  This paper quantifies the intuitive observation that adding noise reduces
available information by means of non-linear strong data processing
inequalities. Consider the random variables $W\to X\to Y$ forming a Markov
chain, where $Y=X+Z$ with $X$ and $Z$ real-valued, independent and $X$ bounded
in $L_p$-norm. It is shown that $I(W;Y) \le F_I(I(W;X))$ with $F_I(t)<t$
whenever $t>0$, if and only if $Z$ has a density whose support is not disjoint
from any translate of itself. A related question is to characterize for what
couplings $(W,X)$ the mutual information $I(W;Y)$ is close to maximum possible.
To that end we show that in order to saturate the channel, i.e. for $I(W;Y)$ to
approach capacity, it is mandatory that $I(W;X)\to\infty$ (under suitable
conditions on the channel). A key ingredient for this result is a deconvolution
lemma which shows that post-convolution total variation distance bounds the
pre-convolution Kolmogorov-Smirnov distance. Explicit bounds are provided for
the special case of the additive Gaussian noise channel with quadratic cost
constraint. These bounds are shown to be order-optimal. For this case
simplified proofs are provided leveraging Gaussian-specific tools such as the
connection between information and estimation (I-MMSE) and Talagrand's
information-transportation inequality.
