math-ph — Mathematical Physics

Holographic entropy inequalities without unscaled contraction proofs

Contributed by Antti Hippeläinen

We show that a valid holographic entropy inequality can lack a proof by contraction until all its coefficients are multiplied by a common integer, even when it is balanced, that is, every party has the same total coefficient on both sides, or superbalanced, that is, this remains true after any party is exchanged with the purifier. Every facet of the holographic entropy cone other than subadditivity is superbalanced. A proof by contraction is a map between Boolean cubes that respects the terms in which each party occurs and does not increase weighted Hamming distances. A balanced seven-party inequality and a superbalanced six-party inequality fail at a single point of the cube. Six of their right-hand side terms correspond to the edges of two disjoint triangles of parties, and at that point an unscaled map would have to choose three of these edges that together touch all six vertices, which is impossible. The seven-party one has a proof after doubling. Superbalanced eight- and seven-party inequalities have no unscaled proof even on their realizable domains, the subsets of the cube on which a map already proves validity, and they keep this property when instances of the monogamy of mutual information are subtracted in five successive steps. Splitting further into valid inequalities leads to a superbalanced facet $F_7$ of the seven-party cone that we did not find in the literature. It has unscaled maps on its realizable domain, all of which break its symmetry. For star inequalities, whose left-hand side contains only single-party entropies, the least scaling is the least $k$ for which an explicit polytope contains a point with coordinates in $\frac1k\mathbb{Z}$. From the weighing matrices of Alon and V\~u we obtain valid $N$-party star inequalities whose least scaling exceeds $N^{(1/4-\varepsilon)N}$ for every $\varepsilon>0$ and large $N$, while every finite least scaling of an $N$-party star inequality is below $N^{N/2}$. Non-existence rests on DRAT proofs, validity on exact rational maps or identities, and the facet property on $126$ explicit graph models, except for the star inequalities of arbitrary size, whose results rest on written proofs.