Maximal correlation and monotonicity of free entropy and of Stein discrepancy

We introduce the maximal correlation coefficient R(M1,M2) between two noncommutative probability subspaces M1and M2 and show that sub-algebras generated by sn:=x1+…+xnand sm:=x1+…+xm equals m∕n for m≤n, where (xi)i∈N is a sequence of free identically distributed random variables. This free-probability analogue result Dembo–Kagan–Shepp in classical probability. As an application, we use this estimate to provide another simple proof monotonicity entropy Fisher information central limit theorem. Moreover, prove Stein Discrepancy introduced Fathi Nelson non-increasing along