Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.03104

Strong Locality as a Tetrahedron: A Symmetry-Reduced Geometric Representation of the (3,3,2,2) Bell Scenario

Marek Gazdzicki, Francesco Giacosa, Pawel Piesowicz

quant-phhep-thSubmitted May 4, 2026• Announced May 6, 2026• 0 citations
v1 · May 4, 2026v2 · May 10, 2026v3 · Aug 10, 2026
arXivHTMLPDF

Abstract

We present a geometric characterisation of strongly-local models in the bipartite Bell scenario with three measurement settings per site and binary outcomes, i.e.\ the (3,3,2,2) case. Restricting attention to indistinguishable sites, we introduce a three-dimensional mixed-moment space in which the mixed moments are calculated under off-diagonal measurement settings. In this reduced representation, the strongly-local region assumes the remarkably simple form of a regular tetrahedron - the 'pyramid'. We prove that only three independent linear inequalities are required to characterise this region. We call them the pyramid inequalities that separate strongly-local ($\mathcal{SL}$) models from their complement, non-strongly-local ($\mathcal{\overline{SL}}$) models. We also clarify the relation between the symmetry-reduced pyramid representation and the full (3,3,2,2) Bell polytope in the 36-dimensional conditional-probability space, which possesses 684 facet-defining inequalities. The reduction from 684 to three reflects normalisation, symmetry reduction, and projection to the mixed-moment space. In the pyramid representation, the hierarchy $\mathcal{SL} \subsetneq \mathcal{Q} \subsetneq \mathcal{NS}$ appears geometrically as a tetrahedron embedded in a somewhat larger curved body of quantum models, $\mathcal{Q}$, which in turn is embedded in a cube of no-signalling models, $\mathcal{NS}$. The qualitative and quantitative advantages of the pyramid representation over the standard CSHS representation for the (2,2,2,2) case are discussed.

Comments
23 pages, 3 figures

Related papers

From breakfast, lunch, and dinner to the Bell-inequality tetrahedron

Francesco Giacosa

A simple pedagogical discussion and visualization of certain Bell inequalities is presented with the help of a system involving two \textquotedblleft twins\textquotedblright, originally discussed in 2605.03104. These twins answer identically when the same `question' is posed, here exemplified with: Did you like breakfast/lunch/dinner? In analogy, for two entangled particles at distant locations, the `twin-like' quantum state is such that the same outcome for the same measurement (i.e. spin direction) is obtained. In the space $(X,Y,Z)$ of mixed moments (i.e. $X$ refers to breakfast-lunch correlation, being $1$ for same answers and $-1$ for opposite ones), the space of Bell local models is contained in a tetrahedron, while the quantum space is bound by an elliptope that embeds the tetrahedron. The whole no-signalling region in the $(X,Y,Z)$ space is the cube $[-1,1]^{3}.$ The twin scenario applies e.g. to the decay of the Higgs into fermions.

quant-pharXiv:2609.14436•Sep 15, 2026•0 citations

Bell inequalities from outcome equalities

Ricardo Faleiro, Flavien Hirsch, Emmanuel Zambrini Cruzeiro +1 more

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.

quant-pharXiv:2603.17030•Physical Review A•Mar 19, 2026•0 citations

Classifying the simplest Bell inequalities beyond qubits and their applications towards self-testing

Palash Pandya, Shubhayan Sarkar, Remigiusz Augusiak

Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest $(2,2,2)$ scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the $(2,2,3)$ scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.

quant-pharXiv:2602.08469•Feb 10, 2026•0 citations

Crouching Symmetry, Hidden Structure: A Geometric Reinterpretation of Bell's Inequality

Kacheong R. Yeung, Lex Yeung

Bell’s inequality relies on a structural feature of the standard Bell-local model: the potential outcomes for all relevant measurement settings are jointly represented on a single hidden-variable space. This joint representation is not enforced by locality alone, and its status becomes nontrivial once the physical realization of measurement events and their associated symmetry structures are taken into account. We develop a symmetry-sensitive framework in which each measurement event is associated with an event-dependent symmetry action on the hidden-variable space. Within this framework, symmetry-compatible joint representability requires that these actions be compatible with a single equivariant hidden-variable model. We show that, for the symmetry structures considered here, no such common realization exists. Consequently, the standard jointly represented CHSH construction cannot be carried out within this symmetry-restricted class. The obstruction is algebraic rather than causal: spacelike separation rules out direct causal influence between the measurement events, but does not ensure compatibility of their symmetry actions. The analysis brings into view a representational structure beneath the standard Bell construction: joint representability is not only a probabilistic requirement, but also a question of symmetry compatibility. This perspective clarifies the assumptions underlying Bell’s theorem and highlights the role of symmetry in determining which hidden-variable models are physically realizable. This is version 1.0 of the manuscript.

