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We introduce the maximal correlation coefficient $R(M_1,M_2)$ between twononcommutative probability subspaces $M_1$ and $M_2$ and show that the maximalcorrelation 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 distributednoncommutative random variables. This is the free-probability analogue of aresult by Dembo--Kagan--Shepp in classical probability. As an application, weuse this estimate to provide another simple proof of the monotonicity of thefree entropy and free Fisher information in the free central limit theorem.Moreover, we prove that the free Stein Discrepancy introduced by Fathi andNelson is non-increasing along the free central limit theorem.

Maximal correlation and monotonicity of free entropy and Stein discrepancy