Local Equivalence of Multipartite Entanglement

Let R be an invariant polynomial ring of a reductive group acting on a vector space, and let d be the minimum integer such that R is generated by those polynomials in R of degree no more than d. To upper bound such d is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups-the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups-the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if r-copies of them are local unitary equivalent for some r.