The structure of ultrafine entanglement witnesses

TL;DR

This paper investigates the structure of ultrafine entanglement witnesses, providing a mathematical framework for their characterization and addressing misconceptions in their original formulation.

Contribution

It introduces a Legendre transformation approach for entanglement quantification and clarifies conditions for product observables to detect entanglement.

Findings

Legendre transformation aids in entanglement quantification

Necessary and sufficient conditions for product observables to detect entanglement

Identification of fallacies in previous ultrafine entanglement witness constructions

Abstract

An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification of entanglement. Second, we present necessary and sufficient conditions that two product observables are able to detect entanglement. Finally, we explain some fallacies in the original construction of ultrafine entanglement witnesses [F. Shahandeh et al., Phys. Rev. Lett. 118, 110502 (2017)].