  On small neighborhoods of the capacity-achieving input distributions, the
decrease of the mutual information with the distance to the capacity-achieving
input distributions is bounded below by a linear function of the square of the
distance to the capacity-achieving input distributions for all channels with
(possibly multiple) linear constraints and finite input sets using an identity
due to Tops{\o}e and Pinsker's inequality. Counter examples demonstrating
non-existence of such a quadratic bound are provided for the case of infinite
many linear constraints and the case of infinite input sets. Using a Taylor
series approximation, rather than Pinsker's inequality, the exact
characterization of the slowest decrease of the mutual information with the
distance to the capacity-achieving input distributions is determined on small
neighborhoods of the capacity-achieving input distributions. Analogous results
are established for classical-quantum channels whose output density operators
are defined on a separable Hilbert spaces. Implications of these observations
for the channel coding problem and applications of the proof technique to
related problems are discussed.
