  Local unitary equivalence is an important ingredient for quantifying and
classifying entanglement. Verifying whether or not two quantum states are local
unitary equivalent is a crucial problem, where only the case of multipartite
pure states is solved. For mixed states, however, the verification of local
unitary equivalence is still a challenging problem. In this paper, based on the
coefficient matrices of generalized Bloch representations of quantum states, we
find a variety of local unitary invariants for arbitrary-dimensional bipartite
quantum states. These invariants are operational and can be used as necessary
conditions for verifying the local unitary equivalence of two quantum states.
Furthermore, we extend the construction to the arbitrary-dimensional
multipartite case. We finally apply these invariants to estimate concurrence, a
vital entanglement measure, showing the practicability of local unitary
invariants in characterizing entanglement.
