  We give a survey of the two remarkable analytical problems of quantum
information theory. The main part is a detailed report of the recent (partial)
solution of the quantum Gaussian optimizers problem which establishes an
optimal property of Glauber's coherent states -- a particular instance of pure
quantum Gaussian states. We elaborate on the notion of quantum Gaussian channel
as a noncommutative generalization of Gaussian kernel to show that the coherent
states, and under certain conditions only they, minimize a broad class of the
concave functionals of the output of a Gaussian channel. Thus, the output
states corresponding to the Gaussian input are "the least chaotic", majorizing
all the other outputs. The solution, however, is essentially restricted to the
gauge-invariant case where a distinguished complex structure plays a special
role. We also comment on the related famous additivity conjecture, which was
solved in principle in the negative some five years ago. This refers to the
additivity or multiplicativity (with respect to tensor products of channels) of
information quantities related to the classical capacity of quantum channel,
such as $(1\rightarrow p)$-norms or the minimal von Neumann or R\'enyi output
entropies. A remarkable corollary of the present solution of the quantum
Gaussian optimizers problem is that these additivity properties, while not
valid in general, do hold in the important and interesting class of the
gauge-covariant Gaussian channels.
