We study the local unitary equivalence of arbitrary dimensional multipartite quantum mixed states. We present a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment. The criterion is shown to be operational, even for particularly degenerated states, by detailed examples. In addition, explicit expressions of the local unitary operators are constructed for locally equivalent states. Complementary to the criterion, an alternative approach based on the partial transposition of matrices is also given, which makes the criterion more effective in dealing with generally degenerated mixed states.Quantum entanglement is one of the most extraordinary features of quantum physics. Multipartite entanglement plays a vital role in quantum information processing [1,2] and interferometry [3]. One fact is that the degree of entanglement of a quantum state remains invariant under local unitary transformations, while two quantum states with the same degree of entanglement, e.g., entanglement of formation [4,5] or concurrence [6,7], may not be equivalent under local unitary transformations. Another fact is that two entangled states are said to be equivalent in implementing quantum information tasks if they can be mutually exchanged under local operations and classical communication (LOCC). LOCC equivalent states are interconvertible also by local unitary transformations [8]. Therefore, it is important to classify and characterize quantum states in terms of local unitary transformations.To deal with this problem, one approach is to construct invariants of local unitary transformations. The method developed in [9,10], in principle, allows one to compute all the invariants of local unitary transformations for bipartite states, though it is not easy to do this operationally. In [11], a complete set of 18 polynomial invariants is presented for the local unitary equivalence of two-qubit mixed states. Partial results have been obtained for three-qubit states [12,13], some generic mixed states [14][15][16], and tripartite pure and mixed states [17]. Reference [18] gives explicit index-free formulas for all of the degree 6 algebraically independent local unitary invariant polynomials for finite-dimensional k-partite pure and mixed quantum states. The local unitary equivalence problem for multipartite pure qubit states has been solved in [19]. By exploiting the high-order singular-value decomposition technique and local symmetries of the states, Ref.[20] presents a practical scheme of classification under local unitary transformations for general multipartite pure states with arbitrary dimensions, which extends the results of n-qubit pure states [19] to those of n-qudit pure states. For mixed states, Ref.[21] solves the local unitary equivalence problem of arbitrary dimensional bipartite nondegenerated quantum systems by presenting a complete set of invariants, such that two density matrices are local unitary equivalent if and only if all of these invariants have equal ...