$\Phi^\ast$ first and then, based on this, free entropy $\chi^\ast$. We want to note that both constructions lead independently to objects $\chi$ and $\chi^*$ (as well as $\Phi$ and $\Phi^*$) which are, in analogy with the classical theory, justifiably called entropy (and Fisher information), but it is still not known whether they coincide. For many questions the actual value of these quantities is not important, essential is whether they are finite or infinite. There exist also more refined quantities, so-called free entropy dimensions (again in various variations), which allow a further distinction of the case of infinite entropy. In particular, finiteness of free entropy or of free Fisher information implies that the microstates free entropy dimension $\delta^*$ takes on its maximal value.

In the classical case, finiteness of entropy or of Fisher information imply some regularity of the corresponding distribution of the variables; in particular, they have a density (with respect to Lebesgue measure). In the non-commutative situation, the notion of a density does not make any direct sense, but still it is believed that the existence of finite free entropy or finite free Fisher information (in any of the two approaches) should correspond to some regularity properties of the 