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arXiv:1603.09120 · Physical Review A

Complementarity of genuine multipartite Bell nonlocality

Sasha Sami, Indranil Chakrabarty, Anubhav Chaturvedi

Published Aug 15, 2017• 11 citations
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Abstract

We introduce a feature of no-signaling (Bell) nonlocal theories: namely, when a system of multiple parties manifests genuine nonlocal correlation, then there cannot be arbitrarily high nonlocal correlation among any subset of the parties. We call this feature complementarity of genuine multipartite nonlocality. We use Svetlichny's criterion for genuine multipartite nonlocality and nonlocal games to derive the complementarity relations under no-signaling constraints. We find that the complementarity relations are tightened for the much stricter quantum constraints. We compare this notion with the well-known notion of monogamy of nonlocality. As a consequence, we obtain tighter nontrivial monogamy relations that take into account genuine multipartite nonlocality. Furthermore, we provide numerical evidence showcasing this feature using a bipartite measure and several other well-known tripartite measures of nonlocality.

DOI
10.1103/physreva.96.022121

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We introduce a feature of no-signaling (Bell) nonlocal theories: namely, when a system of multiple parties manifests genuine nonlocal correlation, then there cannot be arbitrarily high nonlocal correlation among any subset of the parties. We call this feature complementarity of genuine multipartite nonlocality. We use Svetlichny's criterion for genuine multipartite nonlocality and nonlocal games to derive the complementarity relations under no-signaling constraints. We find that the complementarity relations are tightened for the much stricter quantum constraints. We compare this notion with the well-known notion of monogamy of nonlocality. As a consequence, we obtain tighter nontrivial monogamy relations that take into account genuine multipartite nonlocality. Furthermore, we provide numerical evidence showcasing this feature using a bipartite measure and several other well-known tripartite measures of nonlocality.
The notion of complementarity or mutually exclusiveness of physical phenomenon has always been of keen interest to the physicists. In this work we show that there exists a complementary relationship in terms of the genuine non-locality of a multipartite system vs. that of reduced subsystems. We further assert that the same is enforced due to no-signaling condition. To this end we consider Svetlichny games in a multiparty binary input and output scenario with a threshold value of the winning probability as a signature of genuine multiparty non locality. We analytically show that, in the Svetlichny games setup, there exists complementary relations between Svetlichny correlations of nnn party and Svetlichny correlations of any subset of k≤nk \leq nk≤n parties within the no-signaling framework. In other words, in general non local theories, the genuine multiparty non locality for multiparty systems comes at the cost of upper bounding the genuine multiparty non locality of the reduced subsystem. In addition to Svetlichny games we consider many other non local games like Mermin games, Bancal \emph{et al's} IP, Guess Your Neighbors Input game and provide numerical findings ( in case of no-signaling and quantum correlations ) in support of our main result.
Many three-party correlations, including some that are commonly described as genuinely tripartite nonlocal, can be simulated by a network of underlying subsystems that display only bipartite nonsignaling nonlocal behavior. Quantum mechanics predicts three-party correlations that admit no such simulation, suggesting there are versions of nonlocality in nature transcending the phenomenon of bipartite nonsignaling nonlocality. This paper introduces a rigorous framework for analyzing tripartite correlations that can be simulated by bipartite-only networks. We confirm that expected properties of so-obtained correlations, such as no-signaling, indeed hold, and show how to use the framework to derive Bell-inequality-type constraints on these correlations that can be robustly violated by tripartite quantum systems. In particular, we use this framework to rederive a version of one such constraint previously described in a paper of Chao and Reichardt, Test to separate quantum theory from non-signaling theories [arXiv:1706.02008 (2017)].
Local measurements acting on entangled quantum states give rise to a rich correlation structure in the multipartite scenario. We introduce a versatile technique to build families of Bell inequalities witnessing different notions of multipartite nonlocality for any number of parties. The idea behind our method is simple: a known Bell inequality satisfying certain constraints, for example the Clauser–Horne–Shimony–Holt inequality, serves as the seed to build new families of inequalities for more parties. The constructed inequalities have a clear operational meaning, capturing an essential feature of multipartite correlations: their violation implies that numerous subgroups of parties violate the inequality chosen as seed. The more multipartite nonlocal the correlations, the more subgroups can violate the seed. We illustrate our construction using different seeds and designing Bell inequalities to detect k -way nonlocal multipartite correlations, in particular, genuine multipartite nonlocal correlations—the strongest notion of multipartite nonlocality. For one of our inequalities we prove analytically that a large class of pure states that are genuine multipartite entangled (GME) exhibit genuine multipartite nonlocality for any number of parties, even for some states that are almost product. We also provide numerical evidence that this family is violated by all GME pure states of three and four qubits. Our results make us conjecture that this family of Bell inequalities can be used to prove the equivalence between genuine multipartite pure-state entanglement and nonlocality for any number of parties.
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.
The violations of Bell inequalities by measurements on quantum states give rise to the phenomenon of quantum non-locality and express the advantage of using quantum resources over classical ones for certain information-theoretic tasks. The relative degree of quantum violations has been well studied in the bipartite scenario and in the multipartite scenario with respect to fully local behaviours. However, the multipartite setting entails a more complex classification in which different notions on non-locality can be established. In particular, genuine multipartite non-local distributions apprehend truly multipartite effects, given that these behaviours cannot be reproduced by bilocal models that allow correlations among strict subsets of the parties beyond a local common cause. We show here that, while in the so-called correlation scenario the relative violation of bilocal Bell inequalities by quantum resources is bounded, i.e. it does not grow arbitrarily with the number of inputs, it turns out to be unbounded in the general case. We identify Bell functionals that take the form of non-local games for which the ratio of the quantum and bilocal values grows unboundedly as a function of the number of inputs and outputs.
Abstract The violations of Bell inequalities by measurements on quantum states give rise to the phenomenon of quantum non-locality and express the advantage of using quantum resources over classical ones for certain information-theoretic tasks. The relative degree of quantum violations has been well studied in the bipartite scenario and in the multipartite scenario with respect to fully local behaviours. However, the multipartite setting entails a more complex classification in which different notions on non-locality can be established. In particular, genuine multipartite non-local distributions apprehend truly multipartite effects, given that these behaviours cannot be reproduced by bilocal models that allow correlations among strict subsets of the parties beyond a local common cause. We show here that, while in the so-called correlation scenario the relative violation of bilocal Bell inequalities by quantum resources is bounded, i.e. it does not grow arbitrarily with the number of inputs, it turns out to be unbounded in the general case. We identify Bell functionals that take the form of non-local games for which the ratio of the quantum and bilocal values grows unboundedly as a function of the number of inputs and outputs.