An Entanglement Measure And Complete Classification Of Tristate Bipartite Pure State Systems

A simple entanglement measure for multipartite pure states is formulated based on the von Neumann entropy of a series of reduced density matrices. All possible extremal bipartite entangled pure state for tristate system are searched based on the new measure. Firstly, a classification is made according to the structures and types of pure states with number of terms involved. Then, the analysis is thus based on the new measure with the constrained maximization. According to entanglement measure and constrained condition, the corresponding parameters, phase factors of possible maximal entangled states and extremal value of the corresponding measure are found by means of Lagrange Multiplier maximization. Maximal entangled pure states are obtained. The relation of different entangled states is analyzed by the local unitary (LU) transformations. The inequivalent entangled states that belong to different types of entanglement are clarified. It is shown that the procedure can not only be used to distinguish from entangled, partially entangled, and disentangled multipartite pure states, but also be used to calculate the degree of the entanglement and determine how many ways a multipartite pure state can be maximally entangled. Our analysis shows that there are three inequivalent types of entanglement for tristate bipartite pure state systems.