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:2406.11792 · Entropy

Deriving Three-Outcome Permutationally Invariant Bell Inequalities

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

Published Sep 25, 2024• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3390/e26100816

Related papers

Optimizing and Certifying Multipartite Permutationally Invariant Bell Inequalities

Jin-Fu Chen, Mengyao Hu, Jordi Tura

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 $m$ measurements as the number of parties $N$ 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-$N$ limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-$N$ ratios take simple rational values for finite $m$ and converge to $\coth(1)$ as $m\to\infty$. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.

quant-pharXiv:2607.08462•Jul 10, 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

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

Guillem Müller-Rigat, Albert Aloy, Maciej Lewenstein +2 more

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.

arXiv:2406.12823•npj Quantum Information•Jun 13, 2026•2 citations

Trade-offs and experimental feasibility of nonlocal polygamy with two-outcome Bell inequalities

Josep Batle, Tomasz Rybotycki, Tomasz Białecki +2 more

Entanglement and Bell nonlocality have different sharing constraints across subsystems of a multipartite quantum system. We examine simultaneous violations of Bell inequalities with two-outcome observables and two or three measurement settings per party. We identify configurations whose simultaneous violations can be demonstrated on IBM Quantum hardware, and derive bounds and trade-off relations for several polygamous configurations. The results connect experimental capabilities of quantum computers with mathematical methods for studying the limits of quantum correlations.

quant-pharXiv:2609.24835•Sep 22, 2026•0 citations

No-signalling-projection-invariant Bell inequalities

Soumyadip Patra, Jitendra Prakash, Aaditya Paudel +1 more

In this paper, we highlight how any Bell inequality for a configuration involving $n$ parties each performing one of $m$ binary-outcome measurements has a canonical form that is no-signalling-projection invariant. Specifically, the $L^2$-projection of weakly signalling data onto the no-signalling polytope leaves the violation of this canonical Bell inequality unchanged. Our methods allow us to derive a general closed formula for the projection and present a substantially more computationally simple procedure for its evaluation. We also show this can be generalised to non-standard projections of potential interest for certain applications. No-signalling projections serve as a preliminary step before undertaking any device-independent application involving Bell experiment data, such as hypothesis testing against local realism, random number generation and entanglement detection.

quant-pharXiv:2511.06624•Journal of Physics A Mathematical and Theoretical•Nov 11, 2025•0 citations

Experimental certification of multipartite Bell correlations using only few-body symmetric correlations

Jiacheng Sun, Mengyao Hu, Paweł Cieśliński +9 more

Certifying Bell correlations in increasingly larger multipartite quantum systems is challenging because conventional Bell inequalities rely on many-body correlations, which become experimentally demanding to estimate as system size grows. Here we demonstrate that multipartite Bell correlations can be certified using few-body permutation-invariant (PI) correlations with a fixed maximal correlation order. We experimentally test few-body PI Bell correlation witnesses tailored for six- and eight-photon postselected Dicke states. The addition of a few four-body correlators is found to significantly enhance the noise tolerance compared with previously studied two-body witnesses: for instance, for six qubits, the selected witness tolerates a white-noise fraction of $p_c\approx0.301$, compared with $p_c\approx0.048$ for the best two-body PI inequality with the same two settings per party. For six and eight photons respectively, we observe violations of the witness bounds by $32$ and $8.9$ standard deviations, arising from postselected polarisation states with fidelities of $0.933(4)$ and $0.916(11)$ to the corresponding half-excitation Dicke states. The witnesses for the six- and eight-photon cases use the same symmetric two- and four-body correlators, just with different coefficients. These results demonstrate an experimentally accessible approach for certifying multipartite Bell correlations in larger systems without requiring full-body correlation measurements.

quant-pharXiv:2609.37442•Sep 30, 2026•0 citations
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.
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.
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 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.
Entanglement and Bell nonlocality have different sharing constraints across subsystems of a multipartite quantum system. We examine simultaneous violations of Bell inequalities with two-outcome observables and two or three measurement settings per party. We identify configurations whose simultaneous violations can be demonstrated on IBM Quantum hardware, and derive bounds and trade-off relations for several polygamous configurations. The results connect experimental capabilities of quantum computers with mathematical methods for studying the limits of quantum correlations.
In this paper, we highlight how any Bell inequality for a configuration involving nnn parties each performing one of mmm binary-outcome measurements has a canonical form that is no-signalling-projection invariant. Specifically, the L2L^2L2-projection of weakly signalling data onto the no-signalling polytope leaves the violation of this canonical Bell inequality unchanged. Our methods allow us to derive a general closed formula for the projection and present a substantially more computationally simple procedure for its evaluation. We also show this can be generalised to non-standard projections of potential interest for certain applications. No-signalling projections serve as a preliminary step before undertaking any device-independent application involving Bell experiment data, such as hypothesis testing against local realism, random number generation and entanglement detection.
Certifying Bell correlations in increasingly larger multipartite quantum systems is challenging because conventional Bell inequalities rely on many-body correlations, which become experimentally demanding to estimate as system size grows. Here we demonstrate that multipartite Bell correlations can be certified using few-body permutation-invariant (PI) correlations with a fixed maximal correlation order. We experimentally test few-body PI Bell correlation witnesses tailored for six- and eight-photon postselected Dicke states. The addition of a few four-body correlators is found to significantly enhance the noise tolerance compared with previously studied two-body witnesses: for instance, for six qubits, the selected witness tolerates a white-noise fraction of pc≈0.301p_c\approx0.301pc​≈0.301, compared with pc≈0.048p_c\approx0.048pc​≈0.048 for the best two-body PI inequality with the same two settings per party. For six and eight photons respectively, we observe violations of the witness bounds by 323232 and 8.98.98.9 standard deviations, arising from postselected polarisation states with fidelities of 0.933(4)0.933(4)0.933(4) and 0.916(11)0.916(11)0.916(11) to the corresponding half-excitation Dicke states. The witnesses for the six- and eight-photon cases use the same symmetric two- and four-body correlators, just with different coefficients. These results demonstrate an experimentally accessible approach for certifying multipartite Bell correlations in larger systems without requiring full-body correlation measurements.