A neural network oracle for quantum nonlocality problems in networks

The study addresses whether observed correlations in quantum networks can be simulated with only classical resources (membership problem for causal structures). Unlike standard Bell scenarios, network scenarios (e.g., triangle network) have non-convex local sets, making analytical and numerical characterization hard. The authors encode a given network’s causal structure into a feedforward neural network that learns parties’ local response functions to hidden variables. If the target distribution is local, the network should learn it; otherwise, it will fail, revealing nonlocality. By scanning families of noisy target distributions and monitoring qualitative changes (phase transitions) in the distance between target and learned distributions, the method estimates where distributions exit the local set. The approach is applied to the triangle network with quaternary outputs, testing known and conjectured nonlocal distributions to provide evidence of nonlocality and noise robustness.

Foundational work on Bell nonlocality and extensions to networked causal structures highlight the complexity of characterizing local vs nonlocal correlations in networks. Prior analytical tools include generalized Bayesian networks, entropic methods, polynomial/nonlinear Bell inequalities, and especially the inflation technique, which in principle converges but has limited practical success for some distributions (e.g., Elegant distribution) and offers weak noise tolerance bounds. Prior machine learning applications to nonlocality mainly acted as classifiers suggesting locality/nonlocality rather than constructing explicit local models. This work differs by providing a generative approach that constructs local response functions consistent with the causal graph, thus certifying locality when learned.

- Causal-to-neural encoding: For a given network (triangle network with independent sources α, β, γ to parties Alice, Bob, Charlie; outputs a,b,c ∈ {0,1,2,3}), represent each party’s local response function PX(x|·) by a multilayer perceptron (MLP). Inputs to each party’s MLP are only the hidden variables permitted by the network DAG (e.g., PA takes β,γ; PB takes γ,α; PC takes α,β). Outputs are normalized 4-component vectors of conditional probabilities via a softmax layer. - Hidden variables: Assume sources emit i.i.d. uniform random variables on [0,1]; any other distributions can be absorbed into response functions (inverse transform sampling). - Monte Carlo approximation: Sample N_batch i.i.d. triples (α,β,γ) and compute PM(abc) ≈ (1/N_batch) Σi PA(a|βγ)i PB(b|γα)i PC(c|αβ)i (Cartesian product of local conditionals per sample). - Training objective: Minimize a differentiable discrepancy between target pT and model PM; they use KL divergence L = Σabc pT(abc) log[pT(abc)/PM(abc)]. Train with standard optimizers on minibatches; N_batch of several thousands (≈8000) suffices for quaternary outputs. - Phase-transition evaluation: Construct families of targets pT(v) by adding physically meaningful noise (e.g., source visibility for states; detector inefficiency modeled as random outputs). For each v, retrain and compute d(v) = Σabc [pT(abc) − PM(abc)]^2. Look for a qualitative liftoff in d(v) as evidence of leaving the local set. - Geometric fit: Approximate true distance from the local set as da(v) = 0 for v < v*, and da(v) = ||pT(v)−pT(v*)||^2 sin(θ) for v ≥ v*, assuming local boundary is locally flat and pT(v) is nearly straight. Fit da(v) to learned d(v) to estimate threshold v* and exit angle θ. - Response function inspection: Visualize learned local responses by sampling latent space and outputs to infer structure and consistency across v. If θ ≈ 90°, learned responses stabilize for v ≥ v*. - Architectures and resources: MLPs with ReLU/tanh activations; softmax output; example settings include depth 5, width 30 per party; training for minutes on a standard computer; code provided for triangle and Bell scenarios.

- Fritz distribution (triangle-wrapped Bell): Using a Werner-state visibility noise model, the learned distance d(ν) matches the analytic model with an exit angle θ ≈ 90°, consistent with a perpendicular exit from the local set and with known Bell-type structure. Learned response functions remain unchanged for ν ≥ ν* (indicative of θ ≈ 90°). The benchmark validates the method. - Elegant distribution (entangled sources and measurements): For two noise models—(i) source visibility applied to singlets; (ii) detector inefficiency modeled as random outputs—the method shows clear transitions in d(v), providing strong numerical evidence of nonlocality. Estimated robustness: nonlocal for visibility v ≳ 0.80 and for detector efficiency v ≳ 0.86. Estimated exit angles: θ ≈ 50° (visibility) and θ ≈ 60° (detector efficiency). Learned distributions closely track targets below threshold; above threshold, visualized responses exhibit intuitive, often clean partitions of latent space. - Renou et al. distribution (genuine triangle nonlocality): Scanning u^2 without added noise reproduces nonlocality in the known regime u_max < u^2 < 1 (u_max ≈ 0.785), and recovers local behavior at u^2 = 0.5 and near the boundary u^2 ≈ u_max (with minor instability at the boundary). Surprisingly, the network indicates larger distances to the local set for some values in 0.5 < u^2 < u_max, suggesting a new parameter range with nonlocality beyond the proven regime. - Noise robustness for Renou family at u^2 ≈ 0.85: Estimated detector efficiency threshold ≈ 0.91 and visibility threshold ≈ 0.89; small exit angles (≈ 6°). Distances are numerically small and sensitive to local optima, so thresholds are approximate but indicative. - General: The learned d(v) aligns well with the simple geometric model da(v) using only two parameters (v*, θ), supporting the interpretation of phase transitions as crossings of the local boundary. Learned models provide explicit local strategies whenever the target is inside the local set.

The approach contrasts with analytical tools like inflation and entropic methods, which either have limited practical strength on the studied distributions or provide weak noise robustness. Standard nonlinear optimization over discrete hidden-variable cardinalities becomes intractable for quaternary outputs in the triangle (huge discrete configuration space and nonconvexity). The neural approach relaxes discrete-cardinality and determinism assumptions, shifting complexity to learning continuous response functions within the causal DAG. Universal approximation and increased sampling/network size can, in principle, improve precision. Empirically, moderate architectures suffice for the Fritz and Elegant cases; the Renou family remains more delicate due to tiny distances and local optima. The method generalizes to arbitrary networks and can provide practical oracles for classical reproducibility, offering explicit local models when successful and phase-transition-based evidence of nonlocality otherwise. Future directions include extracting certificates (e.g., nonlinear inequalities) from learned models, improving interpretability, and extending to learning quantum strategies.

A neural-network-based oracle that encodes causal structures provides a practical tool to test classical reproducibility of observed distributions in quantum networks. Applied to the triangle network with quaternary outputs, it (i) validates against the Fritz construction, (ii) gives firm numerical evidence that the Elegant distribution is nonlocal and estimates its noise tolerance (visibility ≳ 0.80, detector efficiency ≳ 0.86), and (iii) refines understanding of the Renou et al. family by estimating noise robustness (detector efficiency ≈ 0.91, visibility ≈ 0.89 at u^2 ≈ 0.85) and conjecturing additional nonlocal regions for 0.5 < u^2 < u_max. The method is broadly applicable to other causal networks and motivates derivation of analytic certificates informed by machine learning, as well as exploration of learned quantum strategies.

Findings are numerical and do not constitute formal certificates of nonlocality. The phase-transition fit assumes local flatness of the boundary and near-linearity of the target path, which may only hold approximately. Training can get stuck in local optima, especially when targets lie very close to the local set, leading to run-to-run variability and crude threshold estimates (notably for the Renou family). The Monte Carlo approximation introduces sampling noise; precision depends on network capacity, N_batch, and training time. Affiliations and detailed parameters for some authors/experiments are constrained by available text; code- and data-level specifics beyond what is reported may affect reproducibility.