When different entanglement witnesses detect entangled states simultaneously

TL;DR

This paper investigates conditions under which multiple entanglement witnesses can simultaneously detect entangled states, providing criteria based on convex combinations and linear dependence, with applications to optimal witnesses.

Contribution

It establishes necessary and sufficient conditions for simultaneous detection by multiple witnesses and characterizes optimal and decomposable witnesses.

Findings

Finite witnesses can detect the same states if their convex combinations are still witnesses.

Two witnesses detect the same states iff they are linearly dependent.

Conditions for one witness to detect more states than another.

Abstract

The question under what conditions different witnesses may detect some entangled states simultaneously is answered for both finite- and infinite-dimensional bipartite systems. Finite many different witnesses can detect some entangled states simultaneously if and only if their convex combinations are still witnesses; they can not detect any entangled state simultaneously if and only if the set of their convex combinations contains a positive operator. For two witnesses $W_1$ and $W_2$, some more can be said: (1) $W_1$ and $W_2$ can detect the same set of entangled states if and only if they are linearly dependent; (2) $W_2$ can detect more entangled states than that $W_1$ can if and only if $W_1$ is a linear combination of $W_2$ and a positive operator. As an application, some characterizations of the optimal witnesses are given and some structure properties of the decomposable optimal witnesses are presented.