# Bell inequalities tailored to the Greenberger-Horne-Zeilinger states of arbitrary local dimension

## TL;DR

This paper develops a new class of Bell inequalities tailored to GHZ states of any number of parties and local dimensions, providing analytical bounds and exploring their application to partially entangled states.

## Contribution

It generalizes Bell inequalities to multipartite GHZ states of arbitrary local dimension, including analytical bounds and numerical analysis for partially entangled states.

## Key findings

- Analytical maximal quantum and nonsignaling violations for GHZ states.
- Bounds for detecting genuine nonlocality.
- Numerical classes of inequalities for partially entangled states.

## Abstract

In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in [A. Salavrakos et al., Phys. Rev. Lett. 119, 040402 (2017)] a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger-Horne-Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled GHZ-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.