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arXiv:1402.4453 · Physical Review Letters

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

Ravishankar Ramanathan, Paweł Horodecki

Published Nov 20, 2014• 24 citations
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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 free unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previously known results. Our result connects, for the first time, the property of monogamy with that crucial part of the Bell expression that is necessary for revealing a contradiction with local realistic predictions, thus shifting the paradigm in the field of monogamy of correlations.

DOI
10.1103/physrevlett.113.210403

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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 free unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previously known results. Our result connects, for the first time, the property of monogamy with that crucial part of the Bell expression that is necessary for revealing a contradiction with local realistic predictions, thus shifting the paradigm in the field of monogamy of correlations.
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.
In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator in- equalities (XOR games) beyond binary outputs. We define the sub-polytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural prop- erties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, witnesses of genuine multipartite nonlocality, as well as being relevant to CHSH. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.
Physical principles constrain the way nonlocal correlations can be distributed among distant parties in a Bell-type experiment. These constraints are usually expressed by monogamy relations that bound the amount of Bell inequality violation observed by a set of parties by the violation observed by a different set of parties. Here we show that the no-signaling principle yields simple and tight monogamy relations for an important class of bipartite and multipartite Bell inequalities. We also link these trade-offs to the guessing probability---a key quantity in device-independent information processing.
Abstract Quantum correlations are subject to certain distribution rules represented by so-called monogamy relations. Derivation of monogamy relations for multipartite systems is a non-trivial problem, as the multipartite correlations reveal their behaviors in a way different from bipartite systems. We here show that simple geometric properties of probabilistic spaces, in conjunction with no-signaling principle, lead to genuine monogamy relations for a large class of Bell type inequalities for many qubits. The term of ‘genuine’ implies that only one out of N Bell inequalities exhibits a quantum violation. We also generalize our method to qudits. Using the similar geometric approach with a quasi-distance employed, we derive Svetlichny–Zohren–Gill type Bell inequalities for d-dimensional tripartite systems, and show their monogamous nature.
A unified view of the phenomenon of monogamy exhibited by Bell inequalities and noncontextuality inequalities arising from the no-signaling and no-disturbance principles is presented using the graph-theoretic method introduced by Ramanathan et al. [Phys. Rev. Lett. 109, 050404 (2012)]. We propose a hitherto unexplored tradeoff relation, namely, Bell inequalities that do not exhibit monogamy features of their own can be activated to be monogamous by the addition of a local contextuality term. This is illustrated by means of the well-known I3322{\mathcal{I}}_{3322}I3322​ inequality and reveals a resource trade-off between bipartite correlations and the local purity of a single system. In the derivation of no-signaling monogamies, we uncover a unique feature, namely, that two-party Bell expressions that are trivially classically saturated can become nontrivial upon the addition of an expression involving a third party with a single measurement input.