# Correlations in star networks: from Bell inequalities to network inequalities

## TL;DR

This paper develops a systematic method to derive network Bell inequalities from standard Bell inequalities, demonstrating how quantum violations extend from traditional scenarios to complex star-network configurations.

## Contribution

The authors introduce a technique to map Bell inequalities to network inequalities in star configurations, enabling analysis of quantum correlations in complex networks.

## Key findings

- Network Bell inequalities can be derived systematically from standard Bell inequalities.
- Quantum states violating Bell inequalities also violate the corresponding network inequalities.
- The method is illustrated with a multi-setting Bell inequality, showing quantum violations in star networks.

## Abstract

The problem of characterizing classical and quantum correlations in networks is considered. Contrary to the usual Bell scenario, where distant observers share a physical system emitted by one common source, a network features several independent sources, each distributing a physical system to a subset of observers. In the quantum setting, the observers can perform joint measurements on initially independent systems, which may lead to strong correlations across the whole network. In this work, we introduce a technique to systematically map a Bell inequality to a family of Bell-type inequalities bounding classical correlations on networks in a star-configuration. Also, we show that whenever a given Bell inequality can be violated by some entangled state $\rho$, then all the corresponding network inequalities can be violated by considering many copies of $\rho$ distributed in the star network. The relevance of these ideas is illustrated by applying our method to a specific multi-setting Bell inequality. We derive the corresponding network inequalities, and study their quantum violations.