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arXiv:1308.0131 · Physical Review A

Violation of Bell inequality in perfect translation-invariant systems

Zhao‐Yu Sun, Yuying Wu, Jian Xu, Hailin Huang, Chen Bo-jun, Bo Wang

Published Nov 6, 2013• 16 citations
arXivHTMLPDF

Abstract

Bell inequalities and nonlocality have been widely studied in one-dimensional quantum systems. As a kind of quantum correlation, it is expected that bipartite nonlocality should be present in quantum systems, just as bipartite entanglement does. Surprisingly, for various models, two-qubit states do not violate Bell inequalities, i.e., they are local. Recently, it is realized that the results are related to the monogamy trade-off obeyed by bipartite Bell correlations, thus it is believed that for general translation invariant systems, two-qubit states should not violate the Bell inequality [Oliveira, Europhys. Lett. 100, 60004 (2012)]. In this Brief Report, we demonstrate that in perfect translation-invariant systems with an even number of sites, the Bell inequality can be violated. A nontrivial model is constructed to confirm the conclusion.

DOI
10.1103/physreva.88.054101

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Bell inequalities and nonlocality have been widely studied in one-dimensional quantum systems. As a kind of quantum correlation, it is expected that bipartite nonlocality should be present in quantum systems, just as bipartite entanglement does. Surprisingly, for various models, two-qubit states do not violate Bell inequalities, i.e., they are local. Recently, it is realized that the results are related to the monogamy trade-off obeyed by bipartite Bell correlations, thus it is believed that for general translation invariant systems, two-qubit states should not violate the Bell inequality [Oliveira, Europhys. Lett. 100, 60004 (2012)]. In this Brief Report, we demonstrate that in perfect translation-invariant systems with an even number of sites, the Bell inequality can be violated. A nontrivial model is constructed to confirm the conclusion.
The nature of quantum correlations in strongly correlated systems has been a subject of intense research. In particular, it has been realized that entanglement and quantum discord are present at quantum phase transitions and are able to characterize them. Surprisingly, it has been shown for a number of different systems that qubit pairwise states, even when highly entangled, do not violate Bell's inequalities, being in this sense local. Here we show that such a local character of quantum correlations traces back to the the monogamy trade-off obeyed by bipartite Bell correlations, being in fact general for translation invariant systems. We illustrate this result in a quantum spin chain with a soft breaking of translation symmetry. In addition, we provide an extension of the monogamy inequality to the N -qubit scenario, showing that the bound increases with N and providing examples of its saturation through uniformly generated random pure states.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three and four-partite Bell inequalities constructed from one and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three- and four-partite Bell inequalities constructed from one- and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three and four-partite Bell inequalities constructed from one and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality.
Two overlapping bipartite binary Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signaling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signaling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different numbers of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multiparty nonlocality, as evidenced by a generalized Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.
Non-classical quantum correlations underpin both the foundations of quantum mechanics and modern quantum technologies. Among them, Bell nonlocality is a central example. For bipartite Bell inequalities, nonlocal correlations obey strict monogamy: a violation of one inequality precludes violations of other inequalities on the overlapping subsystems. In the multipartite setting, however, Bell nonlocality becomes inherently polygamous. This was previously shown for subsystems obtained by removing a single particle from an NNN-partite system. Here, we generalize this result to arbitrary (N−k)(N-k)(N−k)-partite subsystems. We demonstrate that a single NNN-qubit state can violate all (Nk)\binom{N}{k}(kN​) relevant Bell inequalities simultaneously. We further construct an NNN-qubit Bell inequality, obtained by symmetrizing the (N−k)(N-k)(N−k)-qubit ones, that is maximally violated by states exhibiting this generalized polygamy. We compare these violations with those achievable by GHZ states and show that polygamy offers an advantage in multipartite scenarios, providing new insights into scalable certification of non-classicality in quantum devices. Our analysis relies on symmetry properties of the MABK inequalities. Finally, we show that this behavior can occur across multiple subsystem sizes, a phenomenon we call hyper-polygamy. These structures reveal the remarkable abundance of nonlocality present in multipartite quantum states and offer perspectives for their applications in quantum technologies.