  We establish a dimension-free improvement of Talagrand's Gaussian
transport-entropy inequality, under the assumption that the measures satisfy a
Poincar\'e inequality. We also study stability of the inequality, in terms of
relative entropy, when restricted to measures whose covariance matrix trace is
smaller than the ambient dimension. In case the covariance matrix is strictly
smaller than the identity, we give dimension-free estimates which depend on its
eigenvalues. To complement our results, we show that our conditions cannot be
relaxed, and that there exist measures with covariance larger than the
identity, for which the inequality is not stable, in relative entropy. To deal
with these examples, we show that, without any assumptions, one can always get
quantitative stability estimates in terms of relative entropy to Gaussian
mixtures. Our approach gives rise to a new point of view which sheds light on
the hierarchy between Fisher information, entropy, and transportation distance,
and may be of independent interest. In particular, it implies that the
described results apply verbatim to the log-Sobolev inequality and improve upon
some known estimates in the literature.
