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We study the equivalence of quantum states under local unitarytransformations by using the singular value decomposition. A complete set ofinvariants under local unitary transformations is presented for several classesof tripartite mixed states in KxMxN composite systems. Two density matrices inthe same class are equivalent under local unitary transformations if and onlyif all these invariants have equal values for these density matrices.

A Complete Set of Local Invariants for a Family of Multipartite Mixed States