Holographic Entropy Cone Measures

We investigate numerically several proxy measures for the number of states contained within the holographic entropy cone, compared to the number contained within the quantum entropy cone, for states with $3$ and $4$ parties. We find an interesting tension: while measures focused on calculating the volume ratios between the two cones indicate that the quantum cone is much larger than the holographic one, measures based on the generation of random states and then calculating the entropies thereof imply that almost all such randomly generated states are also contained within the holographic entropy cone. Also interestingly, the volume measures strongly indicate a difference in the number of states within the quantum or stabiliser cones versus the number in the holographic cone, which is not reproduced by the other classes of measures. We comment on the difference between the two classes of measures, and why each may be preferable.