Is the EMI model a QFT? An inquiry on the space of allowed entropy functions

The mutual information $I(A,B)$ of pairs of spatially separated regions satisfies, for any $d$-dimensional CFT, a set of structural physical properties such as positivity, monotonicity, clustering, or Poincar\'e invariance, among others. If one imposes the extra requirement that $I(A,B)$ is extensive as a function of its arguments (so that the tripartite information vanishes for any set of regions, $I_3(A,B,C)\equiv 0$), a closed geometric formula involving integrals over $\partial A$ and $\partial B$ can be obtained. We explore whether this "Extensive Mutual Information" model (EMI), which in fact describes a free fermion in $d=2$, may similarly correspond to an actual CFT in general dimensions. Using the long-distance behavior of $I_{\rm \scriptscriptstyle EMI}(A,B)$ we show that, if it did, it would necessarily include a free fermion, but also that additional operators would have to be present in the model. Remarkably, we find that $I_{\rm \scriptscriptstyle EMI}(A,B)$ for two arbitrarily boosted spheres in general $d$ exactly matches the result for the free fermion current conformal block $G^d_{\Delta=(d-1),J=1}$. On the other hand, a detailed analysis of the subleading contribution in the long-distance regime rules out the possibility that the EMI formula represents the mutual information of any actual CFT or even any limit of CFTs. These results make manifest the incompleteness of the set of known constraints required to describe the space of allowed entropy functions in QFT.