On counterexamples to the unification of free entropy

It has been a long-standing open problem to determine when two versions of Voiculescu’s free entropy, $\chi$ and ${\chi }^{\ast }$, agree. OpenAI recently generated the first example with $-\mathrm{\infty }<\chi <{\chi }^{\ast }$. We give a detailed presentation of a similar example, and explicitly show the separation between free Fisher information and the large-$n$ limit of classical Fisher information through the relationship with Wasserstein distance. We thus situate the example in the context of probability theory and information geometry.