A critical evaluation of bipartite entanglement and steering measures in a genuinely tripartite entangled system

TL;DR

This paper critically evaluates various bipartite entanglement and steering inequalities in a tripartite continuous-variable system, revealing limitations in their detection capabilities and emphasizing the need for more reliable measures.

Contribution

It systematically assesses the performance of existing inequalities in a cascaded nonlinear optical system, highlighting their sensitivity and the gap in reliable entanglement quantification.

Findings

Detection sensitivity varies with the chosen inequality.

Some criteria fail to detect existing entanglement.

Violation of inequalities is sufficient but not necessary for non-classical correlations.

Abstract

A large number of inequalities have been proposed for the detection of bipartite continuous variable entanglement and Einstein-Podolsky-Rosen steering. Many of these are based on either measured or inferred variances and are relatively easily measured via homodyne detection of optical outputs. In this work we examine a number of bipartite inequalities in the process of cascaded downconversion and sum-frequency generation inside an optical cavity. This has previously been predicted to be a potential source of continuous-variable tripartite entanglement, therefore we know that there are no separable bipartitions. In this work we investigate the performance of the chosen inequalities in the stable, below threshold regime. We show that detection of the existing entanglement is sensitive to the actual inequalities chosen, with some criteria missing it completely. Our results are explained by the fact that violation of the chosen inequalities is sufficient but not necessary to demonstrate the non-classical relationships between the modes, and highlights the outstanding problem of developing a reliable continuous-variable entanglement measure.