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

Veronika E. Hubeny, Massimiliano Rota

TL;DR

This work proves that the chordality condition on the line graph of the correlation hypergraph, previously shown as a necessary condition for realizing an entropy vector ${f S}$ through holographic simple tree models, is also sufficient. The authors provide a constructive proof by showing that the reconstruction algorithm from prior work yields a simple tree graph model ${ m T}$ whose min-cut entropies exactly reproduce ${f S}$, for any irreducible vector obeying SA and SSA. They establish this by aligning cut costs $C^ heta_X$ with ${f S}_X$, and proving these cuts are indeed the min-cuts on ${ m T}$ via a decomposition grounded in the correlation hypergraph structure. The results extend to arbitrary numbers of parties and support the view that the essential data encoding the holographic entropy cone lies in the chordal extreme rays of the subadditivity cone; if all holographic vectors can be realized by tree models, then this chordal data captures the cone’s core geometry, with broad implications for entanglement structure in quantum information 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.

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

TL;DR

This work proves that the chordality condition on the line graph of the correlation hypergraph, previously shown as a necessary condition for realizing an entropy vector through holographic simple tree models, is also sufficient. The authors provide a constructive proof by showing that the reconstruction algorithm from prior work yields a simple tree graph model whose min-cut entropies exactly reproduce , for any irreducible vector obeying SA and SSA. They establish this by aligning cut costs with , and proving these cuts are indeed the min-cuts on via a decomposition grounded in the correlation hypergraph structure. The results extend to arbitrary numbers of parties and support the view that the essential data encoding the holographic entropy cone lies in the chordal extreme rays of the subadditivity cone; if all holographic vectors can be realized by tree models, then this chordal data captures the cone’s core geometry, with broad implications for entanglement structure in quantum information 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.
Paper Structure (13 sections, 11 theorems, 38 equations, 2 figures, 1 algorithm)

This paper contains 13 sections, 11 theorems, 38 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. KC-PMIs and the correlation hypergraph
  4. Holographic graph models
  5. Algorithm for simple tree graph models
  6. Proof of \ref{['thm:main-reduced']}
  7. The cost of the cuts $C^\circ_{X}$ on ${\mathrm{T}}$ reproduces the components of $\vec{{\sf S}}$
  8. (i) $|X|=1$, $\beta(X^\complement)\in\mathfrak{B}^{^{\!+}}$:
  9. (i) $|X|=1$, $\beta(X^\complement)\in\mathfrak{B}^{^{\!+}}$:
  10. (ii) $\min(|X|,|X^\complement|)\geq 2$, $\beta(X)\in\mathfrak{B}^{^{\!+}}$, $\beta(X^\complement)\in\mathfrak{B}^{^{\!+}}$:
  11. (iii) $\min(|X|,|X^\complement|)\geq 2$, $\beta(X)\in\mathfrak{B}^{^{\!+}}$, $\beta(X^\complement)\notin\mathfrak{B}^{^{\!+}}$:
  12. The cuts $C^\circ_{X}$ are min-cuts on ${\mathrm{T}}$
  13. Discussion

Key Result

Theorem 1

An entropy vector that satisfies SA and SSA can be realized by a holographic graph model which is a simple forest if and only if the line graph of its correlation hypergraph is a chordal graph.

Figures (2)

  • Figure 1: An example of an entropy vector which violates SA but for which the cost of the cuts $C^\circ_{X}$ on the tree ${\mathrm{T}}$ resulting from Algorithm \ref{['alg:reconstruction']} reproduces the components of the entropy vector. Accordingly, the cuts $C^\circ_{X}$ are not min-cuts. For the KC-PMI $\mathcal{P}=\;\downarrow\!\{{\sf I}(12:4),{\sf I}(12:0),{\sf I}(04:1),{\sf I}(04:2)\}$ (a) shows the realization of the entropy vector $\vec{{\sf S}}=(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1)$ obtained from Algorithm \ref{['alg:reconstruction']}. (b) is a graph model with the same topology as (a), but with weights assigned from the components of the entropy vector $\vec{{\sf S}}'=(1, 1, 3, 1, 3, 4, 2, 4, 2, 4, 3, 4, 2, 2, 1)$, belonging to the same subspace of entropy space spanned by the face of the SAC corresponding to $\mathcal{P}$. Notice that $\vec{{\sf S}}'$ violates SA, since $2={\sf S}'_1+{\sf S}'_2<{\sf S}'_{12}=3$. Nevertheless, one can easily verify that the cost of the cuts $C^\circ_{X}$ correctly reproduce all components of $\vec{{\sf S}}'$. The resolution of the apparent paradox is that while the cut $C^\circ_{40}=\{0,4,\sigma\}$ is the min-cut for the $40$ subsystem in (a), it is not the min-cut in (b). Accordingly, the min-cut prescription assigns to (b) an entropy vector which is not $\vec{{\sf S}}'$.
  • Figure 2: The two main transformations of a graph ${\mathrm{G}}^{(i)}$ discussed in the main text. Leaves are shown in green, buds in purple and rosettes in orange. The gray blob represents the rest of the graph, which is assumed here not to be a leaf. In one type of transformation, a generic rosette (a) is transformed into a bud (b) by replacing its adjoining $k$ leaves with a single new leaf, with the edge label shown in the figure (see the main text for the transformation of a star-like rosette). In the second type of transformation, a bud (c) is removed and the remaining edge, connecting the leaf (originally connected to the bud) to the rest of the graph, is labeled as shown in (d).

Theorems & Definitions (19)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 9 more