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arXiv:2406.12823 · npj Quantum Information

Three-outcome multipartite Bell inequalities: applications to dimension witnessing and spin-nematic squeezing in many-body systems

Guillem Müller-Rigat, Albert Aloy, Maciej Lewenstein, Matteo Fadel, Jordi Tura

Published Jun 13, 2026• 2 citations
arXivHTMLPDF

Abstract

We present a three-outcome Bell inequality, which we show to be naturally suited to explore nonlocal correlations in many-body spin-1 systems or SU(3) models. In the specific, we demonstrate from this inequality an experimentally practical Bell correlation test based on the measurement of collective spin components. Next, we analyze such an inequality as a dimension witness, namely, criteria whose violation signals the impossibility of reproducing the inferred correlations by Hilbert spaces of a certain dimension. In particular, our approach enables the certification of genuine three-level correlations in systems of arbitrary size, which cannot be reproduced by an ensemble of qubits. Then, we demonstrate how metrological resources, in the form of spin-nematic squeezed states as produced in spin-1 Bose Einstein condensates involving thousands of atoms, maximally violate such a witness. Moreover, we propose that the robustness of the violation can be used to certify the number of high-dimensional parties in the system. In this direction, we demonstrate that, in the thermodynamic limit, the squeezed correlations described above require the system to be composed entirely of qutrits. Our results shed light on the fundamental role of high-dimensional quantum resources in many-body systems and their application for metrological purposes.

DOI
10.1038/s41534-026-01261-8

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We present a three-outcome Bell inequality, which we show to be naturally suited to explore nonlocal correlations in many-body spin-1 systems or SU(3) models. In the specific, we demonstrate from this inequality an experimentally practical Bell correlation test based on the measurement of collective spin components. Next, we analyze such an inequality as a dimension witness, namely, criteria whose violation signals the impossibility of reproducing the inferred correlations by Hilbert spaces of a certain dimension. In particular, our approach enables the certification of genuine three-level correlations in systems of arbitrary size, which cannot be reproduced by an ensemble of qubits. Then, we demonstrate how metrological resources, in the form of spin-nematic squeezed states as produced in spin-1 Bose Einstein condensates involving thousands of atoms, maximally violate such a witness. Moreover, we propose that the robustness of the violation can be used to certify the number of high-dimensional parties in the system. In this direction, we demonstrate that, in the thermodynamic limit, the squeezed correlations described above require the system to be composed entirely of qutrits. Our results shed light on the fundamental role of high-dimensional quantum resources in many-body systems and their application for metrological purposes.
Correlations allowed by quantum mechanics can defy any classical explanation, with Bell nonlocality standing as their most profound and operationally powerful manifestation. While nonlocality has been demonstrated across a wide range of platforms, in many-body systems it has remained limited to ensembles of qubits, leaving the observation of higher-dimensional multipartite Bell correlations an open challenge. Here we report the observation of qutrit Bell correlations in a spin-1 87^{87}87Rb Bose-Einstein condensate via the violation of a Bell witness based on collective spin observables only. Exploiting spin-exchange collisions in an ensemble of N≃3.1×104N \simeq 3.1 \times 10^{4}N≃3.1×104 atoms, we generate spin-nematic squeezing of −11.8(7)-11.8(7)−11.8(7) dB and observe a violation that surpasses the minimum bound achievable by a collection of NNN qubits, providing direct evidence of genuine multipartite qutrit Bell correlations. Our results establish spinor Bose-Einstein condensates as a viable platform for investigating high-dimensional Bell correlations in the many-body regime, and demonstrate that coarse-grained collective measurements suffice to certify the dimensionality of quantum correlations at macroscopic scales.
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.
We present strategies to derive Bell inequalities valid for systems composed of many three-level parties. This scenario is formalized by a Bell experiment with N observers, each of which performs one out of two possible three-outcome measurements on their share of the system. As the complexity of the set of classical correlations prohibits its full characterization in this multipartite scenario, we consider its projection to a lower-dimensional subspace spanned by permutationally invariant one- and two-body observables. This simplification allows us to formulate two complementary methods for detecting nonlocality in multipartite three-level systems, both having a complexity independent of N. Our work can have interesting applications in the detection of Bell correlations in paradigmatic spin-1 models, as well as in experiments with solid-state systems or atomic ensembles.
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.
Abstract Bell nonlocality—the existence of quantum correlations that cannot be explained by classical means—is certainly one of the most striking features of quantum mechanics. Its range of applications in device-independent protocols is constantly growing. Many relevant quantum features can be inferred from violations of Bell inequalities, including entanglement detection and quantification, and state certification applicable to systems of arbitrary number of particles. A complete characterisation of nonlocal correlations for many-body systems is, however, a computationally intractable problem. Even if one restricts the analysis to specific classes of states, no general method to tailor Bell inequalities to be violated by a given state is known. In this work we provide a general construction of Bell expressions tailored to the graph states of any prime local dimension. These form a broad class of multipartite quantum states that have many applications in quantum information, including quantum error correction. We analytically determine their maximal quantum values, a number of high relevance for device-independent applications of Bell inequalities. Importantly, the number of expectation values to determine in order to test the violation of our inequalities scales only linearly with the system size, which we expect to be the optimal scaling one can hope for in this case. Finally, we show that these inequalities can be used for self-testing of multi-qutrit graph states such as the well-known four-qutrit absolutely maximally entangled state AME(4,3).
Multipartite Bell nonlocality provides a device-independent probe of many-body quantum correlations, but its characterization is limited by the rapid growth of the underlying classical and quantum optimization problems. We develop a scalable method for constructing and certifying permutationally invariant Bell inequalities using only one- and two-body correlators. The construction gives families of inequalities with robust quantum violations for general mmm measurements as the number of parties NNN becomes large. To improve robustness against noise, we optimize the ratio of the quantum value to the classical bound for these families in the large-NNN limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-NNN ratios take simple rational values for finite mmm and converge to coth⁡(1)\coth(1)coth(1) as m→∞m\to\inftym→∞. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.