Strong Monogamies of No-Signaling Violations for Bipartite Correlation Bell Inequalities

Abstract

The phenomenon of monogamy of Bell inequality violations is interesting both from the fundamental perspective as well as in cryptographic applications such as the extraction of randomness and secret bits. In this article, we derive new and stronger monogamy relations for violations of Bell inequalities in general no-signaling theories. These relations are applicable to the class of binary output correlation inequalities known as XOR games, and to a restricted set of unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previously known results. The result involves a shift in paradigm towards the importance in monogamies of the number of inputs of one party which lead to a contradiction from local realistic predictions.Introduction. The violation of a Bell inequality is a defining feature of quantum correlations that distinguishes them from classical correlations. Apart from the fundamental interest in the absence of a local hidden variable description of Nature, this feature of quantum theory has led to numerous applications in communication protocols, including key distribution [1,2] and generation of randomness [3,4]. Many interesting properties of this phenomenon dubbed non-locality have been discovered leading to a better understanding of the set of quantum correlations [5,6] which is crucial in the development of further applications. It is known that the set of quantum correlations is a convex set sandwiched between the classical polytope and the no-signaling polytope which is the set of correlations obeying the so-called no-signaling principle (impossibility of faster-than-light communication).The violation of Bell inequalities in general nosignaling theories (which includes quantum theory) displays a very interesting property called monogamy. Strong non-local correlations between two parties in extreme cases lead to weak correlations between these parties and any other no-signaling system. Specifically, there are instances where the violation of a Bell inequality by Alice and Bob precludes its violation by Alice and any other party Charlie, when Alice uses the same measurement results in both experiments. This phenomenon is seen to be important in secure communication protocols for key distribution or randomness generation between these parties, due to the fact that any third party such as an eavesdropper is only able to establish weak correlations with their systems [4,7,8].The monogamy of no-signaling correlations was first discovered for the CHSH inequality in [9] and has since been shown to be a generic feature of all no-signaling theories [5]. A general monogamy relation applicable to any no-signaling correlations in the two-party Bell scenario that only involves the number of settings (inputs) for one party was developed in [10,11]. This approach follows the idea of symmetric extensions and shareability of correlations which can also be recast in terms of the existence of joint probability distributions [12]. See also [13] for a nov...

Trade-offs in multiparty Bell-inequality violations in qubit networks

“…While the no-signaling trade-off relation is general, it is an open question whether it is tight, i.e., whether there exists a Bell expression for which a network with n i=2 mi + 1 parties is needed before such a monogamy relation manifests itself. For instance, it has been found that for general two-party correlation inequalities, monogamy relations hold in networks with far fewer number of parties [39].…”

Energy as a Detector of Nonlocality of Many-Body Spin Systems

“…This inequality cannot be violated, as quantum correlations are monogamous with respect to the chained Bell inequality [37,41]. Loosely speaking, if party B violates I chain with A, it cannot violate it simultaneously with C. For some types of monogamy relations, this result holds for various generalizations of the CHSH inequality to more measurements, outcomes and parties [42].…”

Activation of monogamy in nonlocality using local contextuality

“…While this appears counterintuitive at first sight, we explain that this arises due to an incompatibility between the optimal classical strategies for the sub-expressions into which the commutation graph of the whole Bell expression is decomposed. In [10], it was shown that in the paradigmatic example of correlation inequalities for binary outcomes, the pa-…”

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