  In this paper, a class of indecomposable positive finite rank elementary
operators of order $(n,n)$ are constructed. This allows us to give a simple
necessary and sufficient criterion for separability of pure states in bipartite
systems of any dimension in terms of positive elementary operators of order
$(2,2)$ and get some new mixed entangled states that can not be detected by the
positive partial transpose (PPT) criterion and the realignment criterion.
