Algorithmic construction of SSA-compatible extreme rays of the subadditivity cone and the ${\sf N}=6$ solution

TL;DR

This paper algorithmically constructs all extreme rays of the 6-party subadditivity cone compatible with strong subadditivity, identifying new orbits, some violating holographic inequalities, and providing graph models for most, thus advancing understanding of holographic entropy structures.

Contribution

It introduces a general algorithm for constructing extreme rays of polyhedral cones based on linear inequalities and partial orders, applied to the 6-party subadditivity cone.

Findings

Identified 208 new extreme rays of the 6-party subadditivity cone.

Found 52 rays violate known holographic entropy inequalities.

Constructed holographic graph models for 150 of the remaining rays.

Abstract

We compute the set of all extreme rays of the 6-party subadditivity cone that are compatible with strong subadditivity. In total, we identify 208 new (genuine 6-party) orbits, 52 of which violate at least one known holographic entropy inequality. For the remaining 156 orbits, which do not violate any such inequalities, we construct holographic graph models for 150 of them. For the final 6 orbits, it remains an open question whether they are holographic. Consistent with the strong form of the conjecture in arXiv:2204.00075, 148 of these graph models are trees. However, 2 of the graphs contain a "bulk cycle", leaving open the question of whether equivalent models with tree topology exist, or if these extreme rays are counterexamples to the conjecture. The paper includes a detailed description of the algorithm used for the computation, which is presented in a general framework and can be applied to any situation involving a polyhedral cone defined by a set of linear inequalities and a partial order among them to find extreme rays corresponding to down-sets in this poset.