A Free Analogue of Shannon's Problem on Monotonicity of Entropy

We prove a free probability analog of a result of Artstein-Bally-Barthez-Naor. In particualar we prove that if X_{1},X_{2},… are freely independent identically distributed random variables, then the free entropy chi(X_{1}+…+X_{n}/\sqrt{n}) is monotone increasing for all n. Our proof also leads to a slight simplification of the original argument in the classical case.