Multipartite states under local unitary transformations

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The equivalence problem under local unitary transformation for n--partite pure states is reduced to the one for (n-1)--partite mixed states. In particular, a tripartite system Hₐ, where Hⱼ is a finite dimensional complex Hilbert space for j=A,B,C, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to Hₐ belongs to the class of generic mixed states.