The time evolution of relative density operator in phase diffusion channels

By virtue of the properties of thermal entanglement, we succeed to obtain the solution to the representative phase diffusion master equation and give the operator sum representation of density operator. After analyzing the time evolution process of density operator at different initial state, we find out density operators keep invariable with the initial state being Fock state or thermal state. But the system always maintains coherence with phase diffusion when the initial state is coherent state.