Shannon's monotonicity problem for free and classical entropy.

We give a short unified proof of the following theorem, valid in the context of both classical probability theory and Voiculescu's free probability theory: let (X(j)((1)), ..., X(j)((n))) be independent (resp., freely independent) n-tuples of random variables. Let Z(N)((p)) = N(-1/2)(X(1)((p)) + ... + X(N)((p))) be their central limit sums. Then the entropy (resp., free entropy) of the n-tuple (Z(N)((1)), ..., Z(N)((n))) is a monotone function of N. The classical case (for n = 1) is a celebrated result of Artstein, Ball, Barthe, and Naor, and our proof is an adaptation and simplification of their argument.