AbstractRigorous study of optical communication processes has become interesting since around 1970. Especially, quantum information theory has been developed in both mathematics and engineering. In this paper, by controlling the energy associated with a quantum input state, we obtain an optimal solution (optimal state, optical modulation, etc.) giving the minimum value to the error probability in the attenuation optical communication process. For instance, we find some relations between the energy constraint of the input state and the optimal error probabilities in each modulation.