  We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two
noncommutative probability subspaces $M_1$ and $M_2$ and show that the maximal
correlation coefficient between the sub-algebras generated by $s_n:=x_1+\ldots
+x_n$ and $s_m:=x_1+\ldots +x_m$ equals $\sqrt{m/n}$ for $m\le n$, where
$(x_i)_{i\in \mathbb{N}}$ is a sequence of free and identically distributed
noncommutative random variables. This is the free-probability analogue of a
result by Dembo--Kagan--Shepp in classical probability. As an application, we
use this estimate to provide another simple proof of the monotonicity of the
free entropy and free Fisher information in the free central limit theorem.
Moreover, we prove that the free Stein Discrepancy introduced by Fathi and
Nelson is non-increasing along the free central limit theorem.
