Simple Sufficient Criteria for Optimality of Entanglement Witnesses

TL;DR

This paper introduces new, easier-to-verify criteria for the optimality of bipartite entanglement witnesses, simplifying the process of confirming their effectiveness in quantum information tasks.

Contribution

It presents a novel characterization of the spanning property via entanglement-breaking channels, leading to practical criteria for optimality that are easier to check than existing methods.

Findings

New sufficient criterion based on the kernel of bipartite states

A simple expectation value test on maximally entangled states

Spectral constraints on witnesses and positive maps

Abstract

If one wants to establish optimality of a given bipartite entanglement witness, the current standard approach is to check whether it has the spanning property. Although this is not necessary for optimality, it is most often satisfied in practice, and for small enough dimensions or sufficiently structured witnesses this criterion can be checked by hand. In this work we introduce a novel characterization of the spanning property via entanglement-breaking channels, which in turn leads to a new sufficient criterion for optimality. This criterion amounts to just checking the kernel of some bipartite state. It is slightly weaker than the spanning property, but it is a lot easier to test for -- by hand as well as numerically -- and it applies to almost all witnesses which are known to have the spanning property. A second criterion is derived from this, where one can simply compute the expectation value of the given witness on a maximally entangled state. Finally, this approach implies new spectral constraints on witnesses as well as on positive maps.