Normal forms and entanglement measures for multipartite quantum states

A general mathematical framework is presented to describe local equivalence classes of multipartite quantum states under the action of local unitary and local filtering operations. This yields multipartite generalizations of the singular value decomposition. The analysis naturally leads to the introduction of entanglement measures quantifying the multipartite entanglement (as generalizations of the concurrence for two qubits and the 3-tangle for three qubits), and the optimal local filtering operations maximizing these entanglement monotones are obtained. Moreover, a natural extension of the definition of Greenberger-Horne-Zeilinger states to, e.g., 2×2×N systems is obtained.