immediately the statement of Corollary \[cor-main2\].

We point out that the conclusions of both Theorem \[main2\] and Corollary \[cor-main2\] were shown under the stronger assumption of finite non-microstates free Fisher information in an earlier version of this paper. In the above stated generality they appeared first in [@Shly2014], where the proof is based on results from [@Connes-Shlyakhtenko]. We are going to prove those statements in a more direct way by refining our initial methods.

More precisely, in Subsection \[subsec:finite\_Fisher\], we will give a quantitative version of our key idea that under the assumption of finite non-microstates Fisher information there is a strong relation between kernels of polynomials and the kernels of their derivatives.

Since the semicircular perturbation is inherent in the definition of the non-microstates free entropy as well as the corresponding entropy dimension, we find ourselves in the setting of finite free Fisher information. Therefore, we will study in Subsection \[subsec:entropy\_dimension\] the behavior of the results found in Subsection \[subsec:finite\_Fisher\] under semicircular perturbations – roughly speaking, we will be interested in the case where the perturbation tends to zero.

Finally, in Subsection \[subsec:proof\_thm\], which is dedicated to the proof of Theorem \[main2\], we will deduce from this observation a 