# Exploring Bell inequalities for the device-independent certification of multipartite entanglement depth

## TL;DR

This paper investigates the use of specific Bell inequalities for device-independent certification of multipartite entanglement depth in various quantum states, revealing new nonlocality demonstrations and methods for different graph states.

## Contribution

It introduces a family of Bell inequalities suited for certifying entanglement depth across diverse multipartite states, including unbalanced GHZ, W, and graph states, with novel nonlocality insights.

## Key findings

- Bell inequalities can certify entanglement depth in well-known states.
- Full-correlation Bell inequalities can demonstrate nonlocality of weakly entangled GHZ states.
- Constructed Bell inequalities based on GHZ paradox can witness entanglement in graph states.

## Abstract

Techniques developed for device-independent characterizations allow one to certify certain physical properties of quantum systems without assuming any knowledge of their internal workings. Such a certification, however, often relies on the employment of device-independent witnesses catered for the particular property of interest. In this work, we consider a one-parameter family of multipartite, two-setting, two-outcome Bell inequalities and demonstrate the extent to which they are suited for the device-independent certification of genuine many-body entanglement (and hence the entanglement depth) present in certain well-known multipartite quantum states, including the generalized Greenberger-Horne-Zeilinger (GHZ) states with unbalanced weights, the higher-dimensional generalizations of balanced GHZ states, and the $W$ states. As a by-product of our investigations, we have found that, in contrast with well-established results, provided trivial qubit measurements are allowed, full-correlation Bell inequalities can also be used to demonstrate the nonlocality of weakly entangled unbalanced-weight GHZ states. Besides, we also demonstrate how two-setting, two-outcome Bell inequalities can be constructed, based on the so-called GHZ paradox, to witness the entanglement depth of various graph states, including the ring graph states, the fully connected graph states, and some linear graph states, etc.