  Motivated by the entropy computations relevant to the evaluation of decrease
in entropy in bit reset operations, the authors investigate the deficit in an
entropic inequality involving two independent random variables, one continuous
and the other discrete. In the case where the continuous random variable is
Gaussian, we derive strong quantitative bounds on the deficit in the
inequality. More explicitly it is shown that the decay of the deficit is
sub-Gaussian with respect to the reciprocal of the standard deviation of the
Gaussian variable. What is more, up to rational terms these results are shown
to be sharp.
