Joint separable numerical range and bipartite entanglement witness

TL;DR

This paper links joint separable numerical range with ultrafine entanglement witnessing, providing a new perspective, a sufficient condition for effective pairs, and proving its necessity for optimization.

Contribution

It establishes a direct derivation of ultrafine entanglement witnessing from joint separable numerical range and introduces a simple method for identifying effective pairs.

Findings

Derived ultrafine entanglement witnessing from joint separable numerical range.

Provided a simple sufficient condition for effective pairs.

Proved the condition is necessary for optimization.

Abstract

In 2017 an idea considering a pair of Hermitian operators of product form was published, which is called ultrafine entanglement witnessing. In 2018 some rigorous results were given. Here we improve their work. First we point this idea can be directly derived from an earlier concept named joint separable numerical range and explain how it works as a series of witnesses. Second by a simple method we present a sufficient condition for an effective pair. Finally we prove this condition is necessary for optimization.