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arXiv:2609.37442

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

Jiacheng Sun, Mengyao Hu, Paweł Cieśliński, Jinfu Chen, Fei Shi, Bowen Wang, Hongyu Yang, Jizhou Wu, Jordi Tura, Valerio Scarani, Dian Wu, Jian Wu

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

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.

Comments
36 pages, 5 figures, comments are welcome

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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.
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.
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Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number mmm provides an additional certification dimension complementary to the system size nnn. We show that, for the powers-of-two setting choices considered here, increasing mmm leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing mmm strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.
Quantum nonlocality describes a stronger form of quantum correlation than that of entanglement. It refutes Einstein’s belief of local realism and is among the most distinctive and enigmatic features of quantum mechanics. It is a crucial resource for achieving quantum advantages in a variety of practical applications, ranging from cryptography and certified random number generation via self-testing to machine learning. Nevertheless, the detection of nonlocality, especially in quantum many-body systems, is notoriously challenging. Here, we report an experimental certification of genuine multipartite Bell-operator correlations, which signal nonlocality in quantum many-body systems, up to 24 qubits with a fully programmable superconducting quantum processor. In particular, we employ energy as a Bell-operator correlation witness and variationally decrease the energy of a many-body system across a hierarchy of thresholds, below which an increasing Bell-operator correlation depth can be certified from experimental data. We variationally prepare the low-energy state of a two-dimensional honeycomb model with 73 qubits and certify its Bell-operator correlations by measuring an energy that surpasses the corresponding classical bound with up to 48 standard deviations. In addition, we variationally prepare a sequence of low-energy states and certify their genuine multipartite Bell-operator correlations up to 24 qubits via energies measured efficiently by parity oscillation and multiple quantum coherence techniques. Our results establish a viable approach for preparing and certifying multipartite Bell-operator correlations, which provide not only a finer benchmark beyond entanglement for quantum devices, but also a valuable guide toward exploiting multipartite Bell correlations in a wide spectrum of practical applications.
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.
As Hermitian operators, many-body Bell operators can naturally be identified as many-body Hamiltonians. An important subclass of such Hamiltonians is the stoquastic class, characterized by having nonpositive off-diagonal matrix elements in a given basis. Interestingly, this property is shared by the permutationally invariant (PI) Bell operators underlying the largest Bell-correlation experiments to date. In this work, we explore the connection between many-body PI Bell operators and stoquasticity. We introduce the stoquasticity cone, which allows for a full characterization of the stoquastic parameter regimes for any PI Bell operator. We use this to show that PI Bell operators of the binary-input binary-output scenario consisting of at most three-body correlators can always be rendered stoquastic for any set of measurement parameters. Additionally, we also provide examples that use the stoquasticity cone to optimize for the quantum-classical gap. Numerical evidence suggests that the Bell operator used in the largest experiments to date is optimal with respect to stoquasticity. To the best of our knowledge, this work establishes the first connection between PI Bell operators and stoquasticity.