Necessary conditions for optimality of decomposable entanglement witnesses

TL;DR

This paper establishes additional necessary conditions for the optimality of decomposable entanglement witnesses, providing insights into their structure and identifying examples of non-optimal witnesses in higher-dimensional systems.

Contribution

It introduces two new necessary conditions for optimality of decomposable entanglement witnesses and demonstrates their implications with specific examples.

Findings

Support of optimal witnesses is completely entangled

Orthogonal complement must have a nonzero product vector

Existence of non-optimal witnesses as partial transposes of positive matrices in higher dimensions

Abstract

It is well known that the support of an optimal decomposable entanglement witness is completely entangled. We add two more necessary conditions for the optimality: The orthogonal complement of the support must have a nonzero product vector; another one will be given in terms of related faces of a convex cone. With these necessary conditions, we show that there exist examples of non-optimal decomposable entanglement witnesses which are the partial transposes of positive semi-definite matrices supported on completely entangled spaces, whenever both of the local dimensions are greater than or equal to three.