  We introduce a free version of the Stein kernel, relative to a semicircular
law. We use it to obtain a free counterpart of the HSI inequality of Ledoux,
Peccatti and Nourdin, which is an improvement of the free logarithmic Sobolev
inequality of Biane and Speicher, as well as a rate of convergence in the
(multivariate) entropic free Central Limit Theorem. We also compute the free
Stein kernels for several relevant families of self-adjoint operators.
