  The question under what conditions different witnesses may detect some
entangled states simultaneously is answered for both finite- and
infinite-dimensional bipartite systems. Finite many different witnesses can
detect some entangled states simultaneously if and only if their convex
combinations are still witnesses; they can not detect any entangled state
simultaneously if and only if the set of their convex combinations contains a
positive operator. For two witnesses $W_1$ and $W_2$, some more can be said:
(1) $W_1$ and $W_2$ can detect the same set of entangled states if and only if
they are linearly dependent; (2) $W_2$ can detect more entangled states than
that $W_1$ can if and only if $W_1$ is a linear combination of $W_2$ and a
positive operator. As an application, some characterizations of the optimal
witnesses are given and some structure properties of the decomposable optimal
witnesses are presented.
