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arXiv:2603.24665 · Quantum Science and Technology

A versatile neural-network toolbox for testing Bell locality in networks

Antoine Girardin, Mohammad Massi Rashidi, Géraldine Haack, Nicolas Brunner, Alejandro Pozas-Kerstjens

quant-phSubmitted Mar 25, 2026• Announced Mar 27, 2026• 1 citation• Revised Aug 5, 2026 (v2)
v1 · Mar 25, 2026v2 · Aug 5, 2026
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Abstract

Determining whether an observed distribution of events generated in a quantum network is Bell local, i.e., if it admits an alternative realization in terms of independent local variables, is extremely challenging. Building upon arXiv:1907.10552, we develop a software solution that parameterizes local models in networks via neural networks. This allows one to leverage optimization tools available from the machine learning community in the search of network Bell nonlocality. Our solution applies to arbitrary networks, is easy to use, and includes technical improvements that significantly increase performance compared to previous implementations. We apply it to investigate nonlocality in several networks hitherto unexplored, providing insights on the corresponding quantum nonlocal sets and suggesting concrete, promising realizations of quantum nonlocal correlations.

AI Summary

The paper addresses whether correlations generated by independent quantum sources admit network-local hidden-variable models. It extends an existing neural-network approach from the triangle to arbitrary network topologies, providing a PyTorch toolbox with utilities for computing quantum distributions and inspecting learned response functions. Each party’s response function is represented by a feed-forward network receiving only its connected sources’ independent random variables. Monte Carlo estimates of the resulting distribution are fitted to a target using KL divergence and Euclidean distance. An adaptive sampling rule increases sample counts as the loss decreases, aiming to keep sampling error below the optimization loss. Benchmarking on the triangle-network RGB4 distribution at measurement parameter $u^2=0.85$ revises an earlier numerical estimate of the critical Werner-state visibility from approximately $0.89$ to $0.99$, indicating substantially weaker noise robustness. This remains a numerical estimate; the cited rigorous bound is $V^*<0.9945$. Parameter scans then identify candidate nonlocal correlations in triangle, square and pentagon networks, including coarse-grained three-output pentagon distributions, and in a square with an additional diagonal source and tripartite measurements. Noise scans suggest candidate robustness to roughly $10\%$ source white noise in the triangle and $20\%$ in the four-output square and pentagon. These conclusions rely on optimization residuals and comparisons with known-local baselines. Small residuals establish approximate fits, not exact locality; large residuals can reflect nonlocality, insufficient expressivity or optimization failure. Hyperparameter tuning is incomplete, and attempted inflation tests did not certify the new candidates.

No sign of AI-generated textIncrementalMissing comparison

Error found: The claimed runtime improvement is not supported by reported timing measurements or a controlled speed comparison. The RGB4 benchmark demonstrates improved fitting but does not isolate adaptive sampling as its cause. Appendix A prints N_s = L^2 where its stated Euclidean-error scaling requires N_s proportional to L^(-2), although the main text uses the correct dependence. The sampling-error derivation omits an expectation and presents an RMS bound as though it bounded every empirical realization. The binomial-variance equality does not generally describe the conditional-probability estimator in Eq. (5), although the same RMS upper bound remains valid.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Journal reference
Quantum Sci. Technol. 11 035056 (2026)
DOI
10.1088/2058-9565/ae8df7
Comments
11+4 pages, 4+3 figures, 1 table, RevTeX 4.2. The computational appendix is available at https://www.github.com/Antoine0Girardin/local_model_in_networks V2: Added section V.B with new results, close to published version

