  Optical channels, such as fibers or free-space links, are ubiquitous in
today's telecommunication networks. They rely on the electromagnetic field
associated with photons to carry information from one point to another in
space. As a result, a complete physical model of these channels must
necessarily take quantum effects into account in order to determine their
ultimate performances. Specifically, Gaussian photonic (or bosonic) quantum
channels have been extensively studied over the past decades given their
importance for practical purposes. In spite of this, a longstanding conjecture
on the optimality of Gaussian encodings has yet prevented finding their
communication capacity. Here, this conjecture is solved by proving that the
vacuum state achieves the minimum output entropy of a generic Gaussian bosonic
channel. This establishes the ultimate achievable bit rate under an energy
constraint, as well as the long awaited proof that the single-letter classical
capacity of these channels is additive. Beyond capacities, it also has broad
consequences in quantum information sciences.