•Aug 13, 2026•0 citations

Tight Bell inequalities from polytope slices

José Jesus, Emmanuel Zambrini Cruzeiro

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.

arXiv:2212.03212•Physical Review A•Nov 27, 2023•4 citations

Platonic Bell inequalities for all dimensions

Károly F. Pál, Tamás Vértesi

In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regular polyhedra, we use the correspondence of the vertices to the measurements in the abstract Tsirelson space. We give a remarkably simple formula for the quantum violation of all the Platonic Bell inequalities, which we prove to attain the maximum possible quantum violation of the Bell inequalities, i.e. the Tsirelson bound. To construct Bell inequalities with a large number of settings, it is crucial to compute the local bound efficiently. In general, the computation time required to compute the local bound grows exponentially with the number of measurement settings. We find a method to compute the local bound exactly for any bipartite two-outcome Bell inequality, where the dependence becomes polynomial whose degree is the rank of the Bell matrix. To show that this algorithm can be used in practice, we compute the local bound of a 300-setting Platonic Bell inequality based on the halved dodecaplex. In addition, we use a diagonal modification of the original Platonic Bell matrix to increase the ratio of quantum to local bound. In this way, we obtain a four-dimensional 60-setting Platonic Bell inequality based on the halved tetraplex for which the quantum violation exceeds the 2 ratio.

arXiv:2112.03887•Quantum•Jul 7, 2022•7 citations
We present a geometric characterisation of strongly-local models in the bipartite Bell scenario with three measurement settings per site and binary outcomes, i.e.\ the (3,3,2,2) case. Restricting attention to indistinguishable sites, we introduce a three-dimensional mixed-moment space in which the mixed moments are calculated under off-diagonal measurement settings.
In this reduced representation, the strongly-local region assumes the remarkably simple form of a regular tetrahedron - the 'pyramid'. We prove that only three independent linear inequalities are required to characterise this region. We call them the pyramid inequalities that separate strongly-local (SL\mathcal{SL}SL) models from their complement, non-strongly-local (SL‾\mathcal{\overline{SL}}SL) models. We also clarify the relation between the symmetry-reduced pyramid representation and the full (3,3,2,2) Bell polytope in the 36-dimensional conditional-probability space, which possesses 684 facet-defining inequalities. The reduction from 684 to three reflects normalisation, symmetry reduction, and projection to the mixed-moment space.
In the pyramid representation, the hierarchy SL⊊Q⊊NS\mathcal{SL} \subsetneq \mathcal{Q} \subsetneq \mathcal{NS}SL⊊Q⊊NS appears geometrically as a tetrahedron embedded in a somewhat larger curved body of quantum models, Q\mathcal{Q}Q, which in turn is embedded in a cube of no-signalling models, NS\mathcal{NS}NS. The qualitative and quantitative advantages of the pyramid representation over the standard CSHS representation for the (2,2,2,2) case are discussed.
A simple pedagogical discussion and visualization of certain Bell inequalities is presented with the help of a system involving two \textquotedblleft twins\textquotedblright, originally discussed in 2605.03104. These twins answer identically when the same `question' is posed, here exemplified with: Did you like breakfast/lunch/dinner? In analogy, for two entangled particles at distant locations, the `twin-like' quantum state is such that the same outcome for the same measurement (i.e. spin direction) is obtained. In the space (X,Y,Z)(X,Y,Z)(X,Y,Z) of mixed moments (i.e. XXX refers to breakfast-lunch correlation, being 111 for same answers and −1-1−1 for opposite ones), the space of Bell local models is contained in a tetrahedron, while the quantum space is bound by an elliptope that embeds the tetrahedron. The whole no-signalling region in the (X,Y,Z)(X,Y,Z)(X,Y,Z) space is the cube [−1,1]3.[-1,1]^{3}.[−1,1]3. The twin scenario applies e.g. to the decay of the Higgs into fermions.
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.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
Bell’s inequality relies on a structural feature of the standard Bell-local model: the potential outcomes for all relevant measurement settings are jointly represented on a single hidden-variable space. This joint representation is not enforced by locality alone, and its status becomes nontrivial once the physical realization of measurement events and their associated symmetry structures are taken into account. We develop a symmetry-sensitive framework in which each measurement event is associated with an event-dependent symmetry action on the hidden-variable space. Within this framework, symmetry-compatible joint representability requires that these actions be compatible with a single equivariant hidden-variable model. We show that, for the symmetry structures considered here, no such common realization exists. Consequently, the standard jointly represented CHSH construction cannot be carried out within this symmetry-restricted class. The obstruction is algebraic rather than causal: spacelike separation rules out direct causal influence between the measurement events, but does not ensure compatibility of their symmetry actions. The analysis brings into view a representational structure beneath the standard Bell construction: joint representability is not only a probabilistic requirement, but also a question of symmetry compatibility. This perspective clarifies the assumptions underlying Bell’s theorem and highlights the role of symmetry in determining which hidden-variable models are physically realizable. This is version 1.0 of the manuscript.
We derive tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario (X,Y,A,B)(X,Y,A,B)(X,Y,A,B) is defined by the numbers of settings and outcomes per party, X,AX,AX,A, and Y,BY,BY,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.
In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regular polyhedra, we use the correspondence of the vertices to the measurements in the abstract Tsirelson space. We give a remarkably simple formula for the quantum violation of all the Platonic Bell inequalities, which we prove to attain the maximum possible quantum violation of the Bell inequalities, i.e. the Tsirelson bound. To construct Bell inequalities with a large number of settings, it is crucial to compute the local bound efficiently. In general, the computation time required to compute the local bound grows exponentially with the number of measurement settings. We find a method to compute the local bound exactly for any bipartite two-outcome Bell inequality, where the dependence becomes polynomial whose degree is the rank of the Bell matrix. To show that this algorithm can be used in practice, we compute the local bound of a 300-setting Platonic Bell inequality based on the halved dodecaplex. In addition, we use a diagonal modification of the original Platonic Bell matrix to increase the ratio of quantum to local bound. In this way, we obtain a four-dimensional 60-setting Platonic Bell inequality based on the halved tetraplex for which the quantum violation exceeds the 2 ratio.