# Maximal violation of n-locality inequalities in a star-shaped quantum network

## TL;DR

This paper investigates the maximum quantum violations of n-locality inequalities in star-shaped quantum networks, providing new bounds and insights into quantum correlations in complex network structures.

## Contribution

It derives the maximal violation of bilocality inequalities for arbitrary two-qubit states and extends the analysis to n-locality in star-shaped networks, advancing understanding of quantum correlations.

## Key findings

- Maximal violation of bilocality inequality for two-qubit states
- Relation between bilocality violation and CHSH inequality
- Extension to n-locality in star-shaped networks

## Abstract

Bell's theorem was a cornerstone for our understanding of quantum theory, and the establishment of Bell non-locality played a crucial role in the development of quantum information. Recently, its extension to complex networks has been attracting a growing attention, but a deep characterization of quantum behaviour is still missing for this novel context. In this work we analyze quantum correlations arising in the bilocality scenario, that is a tripartite quantum network where the correlations between the parties are mediated by two independent sources of states. First, we prove that non-bilocal correlations witnessed through a Bell-state measurement in the central node of the network form a subset of those obtainable by means of a separable measurement. This leads us to derive the maximal violation of the bilocality inequality that can be achieved by arbitrary two-qubit quantum states and arbitrary projective separable measurements. We then analyze in details the relation between the violation of the bilocality inequality and the CHSH inequality. Finally, we show how our method can be extended to n-locality scenario consisting of n two-qubit quantum states distributed among n+1 nodes of a star-shaped network.