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Introduction
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In a groundbreaking series of papers [@Voi-Entropy-I; @Voi-Entropy-II; @Voi-Entropy-III; @Voi-Entropy-IV; @Voi-Entropy-V; @Voi-Entropy-VI] (see also the survey [@Voi-Entropy-Surv]), Voiculescu transferred the notion of entropy and Fisher information to the world ofnon-commutative probability theory. Free entropy and free Fisher information are some of the core quantities in free probability theory, with fundamental importance both for operator algebraic and random matrix questions. One of the most striking results which came out of this program is arguably the proof of the fact [@Voi-Entropy-III] that the free group factors do not possess Cartan subalgebras. This gave in particular the solution of the by then longstanding open question, whether every separable $\text{II}_1$-factor contains Cartan subalgebras. But despite such deep results and applications, still many of the basic analytic properties of free entropy and Fisher information are poorly understood.

Voiculescu gave actually two different approaches to entropy and Fisher information in the non-commutative setting. The first one is based on the notion of matricial microstates and defines free entropy $\chi$ first and then, based on this, the free Fisher information $\Phi$; the second approach is based on the notion of conjugate variables with respect to certain non-commutative derivatives and defines free Fisher information 