We study the invariants of arbitrary dimensional multipartite quantum states under local unitary transformations. For multipartite pure states, we give a set of invariants in terms of singular values of coefficient matrices. For multipartite mixed states, we propose a set of invariants in terms of the trace of coefficient matrices. For full ranked mixed states with nondegenerate eigenvalues, this set of invariants is also the necessary and sufficient conditions for the local unitary equivalence of such two states.
@article{arxiv.1307.6659,
title = {Local unitary invariants for multipartite states},
author = {Ting-Gui Zhang and Ming-Jing Zhao and Xianqing Li-Jost and Shao-Ming Fei},
journal= {arXiv preprint arXiv:1307.6659},
year = {2015}
}