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arXiv:2603.17030 · Physical Review A

Bell Inequalities for Smells

Ricardo Faleiro, Flavien Hirsch, Emmanuel Zambrini Cruzeiro, Nicolas Gisin

quant-phSubmitted Mar 17, 2026• Announced Mar 19, 2026• 0 citations

v1 · Mar 17, 2026 v2 · Oct 5, 2026

Abstract

In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. This arises naturally when outcomes are difficult to characterize. For instance, if measurements yield smells, it may be impractical to process them individually, while still being reasonable to judge whether two smells are identical or not. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator inequalities (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 properties. 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.

Journal reference: Phys. Rev. A 114, 032448, 22 September, 2026
DOI: 10.1103/wccm-zgnl
Comments: 16 pages, 6 tables, 2 figures

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We derive tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario

(X,Y,A,B)

is defined by the numbers of settings and outcomes per party,

X,A

, and

Y,B

for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of (6,3,2,2), (3,3,3,2), (3,2,3,3), and (2,2,3,5). We provide extensive lists of facets for (2,2,4,4), (3,3,4,2), and (4,3,3,2). For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits, and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to the Clauser-Horne-Shimony-Holt inequality. Such scenarios could find important applications in quantum communication.

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Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.