  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.

Quantum Physics