Local Unitary Equivalence of Arbitrary Dimensional Bipartite Mixed   Quantum States
  The nonlocal properties of arbitrary dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. These invariants give rise to both sufficient and necessary conditions for the equivalence of quantum states under local unitary transformations: two density matrices are locally equivalent if and only if all these invariants have equal values. 
Chunqin Zhou, Tinggui Zhang, Shao-Ming Fei, Naihuan Jing and Xianqing   Li-Jost
quant-ph