Abstract: Quantum entanglement is the most important physical resource in quantum information and quantum computation. Based on the basic theory of two-body quantum entanglement measurement, the important properties of the entanglement measurement of multi-body quantum pure state is discussed by the topological analysis and control theory in this paper. Firstly, the concavity of entanglement measurement of two-body quantum pure states is obtained by Schmidt decomposition. Secondly, the concavity of the convex combination of the entanglement measurement of multi-body quantum pure states is considered by the topological analysis and inequality theory. Finally, the upper bound of two-body quantum pure states is precisely studied by the control theory and Schur convex function. The concavity is analytic and able to calculate, and the topological order of states is described more successfully in the Kitaev model. Therefore, this work is broadened to topological quantum computation and quantum precision measurement.

Key words: quantum state, pure state, mixed state, entanglement measurement