This is the definition of a PMI given in [21], whereas the original definition from [23] was weaker.

Notice that this implies that to reconstruct the extreme rays of the $\mathsf{N}$-party HEC, one needs to know the extreme rays of the SAC for some ${\mathsf{N}}^{\prime }\ge \mathsf{N}$. We will return to this point in the next section.

Since no efficient algorithm is known, such explicit derivation of the inequalities remains computationally unfeasible for large $\mathsf{N}$.

This is not a priori obvious because the original definition of the HEC in [8] is a purely geometric one, and it does not assume that the bulk geometry corresponds to a CFT state [9].

Most of the machinery developed in [1] was used to show that Conjecture 1 would follow from other purely graph theoretic conjectures, but this machinery is not necessary for the purpose of this Letter.

The precise expression of ${\mathrm{\Lambda }}_{{\mathsf{N}}^{\prime }\to \mathsf{N}}$ is not necessary for this discussion, but for clarity we give a simple example of such a projection. Consider the three-party entropy vector $\overrightarrow{\mathsf{S}}=(1,1,1;2,2,2;1)$ for $A$, $B$, $C$ [the entropies are ordered conventionally as exemplified in (4) for $\mathsf{N}=6$], and define the new parties $A^{\prime }=A,B^{\prime }=BC$. The two-party entropy vector for these coarse-grained parties is then ${\overrightarrow{\mathsf{S}}}^{\prime }=(1,2;1)$, and it is obtained from $\overrightarrow{\mathsf{S}}$ by simply dropping the components that separate $B$ from $C$, i.e., $B,C,AB,AC$.

The computation of all the extreme rays of the SAC for $\mathsf{N}=5$ takes several days on a standard laptop, but the extreme rays of the face identified by Theorem 1 only takes a few minutes.