  We introduce a class of inequalities based on low order correlations of
operators to detect entanglement in bipartite systems. The operators may either
be Hermitian or non-Hermitian and are applicable to any physical system or
class of states. Testing the criteria on example systems reveals that they
outperform other common correlation based criteria, such as those by
Duan-Giedke-Cirac-Zoller and Hillery-Zubairy. One unusual feature of the
criteria is that it includes a term involving the average of the density matrix
squared, related to the purity of the system. We discuss how such a term could
be measured in relation to the criteria.
