Quantum nonlocality, the ability to create correlations stronger than classically possible, has significant implications for both the foundations and applications of quantum theory. Bell's theorem initiated this research, focusing on scenarios where multiple parties share a single source. However, recent interest has shifted towards more decentralized causal structures involving multiple independent sources shared among parties in a network. Analyzing these networks is significantly more complex due to non-linear boundaries between local and nonlocal correlations and non-convex local sets. While some progress has been made, robust tools for investigating generic networks analytically and numerically are lacking. This paper explores the use of machine learning, specifically neural networks, to address this challenge. The central problem is the membership problem: given a network and a distribution, determine if it can be produced using only local resources. The approach encodes the causal structure into a neural network and trains it to reproduce the target distribution. If a local causal model exists, a sufficiently expressive network should learn it; otherwise, it should fail to approximate the distribution. This leverages the similarity between causal structures and feedforward neural networks, both of which have information flow determined by a directed acyclic graph. The triangle network, a relatively simple but theoretically and numerically challenging tripartite network, serves as a testbed for this neural network oracle.

The existing literature on characterizing quantum nonlocality in networks highlights the difficulties in analytically and numerically determining whether a given probability distribution is local or nonlocal. Analytical methods like the inflation technique, while converging toward a perfect oracle, have limitations in witnessing nonlocality for certain distributions and extracting noise robustness. Other approaches, such as using entropy, are relatively weak in detecting nonlocality. Numerical methods, based on nonlinear optimization, become computationally intractable even for simple networks with multiple outputs per party. The complexity stems from the non-convex optimization space involved in searching for local models. Existing causal modeling and Bayesian network methods are either inapplicable or make assumptions that limit their usefulness in this specific quantum context.

The core methodology involves encoding the causal structure of the network into a neural network. For the triangle network with quaternary outputs, three multilayer perceptrons represent the response functions of Alice, Bob, and Charlie. The inputs to these perceptrons are uniformly distributed random numbers representing the hidden variables from the sources. The network architecture reflects the network's causal constraints, ensuring that only local models can be generated. The network is trained to reproduce a target probability distribution using a differentiable loss function, such as the Kullback-Leibler divergence. The training process involves generating random inputs and adjusting the network's weights to minimize the discrepancy between the target and the generated distributions. The output of the neural network is a Monte Carlo approximation of the joint probability distribution. To evaluate the network's performance, the authors introduce a noise parameter into the target distribution, creating a family of distributions. The distance between the target and learned distribution is used as a metric, and a qualitative change in this distance—a 'phase transition'—indicates the boundary between the local and nonlocal sets. The authors also analyze the learned response functions to gain further insight into the local strategies learned by the neural network. The analytic distance from the local set is approximated using geometric considerations, which helps to refine the analysis and provide more confidence in the results. Visualization of response functions is also used to understand the learned strategies.

The authors tested their method on three distributions: the Fritz distribution, the Elegant distribution, and a distribution proposed by Renou et al. For the Fritz distribution, the neural network accurately reproduced the known nonlocality threshold and the behavior of response functions, validating the method. For the Elegant distribution, the neural network provided strong numerical evidence supporting the conjecture that it is nonlocal, also providing estimates for its noise robustness. For the Renou et al. distribution, the neural network reproduced the already known nonlocal regime and surprisingly suggested nonlocality in a new range of parameters. Estimates for the noise robustness of this distribution were also obtained, although with slightly less precision due to the proximity of the target distributions to the local set. The authors note that while the local set is close to the target distributions in this case, the learned and target distributions are nearly indistinguishable to the human eye, emphasizing the value of their method over threshold-based approaches.

The proposed neural network method offers several advantages over existing analytic and numerical techniques. It relaxes the discrete hidden variable and deterministic response function assumptions made by traditional numerical optimization methods, making it applicable to more complex scenarios. The method's robustness comes from observing qualitative changes in the neural network's performance when transitioning between local and nonlocal sets, rather than relying on arbitrary confidence levels. The results obtained for the Elegant and Renou et al. distributions demonstrate the method's effectiveness in addressing open problems in quantum nonlocality, providing numerical evidence and estimations of noise robustness that were previously inaccessible.

This paper introduces a novel method for testing the classical reproducibility of probability distributions in networks using neural networks. The method, applicable to various causal structures, overcomes limitations of existing techniques. The authors successfully applied the method to several benchmark distributions, obtaining insightful results and suggesting new conjectures about nonlocality. Future research directions include exploring ways to obtain certificates of nonlocality from machine learning, investigating the boundary of the local set, and extending the method to networks with quantum sources.

The method is approximate, although its precision can be improved by increasing the neural network size and training time. The neural network might get stuck in local optima, particularly when the target distribution is very close to the local set, affecting the accuracy of noise robustness estimations. Furthermore, the results are numerical evidence, not rigorous proofs of nonlocality. The interpretation of the results relies on the assumption of a relatively flat local set near the boundary.