An Entanglement Measurer For Multipartite Pure States And A Complete Classification Of Tripartite Pure States

A simple entanglement measure for multipartite pure states is formulated based on the Von Neumann entropy of a series of reduced density matrices, which can not only be used to distinguish entangled, partially entangled, and disentangled multipartite pure states, but also be used to calculate the degree of entanglement and determine how many ways a multipartite pure state can be maximally entangled. Based on this new measure, all possible ways of maximal entanglement of the bipartite pure states with up to four terms and those of the tripartite pure states with up to eight terms are studied in detail. Firstly, a classification is carried out according to the structures and types of entangled states. Then, the constrained maximization for the defined entanglement measure is carried out for concerned state in finding the corresponding parameters and phase factors. Maximal entangled bipartite and tripartite pure states are found. It is verified that entangled states can be transformed into one another by local unitary (LU) transformations, if their entanglement measure are the same. Complete basis vectors for bipartite and tripartite systems are constructed by using the corresponding newly derived maximal entangled states. Their transformation relations under the local operations are also discussed. Thus, any pure state can also be expanded in terms of the corresponding entangled complete basis vectors. The most important conclusion of this thesis is that there are three inequivalent classes of entangled states for tripartite pure state system.