Structural physical approximations and entanglement witnesses

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.