  We establish a quantitative bound on the entropy jump associated to the sum
of independent, identically distributed (IID) radially symmetric random vectors
having dimension greater than one. Following the usual approach, we first
consider the analogous problem of Fisher information dissipation, and then
integrate along the Ornstein-Uhlenbeck semigroup to obtain an entropic
inequality. In a departure from previous work, we appeal to a result by
Desvillettes and Villani on entropy production associated to the Landau
equation. This obviates strong regularity assumptions, such as presence of a
spectral gap and log-concavity of densities, but comes at the expense of radial
symmetry. As an application, we give a quantitative estimate of the deficit in
the Gaussian logarithmic Sobolev inequality for radially symmetric functions.
