Almost all modern communication systems rely on electromagnetic fields. The additive white Gaussian noise (AWGN) channel is often a good approximate description of such a system, and its information-carrying capacity is given by a simple formula. The quantum analogue of AWGN channels, the bosonic Gaussian noise channel, accurately describes many quantum optical communication systems of interest. Estimating its capacity is significantly more difficult; although some simple coding strategies are known, whether or not more sophisticated techniques could dramatically improve communication rates has been unknown. Here, we present strong new upper bounds for the classical capacity of bosonic Gaussian noise channels. These results imply that known coding techniques are typically close to optimal. Our main technical tool is an entropy power inequality bounding the entropy produced as two quantum signals combine at a beamsplitter. Its proof relies on a quantum diffusion process which smooths arbitrary states towards Gaussians. © 2013 Macmillan Publishers Limited. All rights reserved.