Local invariants for multi-partite entangled states allowing for a simple entanglement criterion

We present local invariants of multi-partite pure or mixed states, which can be easily calculated and have a straight-forward physical meaning. As an application, we derive a new entanglement criterion for arbitrary mixed states of n parties. The new criterion is weaker than the partial transposition criterion but offers advantages for the study of multipartite systems. A straightforward generalization of these invariants allows for the construction of a complete set of observable polynomial invariants.