Classical capacity of the noisy bosonic channel and the bosonic minimum output entropy conjecture

We consider a line with noise in the simplest case. Loss does not add noise. Amplification via phase insensitive amplifiers do add noise. A lower bound of this capacity is the quantum analog to the Shannon capacity of a linear channel with additive white Gaussian noise, namely the difference of the Von Neumann entropy of the signal plus noise at the output of the line and the entropy of the noise alone. We show that this expression is indeed the capacity for the case of an amplifier with infinitesimal gain $G = 1+\epsilon$, and for a cascade of an amplifier with arbitrary gain and a large loss, such that the overall gain of the cascade is infinitesimal.