The structural physical approximations and optimal entanglement witnesses

TL;DR

This paper explores the properties of entanglement witnesses, introducing new classifications based on their proximity to positive and copositive parts, and examines the validity of the SPA conjecture within these classifications.

Contribution

It introduces positive and copositive types for entanglement witnesses and analyzes the conditions under which the SPA conjecture holds or fails for these types.

Findings

SPA of $W$ is separable only if $W$ is of copositive type

SPA conjecture fails even for copositive types

Partial transpose relates positive and copositive types

Abstract

We introduce the notions of positive and copositive types for entanglement witnesses, depending on the distance to the positive part and copositive part. An entanglement witness $W$ is of positive type if and only if its partial transpose $W^\Gamma$ is of copositive type. We show that if the structural physical approximation of $W$ is separable then $W$ should be of copositive type, and the SPA of $W^\Gamma$ is never separable unless $W$ is of both positive and copositive type. This shows that the SPA conjecture is meaningful only for those of copositive type. We provide examples to show that the SPA conjecture fails even for the case of copositive types.