LPP elementary operator criterion of full separability for states in multipartite quantum systems

Let H-1, H-2, ..., H-n, K (n >= 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state rho in H-1 circle times H-2 circle times ... circle times H-n circle times K is fully separable if and only if (Lambda circle times Id)rho >= 0 for all (finite-rank) LPP elementary operators Lambda from B(H-1, H-2, ..., H-n) into B(K). Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.