Is genuine nonlocality in the triangle network exclusive to pure states?

Abstract: Genuine network non-locality (GNN) refers to the existence of quantum correlations in a network with independent sources that cannot be explained by local hidden-variables (LHV) models. Even in the simplest scenario, determining whether these quantum correlations remain genuinely network non-local when derived from entangled states that deviate from their ideal forms is highly challenging due to the non-convex nature of local correlations. Understanding the boundary of these correlations thus becomes a hard problem, but one that raises academic interest specifically its robustness to noise. To address this problem, we introduce a causal domain-informed learning algorithm called the LHV k-rank neural network, which assesses the rank parameter of the non-ideal combined state produced by sources. Applied to the triangle network scenario with the three sources generating a class of quantum states known as X states, the neural network reveals that non-locality persists only if the states remain pure. Remarkably, we find that even slight deviations from ideal Bell states due to noise cause GNN to vanish, exhibiting a discrete behavior that hasn't been witnessed in the standard bell scenario. This finding thus raises a fundamental question as to whether GNN in the triangle network is exclusive to pure states or not. Additionally, we explore the case of the three sources producing dissimilar states, indicating that GNN requires all its sources to send pure entangled states with joint entangled measurements as resources. Apart from these results, this work succeeds in showing that machine learning approaches with domain-specific constraints can greatly benefit the field of quantum foundations.