Calculation of the channel capacity of optical phase damping channel with iteration

The optical signal transmitted on a fiber with phase damping is studied. At the limitation of complete damping, we prove that the channel is in fact a classical channel. The complete phase damping channel maps the coherent state to a Poisson distributed state. Classical information is encoded in the parameter of the coherent state, the general input is a probability distribution of coherent states. The transfer matrix is given after the discretion of the parameter of the coherent state. The quantum Holevo quantity (whose maximum gives the channel capacity) is simplified to Shannon mutual information. We calculate the capacity with iteration. The optimal distribution of the input achieve the channel capacity is a probabilistic mixture of vacuum state (zero input) and a wave packet when the total energy is small, other wave packets appear when the total energy increases. The wave packet phenomenon is apparent