In this paper, we give an analytical relation between two measures of quantum correlations. One of them is related to the monogamy of squared bipartite quantum discord, which is a information-theoretic multipartite quantum correlation measure, while the other is the generalized geometric measure which lies in the entanglement-separability paradigm. We find a certain cone-like region on the two-dimensional spaces spanned by the two measures. Moreover, we have investigated the quantum phase transition with the two measures in the anisotropic spin XXZ model by exploiting the quantum renormalization group method.\n
Violation of a Bell inequality guarantees the existence of quantum correlations in a shared quantum state. A pure bipartite quantum state, having nonvanishing quantum correlation, always violates a Bell inequality. Such correspondence is absent for multipartite pure quantum states in the case of multipartite correlation function Bell inequalities with two settings at each site. We establish a connection between the monogamy of Bell-inequality violation and multiparty quantum correlations for shared multisite quantum states. We believe that the relation is generic, as it is true for a number of different multisite measures that are defined from radically different perspectives. Precisely, we quantify the multisite-quantum-correlation content in the states by generalized geometric measure, a genuine multisite entanglement measure, as well as three monogamy-based multiparty-quantum-correlation measures, viz., 3-tangle, quantum-discord score, and quantum-work-deficit score. We find that generalized Greenberger-Horne-Zeilinger states and another single-parameter family of states, which we refer to as the special Greenberger-Horne-Zeilinger states, have the status of extremal states in such relations.
The shareability of quantum correlations among the constituent parties of a multiparty quantum system is restricted by the quantum information theoretic concept called monogamy. Depending on the multiparty quantum systems, different measures of quantum correlations show disparate signatures for monogamy. We characterize the shareability of quantum correlations, from both entanglement-separability and information-theoretic kinds, in a multiparty quantum spin system containing two- and three-body interactions with respect to its system parameters and external applied magnetic field. Monogamy score in this system exhibits both monogamous and nonmonogamous traits depending on the quantum correlation measure, strengths of system parameters, and external magnetic field. The percentage of nonmonogamous states when the information-theoretic quantum correlations are considered is higher than that of the entanglement-separability kind in allowed ranges of these variables. The integral powers of the quantum correlation measures for which the nonmonogamous states become monogamous are identified.
Quantum correlations are expected to respect all the conditions required for them to be good measures of quantumness in the bipartite scenario. In a multipartite setting, sharing entanglement between several parties is restricted by the monogamy of entanglement. We take over the concept of monogamy to an information-theoretic quantum correlation measure, and find that it violates monogamy in general. Using the notion of interaction information, we identify necessary and sufficient conditions for the measure to obey monogamy, for arbitrary pure and mixed quantum states. We show that while three-qubit generalized Greenberger-Horne-Zeilinger states follow monogamy, generalized WWW states do not.
"Is entanglement monogamous?" asks the title of a popular article [B. Terhal, IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett's legacy on quantum information theory. While the answer is affirmative in the qualitative sense, the situation is less clear if monogamy is intended as a quantitative limitation on the distribution of bipartite entanglement in a multipartite system, given some particular measure of entanglement. Here, we formalize what it takes for a bipartite measure of entanglement to obey a general quantitative monogamy relation on all quantum states. We then prove that an important class of entanglement measures fail to be monogamous in this general sense of the term, with monogamy violations becoming generic with increasing dimension. In particular, we show that every additive and suitably normalized entanglement measure cannot satisfy any nontrivial general monogamy relation while at the same time faithfully capturing the geometric entanglement structure of the fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of such entanglement measures can be recovered if one allows for dimension-dependent relations, as we show explicitly with relevant examples.
Multipartite entanglement includes not only the genuine entanglement but also the kkk-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 kkk-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the kkk-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete kkk-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 kkk-entanglement measures in detail. We find that all these kkk-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.