  We establish an improved form of the classical logarithmic Sobolev inequality
for the Gaussian measure restricted to probability densities which satisfy a
Poincar\'e inequality. The result implies a lower bound on the deficit in terms
of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the
deficit in the Talagrand quadratic transportation cost inequality this time by
means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product
measures, and deduce a lower bound on the deficit in the logarithmic Sobolev
inequality in terms of this metric. Applications are given in the context of
the Bakry-\'Emery theory and the coherent state transform. The proofs combine
tools from semigroup and heat kernel theory and optimal mass transportation.
