  We construct optimal PPTES witnesses to detect $3\otimes 3$ PPT entangled
edge states of type $(6,8)$ constructed recently \cite{kye_osaka}. To do this,
we consider positive linear maps which are variants of the Choi type map
involving complex numbers, and examine several notions related to optimality
for those entanglement witnesses. Through the discussion, we suggest a method
to check the optimality of entanglement witnesses without the spanning
property.
