Classification Of High Dimensional Multipartite Entangled States

Quantum entanglement lies at the heart of quantum information theory (QIT), and it also plays a very important role in many kinds of interdisciplinary subjects of science. Today, more and more people do research on entanglement because of its importance. In this thesis, we focus on some subjects of classifica-tion of quantum entangled states, including that:(1) we study the symmetry and equivalency of high dimensional multipartite entangled states under the condition of local operation and classical communication (LOCC); (2)under stochastic lo-cal operation and classical communication (SLOCC), we study the classification and parametric symmetry of the 2 x M x N pure entangled system.First, based on local operation and classical communication (LOCC), we make use of high order singular value decomposition (HOSVD) to study the clas-sification of high dimensional multipartite entangled states. By introducing the LOCC invariant quantum states, we classify the multipartite entangled states. And we also discuss some relations between this approach and the former re-search, which could verify the efficiency of our work further.Second, we further develop the method in classifying the multipartite en-tanglement state of 2 x M x N under stochastic local operation and classical communication (SLOCC). Using this method, all inequivalent classes of true entangled state can be assorted directly without knowing the classification infor-mation of lower dimension ones. For any given state we can determine whether it is fully entangled or not. If it is, we can further determine which class it belongs to. We give the analytic expression for the total numbers of classes of the true 2 x M x N entangled states. And we study the internal symmetric properties of the nonlocal parameters in the continuous entangled class, which could help us to know more about the nature of the 2 x M x N entangled system.