Purification and correlated measurements of bipartite mixed states

We prove that all purifications of a nonfactorable state (i.e., the state that cannot be expressed in a form $\rho_{AB}=\rho_A\otimes\rho_B$) are entangled. We also show that for any bipartite state there exists a pair of measurements that are correlated on this state if and only if the state is nonfactorable.