Gaussian optimizers and the additivity problem in quantum information theory

We survey two remarkable analytical problems of quantum information theory. We report on the recent solution of the quantum Gaussian optimizers problem which establishes an optimal property of Glauber's coherent states as a particular instance of pure quantum Gaussian states. Namely, it is shown that the coherent states, and under certain conditions only they, minimize a broad class of the concave functionals of the output of a gauge-covariant or contravariant Gaussian channel. A remarkable corollary of this solution in the multimode case is the additivity of the minimal output entropy and the classical capacity of Gaussian channels (which is not valid for general quantum channels). This in particular allows for explicit computation of the classical capacity for the mathematical models of phase-insensitive channels in quantum optics, such as attenuators, amplifiers and additive classical noise channels.