Higher-order EPR correlations and inseparability conditions for continuous variables

TL;DR

This paper develops higher-order inseparability criteria for continuous-variable bipartite entanglement, extending known inequalities and identifying conditions that detect both Gaussian and non-Gaussian entangled states.

Contribution

It introduces hierarchical higher-order EPR correlation conditions for continuous variables, including tight inequalities that detect non-Gaussian entanglement.

Findings

Derived higher-order EPR inequalities extending Duan's criteria

Identified a fourth-order condition specific to non-Gaussian states

Provided examples of states violating these conditions for entanglement detection

Abstract

We derive two types of sets of higher-order conditions for bipartite entanglement in terms of continuous variables. One corresponds to an extension of the well-known Duan inequalities from second to higher moments describing a kind of higher-order Einstein-Podolsky-Rosen (EPR) correlations. Only the second type, however, expressed by powers of the mode operators leads to tight conditions with a hierarchical structure. We start with a minimization problem for the single-partite case and, using the results obtained, establish relevant inequalities for higher-order moments satisfied by all bipartite separable states. A certain fourth-order condition cannot be violated by any Gaussian state and we present non-Gaussian states whose entanglement is detected by that condition. Violations of all our conditions are provided, so they can all be used as entanglement tests.