Structural physical approximations and entanglement witnesses

TL;DR

This paper investigates the properties of structural physical approximations (SPAs) to positive maps and entanglement witnesses, providing conditions under which SPAs do not produce entanglement-breaking channels and addressing the SPA conjecture.

Contribution

It establishes new conditions for SPAs of entanglement witnesses and their partial transposes, and clarifies the independence of the SPA conjecture from map optimality.

Findings

SPA of an EW or its partial transposition can be entanglement-breaking in low dimensions

Sufficient conditions for violating the SPA conjecture are provided

SPA conjecture's validity is independent of positive map optimality

Abstract

The structural physical approximation (SPA) to a positive map is considered to be one of the most important method to detect entanglement in the real physical world. We first show that an arbitrary entanglement witness (EW) $W$ can be constructed from a separable density matrix $\sigma$ in the form of $W=\sigma-c_{\sigma} I$, where $c_{\sigma}$ is a non-negative number and $I$ is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [\emph{Phys. Rev. A}{\bf 78,} 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.