Multiparticle covariance matrices and the impossibility of detecting graph state entanglement with two-particle correlations

TL;DR

This paper introduces a covariance matrix-based criterion for multiparticle entanglement detection, capable of identifying certain bound entangled states but unable to detect entanglement in GHZ and graph states due to fundamental limitations.

Contribution

It proposes a new multiparticle entanglement criterion and proves the impossibility of detecting graph state entanglement with only two-particle correlations.

Findings

Detects bound entangled states not identified by other criteria

Fails to detect entanglement in GHZ and graph states

Shows fundamental limitations of two-particle correlation-based criteria

Abstract

We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows to detect bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the GHZ states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states, to which the GHZ states belong.