The holographic multi-entropy cone

A bstract We generalize the holographic entropy cone (HEC) to the holographic multi-entropy cone (HMEC) by adjoining multi-entropy coordinates to the standard entropy coordinates. We show that holographic states, through their multi-entropy vectors, form a rational polyhedral cone in the multi-entropy space, and multicontraction maps provide exact certificates for holographic multi-entropy inequalities (HMEIs). We determine all facet inequalities of the n = 3, 4 HMECs, where n includes the purifier. These inequalities are organized into seven fundamental HMEI orbits: two for n = 3 and five for n = 4. We further propose two structural conjectures: HEC facet inequalities are convex combinations of HMEC facet inequalities, and HMEC facet inequalities obey a balanced-but-not-too-balanced principle.