# Violation of Bell inequalities for multipartite systems

## TL;DR

This paper extends the application of finite group representation theory to analyze Bell inequality violations in complex multipartite quantum systems, deriving new inequalities and exploring their nonlocal game implications.

## Contribution

It introduces new Bell inequalities for multipartite systems based on group orbits, expanding the analysis to more complicated measurement scenarios.

## Key findings

- Quantum bounds depend on the number of outcomes and measurements.
- Derived Bell inequalities for four-party systems with orbit-based probability subsets.
- Analyzed nonlocal games associated with the new Bell inequalities.

## Abstract

In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two partite system, Alice and Bob each make one of $d$ possible measurements, each measurement has $n$ outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements $d$, but the quantum bound is dependent both on $n$ and $d$. Even so, when $d$ is large enough, the quantum bound is only dependent on $d$. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. Moreover, all the corresponding nonlocal games are analyzed.