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Determining whether an observed distribution of events generated in a quantum network is Bell local, i.e., if it admits an alternative realization in terms of independent local variables, is extremely challenging. Building upon arXiv:1907.10552, we develop a software solution that parameterizes local models in networks via neural networks. This allows one to leverage optimization tools available from the machine learning community in the search of network Bell nonlocality. Our solution applies to arbitrary networks, is easy to use, and includes technical improvements that significantly increase performance compared to previous implementations. We apply it to investigate nonlocality in several networks hitherto unexplored, providing insights on the corresponding quantum nonlocal sets and suggesting concrete, promising realizations of quantum nonlocal correlations.
The paper addresses whether correlations generated by independent quantum sources admit network-local hidden-variable models. It extends an existing neural-network approach from the triangle to arbitrary network topologies, providing a PyTorch toolbox with utilities for computing quantum distributions and inspecting learned response functions.
Each party’s response function is represented by a feed-forward network receiving only its connected sources’ independent random variables. Monte Carlo estimates of the resulting distribution are fitted to a target using KL divergence and Euclidean distance. An adaptive sampling rule increases sample counts as the loss decreases, aiming to keep sampling error below the optimization loss.
Benchmarking on the triangle-network RGB4 distribution at measurement parameter u2=0.85u^2=0.85u2=0.85 revises an earlier numerical estimate of the critical Werner-state visibility from approximately 0.890.890.89 to 0.990.990.99, indicating substantially weaker noise robustness. This remains a numerical estimate; the cited rigorous bound is V∗<0.9945V^*<0.9945V∗<0.9945. Parameter scans then identify candidate nonlocal correlations in triangle, square and pentagon networks, including coarse-grained three-output pentagon distributions, and in a square with an additional diagonal source and tripartite measurements. Noise scans suggest candidate robustness to roughly 10%10\%10% source white noise in the triangle and 20%20\%20% in the four-output square and pentagon.
These conclusions rely on optimization residuals and comparisons with known-local baselines. Small residuals establish approximate fits, not exact locality; large residuals can reflect nonlocality, insufficient expressivity or optimization failure. Hyperparameter tuning is incomplete, and attempted inflation tests did not certify the new candidates.
In recent years, the study of Bell nonlocality has been generalized to quantum networks, where multiple independent sources distribute physical systems to distant parties who perform local measurements. In this context, a central open question is to identify the minimal network configuration in which quantum resources produce Bell nonlocal correlations. Here we address this question and show that quantum nonlocality is possible in the triangle network where the parties have no input choices and produce only binary-valued outcomes. To do so, we start by identifying a family of target distributions and proving their nonlocality. Next, we construct an explicit quantum model that reproduces the target distributions to machine precision. For this, we develop an efficient method for parameterizing quantum distributions in networks, inspired by the formalism of higher-order quantum operations. When considering the number of observed variables and their cardinality, this constitutes the smallest scenario possible that supports quantum nonlocality in networks. Moreover, by analyzing the explicit quantum model, we obtain new insights into how nonlocal distributions can be generated in quantum networks.
Abstract Characterizing quantum nonlocality in networks is a challenging, but important problem. Using quantum sources one can achieve distributions which are unattainable classically. A key point in investigations is to decide whether an observed probability distribution can be reproduced using only classical resources. This causal inference task is challenging even for simple networks, both analytically and using standard numerical techniques. We propose to use neural networks as numerical tools to overcome these challenges, by learning the classical strategies required to reproduce a distribution. As such, a neural network acts as an oracle for an observed behavior, demonstrating that it is classical if it can be learned. We apply our method to several examples in the triangle configuration. After demonstrating that the method is consistent with previously known results, we give solid evidence that a quantum distribution recently proposed by Gisin is indeed nonlocal as conjectured. Finally we examine the genuinely nonlocal distribution recently presented by Renou et al., and, guided by the findings of the neural network, conjecture nonlocality in a new range of parameters in these distributions. The method allows us to get an estimate on the noise robustness of all examined distributions.
Network nonlocality extends Bell nonlocality to settings with multiple independent sources and parties. Certifying it in quantum information processing tasks requires suitable witnesses. However, in contrast to local correlations, the set of network-local correlations is non-convex. This non-convexity makes certifying network nonlocality a highly non-trivial task. Existing approaches involve leveraging network-specific properties, or inflation-based methods whose constraints grow combinatorially in the number of local variables. In this work, we introduce a linear programming witness for network nonlocality built from five classes of linear constraints. These classes are network-agnostic, although the explicit forms of the constraints must be tailored to a specific network's structure. We use the procedure to construct network nonlocality witnesses for a family of ring networks and certify network nonlocality for a concrete example, relying only on observed probabilities and a tunable experimental parameter. Our work advances the search for efficient witnesses to certify network nonlocality across diverse quantum network architectures.
The study of nonlocality in scenarios that depart from the bipartite Einstein-Podolsky-Rosen setup is allowing one to uncover many fundamental features of quantum mechanics. Recently, an approach to building network-local models based on machine learning led to the conjecture that the family of quantum triangle distributions of [Renou et al., Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] did not admit triangle-local models in a larger range than the original proof. We prove part of this conjecture in the affirmative. Our approach consists of reducing the family of original, four-outcome distributions to families of binary-outcome ones, and then using the inflation technique to prove that these families of binary-outcome distributions do not admit triangle-local models. This constitutes the first successful use of inflation in a proof of quantum nonlocality in networks for distributions whose nonlocality could not be proved with alternative methods. Moreover, we provide a method to extend proofs of network nonlocality in concrete distributions of a parametrized family to continuous ranges of the parameter. In the process, we produce a large collection of network Bell inequalities for the triangle scenario with binary outcomes, which are of independent interest.
In a quantum network, distant observers sharing physical resources emitted by independent sources can establish strong correlations, which defy any classical explanation in terms of local variables. We discuss the characterization of nonlocal correlations in such a situation, when compared to those that can be generated in networks distributing independent local variables. We present an iterative procedure for constructing Bell inequalities tailored for networks: starting from a given network, and a corresponding Bell inequality, our technique provides new Bell inequalities for a more complex network, involving one additional source and one additional observer. We illustrate the relevance of our method on a variety of networks, demonstrating significant quantum violations, which could not have been detected using standard Bell inequalities.
As the demand for quantum networks and communication grows, identifying quantum nonlocality in these networks has become a crucial and challenging task. The nonconvex nature of the network local set makes traditional Bell-type inequalities inadequate for detection. Machine learning algorithms have emerged as a promising approach to address this issue. In this study we introduce a quantifier for nonlocality in chain-shaped quantum networks and employ deep-learning algorithms for its detection. We validate this method using the trilocal network, adopting an ensemble learning strategy called the stacking method, which combines four distinct deep neural networks to train a linear regression model. Our results show that machine learning effectively quantifies nonlocality in chain networks and exhibits strong generalization capabilities for Werner states.