  We develop a new method for bounding the relative entropy of a random vector
in terms of its Stein factors. Our approach is based on a novel representation
for the score function of smoothly perturbed random variables, as well as on
the de Bruijn's identity of information theory. When applied to sequences of
functionals of a general Gaussian field, our results can be combined with the
Carbery-Wright inequality in order to yield multidimensional entropic rates of
convergence that coincide, up to a logarithmic factor, with those achievable in
smooth distances (such as the 1-Wasserstein distance). In particular, our
findings settle the open problem of proving a quantitative version of the
multidimensional fourth moment theorem for random vectors having chaotic
components, with explicit rates of convergence in total variation that are
independent of the order of the associated Wiener chaoses. The results proved
in the present paper are outside the scope of other existing techniques, such
as for instance the multidimensional Stein's method for normal approximations.
