Exploring the Local Landscape in the Triangle Network

TL;DR

This paper investigates the set of classical distributions in the triangle network, using neural networks and analytical methods to approximate and analyze their properties, providing evidence that the Elegant Joint Measurement distribution is nonlocal.

Contribution

It introduces neural-network-based inner approximations for local distributions in the triangle network and conjectures new Bell inequalities related to symmetry and correlation levels.

Findings

Neural network and analytical methods agree on approximations.

Conjecture that the Elegant Joint Measurement distribution is nonlocal.

Proposed Bell inequalities relate correlation levels to symmetry.

Abstract

Characterizing the set of distributions that can be realized in the triangle network is a notoriously difficult problem. In this work, we investigate inner approximations of the set of local (classical) distributions of the triangle network. A quantum distribution that appears to be nonlocal is the Elegant Joint Measurement (EJM) [Entropy. 2019; 21(3):325], which motivates us to study distributions having the same symmetries as the EJM. We compare analytical and neural-network-based inner approximations and find a remarkable agreement between the two methods. Using neural network tools, we also conjecture network Bell inequalities that give a trade-off between the levels of correlation and symmetry that a local distribution may feature. Our results considerably strengthen the conjecture that the EJM is nonlocal.