Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems

TL;DR

This paper develops criteria based on inequalities to detect entanglement, EPR steering, and Bell-nonlocality in multipartite higher-spin systems, applicable to qudits, and demonstrates their effectiveness through violations with optimized states.

Contribution

It introduces new inequality-based criteria for nonlocality classes in higher-spin multipartite systems, extending previous continuous-variable approaches to discrete spins.

Findings

Bell inequalities can be violated for arbitrary d with enough sites and optimized states.

Entanglement can be detected for any d and N, with violations increasing with N.

The criteria can identify entanglement in generalized GHZ states.

Abstract

We develop criteria to detect three classes of nonlocality that have been shown by Wiseman et al. [Phys. Rev. Lett. 98, 140402 (2007)] to be nonequivalent: entanglement, EPR steering, and the failure of local hidden-variable theories. We use the approach of Cavalcanti et al. [Phys. Rev. Lett. 99, 210405 (2007)] for continuous variables to develop the nonlocality criteria for arbitrary spin observables defined on a discrete Hilbert space. The criteria thus apply to multisite qudits, i.e., systems of fixed dimension d, and take the form of inequalities. We find that the spin moment inequalities that test local hidden variables (Bell inequalities) can be violated for arbitrary d by optimized highly correlated nonmaximally entangled states provided the number of sites N is high enough. On the other hand, the spin inequalities for entanglement are violated and thus detect entanglement for such states, for arbitrary d and N, and with a violation that increases with N.We show that one of the moment entanglement inequalities can detect the entanglement of an arbitrary generalized multipartite Greenberger-Horne-Zeilinger state. Because they involve the natural observables for atomic systems, the relevant spin-operator correlations should be readily observable in trapped ultracold atomic gases and ion traps.