1106.5233 (Bang-Hai Wang et al.)

All structural physical approximations to positive maps of low
dimensions defining entanglement-breaking channels    [PDF]

Bang-Hai Wang, Dong-Yang Long

The structural physical approximation to a positive map is considered to be one of the most important method to detect entanglement in the real physical world. In this paper, we first show that any entanglement witness $W$ can be constructed from a separable density matrix $\sigma$ in the form $W=\sigma-c_{\sigma} I$, where $\sigma$ is a separable density matrix, $c_{\sigma}$ is a non-negative number and $I$ is the identity matrix. Based on this result, we show that the Korbicz et al. conjecture [\emph{Phys. Rev. A}{\bf 78,} 062105 (2008)] does not need to be based on the optimality of positive maps. We show that all structural approximations to positive maps of low dimensions define entanglement-breaking channels. This generalizes the result in the Fiur\'{a}\u{s}ek [\emph{Phys. Rev. A}{\bf 66,} 052315 (2002)], which states that the SPA of the partial transposition map $I\otimes T$ in the two-qubit case is an entanglement breaking (EB) channel.

View original: http://arxiv.org/abs/1106.5233