Capacity of optical transmission with attenuation and amplification

The optical signal transmitted on a fiber with thermal noise, attenuation and amplification is studied. The channel is considered as a quantum channel while the signal to be transmitted is classical. The classical information is encoded in a quantum state then it is transmitted in optical fiber. At the receiving end, measurement is performed on the output quantum state to retrieve the classical information. The optimal distribution of the input quantum state achieving the channel capacity is Gaussian. The capacity is characterized by the attenuation/amplification coefficient and thermal noise in the transmission and at the measurement process together by the power of the input. The loss (or attenuation) of an optical fiber usually should be compensated by the amplification. We show that when they coexist, additional noise occurs