Family of Bell-like inequalities as device-independent witnesses for entanglement depth

TL;DR

This paper introduces a family of Bell inequalities that serve as device-independent witnesses for entanglement depth, enabling certification of genuine k-partite entanglement in multi-party quantum systems without assumptions on measurements or system dimension.

Contribution

The authors present a new family of Bell inequalities that are complete, facet-defining, and naturally quantify entanglement depth in many-body quantum systems.

Findings

Bell inequalities are applicable to arbitrarily many parties with binary measurements.

Quantum violations of these inequalities indicate the extent of genuine many-body entanglement.

The inequalities serve as device-independent witnesses for entanglement depth.

Abstract

We present a simple family of Bell inequalities applicable to a scenario involving arbitrarily many parties, each of which performs two binary-outcome measurements. We show that these inequalities are members of the complete set of full-correlation Bell inequalities discovered by Werner-Wolf-Zukowski-Brukner. For scenarios involving a small number of parties, we further verify that these inequalities are facet-defining for the convex set of Bell-local correlations. Moreover, we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many-body entangled. In other words, our Bell inequalities, when supplemented with the appropriate quantum bounds, naturally serve as device-independent witnesses for entanglement depth, allowing one to certify genuine k-partite entanglement in an arbitrary $n\ge k$-partite scenario without relying on any assumption about the measurements being performed, nor the dimension of the underlying physical system. A brief comparison is made between our witnesses and those based on some other Bell inequalities, as well as the quantum Fisher information. A family of witnesses for genuine k-partite nonlocality applicable to an arbitrary $n\ge k$-partite scenario based on our Bell inequalities is also presented.