  Quantum communication theory explores the implications of quantum mechanics
to the tasks of information transmission. Many physical channels can be
formally described as quantum Gaussian operations acting on bosonic quantum
states. Depending on the input state and on the quality of the channel, the
output suffers certain amount of noise. For a long time it has been
conjectured, but never proved, that output states of Gaussian channels
corresponding to coherent input signals are the less noisy ones (in the sense
of a majorization criterion). In this work we prove this conjecture.
Specifically we show that every output state of a phase insensitive Gaussian
channel is majorized by the output state corresponding to a coherent input. The
proof is based on the optimality of coherent states for the minimization of
strictly concave output functionals. Moreover we show that coherent states are
the unique optimizers.
