Calculation of mutual information for nonlinear communication channel at large SNR

Using the path-integral technique we examine the mutual information for the communication channel modelled by the nonlinear Schr\"odinger equation with additive Gaussian noise. The nonlinear Schr\"odinger equation is one of the fundamental models in nonlinear physics, and it has a broad range of applications, including fiber optical communications --- the backbone of the Internet. At large signal-to-noise ratio ($\mathrm{SNR}$) we present the mutual information through the path-integral which is convenient for the perturbative expansion in nonlinearity. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel.