Quantifying fidelity and purity for some quantum states in dissipative channel by virtue of trace rule in the phase space

By virtue of trace rule in the phase space, we investigate the time evolution of fidelity and purity for some quantum states (including Fock state, coherent state, squeezed state, and thermal state) in dissipative channel. All the relations are calculated by directly applying the Wigner function evolution formula in channel. The method is more convenient and concise than other method such as density operators’ trace. It is found that the dissipation induces all the quantum state eventually to vacuum state (a pure Gaussian state).