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Monogamy Inequality in Terms of Higher Powers of a Correlation Measure: An Analysis

A. R. Usha Devi, S. Sudha, P. J. Geetha

Published Feb 27, 2021• 0 citations
Publisher

Abstract

We begin with the basic notions of limited shareability of quantum correlations, the so-called monogamous nature of quantum states and outline how this feature is manifested through a monogamy inequality. Not all measures of quantum correlations are suitable for bringing out the restricted shareability of correlations in a quantum state. While measures of bipartite correlations such as concurrence, quantum discord do not satisfy monogamy inequality for all categories of correlated quantum states, their second powers are found to do. There has also been a proposition that all multiparty quantum states can be made monogamous by considering positive integral powers of any quantum correlation measure (Ann. Phys. 348, 297 (2014)). We focus on this proposition and analyze whether satisfying monogamy inequality in terms of higher powers of a correlation measure implies limited shareability of correlations.

DOI
10.9734/bpi/nips/v13/6643d

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We begin with the basic notions of limited shareability of quantum correlations, the so-called monogamous nature of quantum states and outline how this feature is manifested through a monogamy inequality. Not all measures of quantum correlations are suitable for bringing out the restricted shareability of correlations in a quantum state. While measures of bipartite correlations such as concurrence, quantum discord do not satisfy monogamy inequality for all categories of correlated quantum states, their second powers are found to do. There has also been a proposition that all multiparty quantum states can be made monogamous by considering positive integral powers of any quantum correlation measure (Ann. Phys. 348, 297 (2014)). We focus on this proposition and analyze whether satisfying monogamy inequality in terms of higher powers of a correlation measure implies limited shareability of correlations.
We examine here the proposition that all multiparty quantum states can be made monogamous by considering positive integral powers of any quantum correlation measure. With Rajagopal-Rendell quantum deficit as the measure of quantum correlations for symmetric 3-qubit pure states, we illustrate that monogamy inequality is satisfied for higher powers of quantum deficit. We discuss the drawbacks of this inequality in quantification of correlations in the state. We also prove a monogamy inequality in higher powers of classical mutual information and bring out the fact that such inequality needs not necessarily imply restricted shareability of correlations. We thus disprove the utility of higher powers of any correlation measure in establishing monogamous nature in multiparty quantum states.
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.
Unlike classical correlation, quantum entanglement cannot be freely shared among many parties. This restricted shareability of entanglement among multi-party systems is known as monogamy of entanglement, which is one of the most fundamental properties of entanglement. Here, we summarize recent theoretical progress in the field of monogamy of entanglement. We firstly review the standard CKW-type monogamy inequalities in terms of various entanglement measures. In particular, the squashed entanglement and one-way distillable entanglement are monogamous for arbitrary dimensional systems. We then introduce some generalized version of monogamy inequalities which extend and sharpen the traditional ones. We also consider the dual polygamy inequalities for multi-party systems. Moreover, we present two new definitions to define monogamy of entanglement. Finally, some challenges and future directions for monogamy of entanglement are highlighted.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
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.
Abstract It is well known that entanglement is one of the distinctive aspects of the quantum world. The measurement and quantification of entanglement is one of the major problems in understanding and applying this resource. This research aims to investigate entanglement measures and their properties in high-dimensional systems. One such property is entanglement monogamy, viz. quantum correlations are not shareable, different from the classical ones. On the other hand, the study of entanglement depends on the definitions of its measures. Here we examine the applicability of concurrence, an entanglement measure known for its effectiveness in the analysis of qubit entanglement. While the entanglement of two-level systems or qubits is widely understood, its extension to higher dimensions remains challenging. Our research establishes equivalence between different concurrence definitions for bipartite qubit pure states and extends them to qudits. Subsequently, we investigate the monogamous relations of these measurements to explore their characteristics. This approach might enhance our comprehension of quantum traits in qudit systems and advance our understanding of entanglement in broader quantum contexts.