  In this note we prove that, under some minimal regularity assumptions on the
initial datum, solutions to the spatially homogenous Boltzmann and Landau
equations for hard potentials uniformly propagate the Fisher information. The
proof of such a result is based upon some explicit pointwise lower bound on
solutions to Boltzmann equation and strong diffusion properties for the Landau
equation. We include an application of this result related to emergence and
propagation of exponential tails for the solution's gradient. These results
complement estimates provided in the literature.
