Necessary and sufficient conditions for entropy vector realizability by holographic simple tree graph models

TL;DR

This paper proves that the chordality condition is both necessary and sufficient for an entropy vector to be realizable by holographic simple tree graph models, providing a constructive algorithm applicable to any number of parties.

Contribution

It establishes the sufficiency of the chordality condition and demonstrates a constructive method for realizing entropy vectors with simple tree graph models.

Findings

Chordality condition is sufficient for realizability.

Constructive algorithm always succeeds for any number of parties.

Insights into the structure of holographic and stabilizer entropy cones.

Abstract

We prove that the ``chordality condition'', which was established in arXiv:2412.18018 as a necessary condition for an entropy vector to be realizable by a holographic simple tree graph model, is also sufficient. The proof is constructive, demonstrating that the algorithm introduced in arXiv:2512.18702 for constructing a simple tree graph model realization of a given entropy vector that satisfies this condition always succeeds. We emphasize that these results hold for an arbitrary number of parties, and, given that any entropy vector realizable by a holographic graph model can also be realized, at least approximately, by a stabilizer state, they highlight how techniques originally developed in holography can provide broad insights into entanglement and information theory more generally, and in particular, into the structure of the stabilizer and quantum entropy cones. Moreover, if the strong form of the conjecture from arXiv:2204.00075 holds, namely, if all holographic entropy vectors can be realized by (not necessarily simple) tree graph models, then the result of this work demonstrates that the essential data that encodes the structure of the holographic entropy cone for an arbitrary number of parties, is the set of ``chordal'' extreme rays of the subadditivity cone.