  In the present paper we give proof that the information-transmission capacity
of the approximate position measurement with the oscillator energy constraint,
which underlies noisy Gaussian homodyning in quantum optics, is attained on
Gaussian encoding. The proof is based on general principles of convex
programming. Rather remarkably, for this particular model the method reduces
the solution of the optimization problem to a generalization of the celebrated
log-Sobolev inequality. We hope that this method should work also for other
models lying out of the scope of the "threshold condition" ensuring that the
upper bound for the capacity as a difference between the maximum and the
minimum output entropies is attainable.
