On the construction of graph models realizing given entropy vectors

TL;DR

The paper develops a constructive framework for realizing given entropy vectors with holographic graph models, focusing on simple tree topologies under a chordality (KC-based) condition. It introduces a comprehensive correlation-hypergraph toolkit, including coarse- and fine-graining, and reinterprets KC-PMIs through the MI-poset and Gamma-partitions, enabling efficient candidate-model construction. A central contribution is an explicit algorithm for building simple-tree holographic graphs when the line graph is chordal, with reductions to strictly positive entries and $|Q^igcap|=0$; it also outlines strategies for non-chordal cases via fine-graining and unrealizability criteria, aiming to illuminate the structure of the holographic entropy cone and test conjectures at larger N. The work provides a structured path toward systematically constructing holographic models for entropy vectors and discusses how to detect unrealizability independent of holographic entropy inequalities, with broader implications for the geometry-entropy correspondence in gauge-gravity duality.

Abstract

We present an efficient algorithm for the construction of a holographic simple tree graph model that realizes a given entropy vector, subject to a specific ``chordality'' condition first introduced in arXiv:2412.18018. We further develop the toolkit of the correlation hypergraph, particularly in relation to coarse-graining and fine-graining of subsystems. We then use these techniques to take the first steps towards the generalization of this new algorithm to arbitrary (not necessarily simple) holographic tree graph models, and the ``detection'' of unrealizability of an entropy vector independently from the knowledge of holographic entropy inequalities.

Figures (17)

• Figure 1: The Hasse diagram of the ${\sf N}=3$ MI-poset.
• Figure 2: An example (from Hubeny:2024fjn) of a down-set $\mathcal{D}$ in the ${\sf N}=3$ MI-poset which is not a PMI, with the corresponding $\beta\text{-set}$ classification. $\mathcal{D}$ is the set of MI instances indicated by the lower index $0$, while $+$ labels the elements of the complementary up-set $\mathcal{D}^\complement$. The elements of the antichain $\Check{\mathcal{A}}$ that generates $\mathcal{D}^\complement$ are underlined. Each $\beta\text{-set}$ is represented by a colored box: positive but not essential (red), essential but not completely essential (orange), partial and not essential (teal), partial and essential (green), completely essential (yellow), and vanishing (blue). The gray lines indicate the order relations among the $\beta\text{-sets}$, which is simply given by inclusion. The key aspect of this example is that $\beta(012)$ is both essential and partial (green), since it contains elements of both $\Check{\mathcal{A}}$ and $\mathcal{D}$, which as discussed in the main text, is not possible for KC-PMIs. To see that $\mathcal{D}$ is indeed not a PMI, it suffices to notice that it is not consistent with the linear dependence among the MI instances. Imposing that all MI instances in $\mathcal{D}$ vanish, it is immediate to verify that ${\sf I}(0:1)=0$, ${\sf I}(0:2)=0$, and ${\sf I}(02:1)=0$ imply ${\sf I}(01:2)=0$, which should therefore be included in $\mathcal{D}$, while it is not. Similar relations further imply that ${\sf I}(12:0)=0$, and that in any KC-PMI $\mathcal{P}$ such that $\mathcal{P}\supseteq\mathcal{D}$, the $\beta\text{-set}$$\beta(012)$ is actually vanishing and no longer essential.
• Figure 3: An example of an ${\sf N}=3$ KC-PMI with the corresponding classification of $\beta\text{-sets}$. The notation and color coding are the same as in \ref{['fig:D-not-P']}. Notice that $\beta(123)$ is essential, and that $\Gamma(123)=\{12,3\}$, since the only $X_i\subset X$ with positive $\beta(X_i)$ is $X_i=12$. Accordingly, the MI instance in $\beta(123)$ which belongs to $\Check{\mathcal{A}}$ is ${\sf I}(12:3)$. For any other positive $\beta(X)$ instead, $\Gamma(X)$ is the trivial partition since the maximal $X_i\subset X$ with positive $\beta(X_i)$ do not form a partition of $X$. For example, for $X=012$ we have $i=3$ and $X_1=01$, $X_2=02$ and $X_3=12$. Furthermore, notice that as discussed in the main text, these subsystems intersect pairwise, and the union of any pair is $X=012$.
• Figure 4: A simple example of two KC-PMIs $\mathcal{P}_1$ and $\mathcal{P}_2$ (whose correlation hypergraphs are indicated respectively in (a) and (b)) such that for their meet (c), $\mathfrak{B}^{^{\!+}}(\mathcal{P}_1\wedge\mathcal{P}_2)$ strictly contains the union of $\mathfrak{B}^{^{\!+}}(\mathcal{P}_1)$ and $\mathfrak{B}^{^{\!+}}(\mathcal{P}_2)$. Notice that in (c) the hyperedge $h_{012}$ is added to the set of hyperedges of the correlation hypergraph of the meet by the week union closure of the union of $\mathfrak{B}^{^{\!+}}(\mathcal{P}_1)$ and $\mathfrak{B}^{^{\!+}}(\mathcal{P}_2)$, since $h_{01}\cap h_{02}=\{v_0\}\neq\varnothing$. The color of the hyperedges corresponds to the classification of the associated $\beta\text{-sets}$ as in \ref{['fig:D-not-P']}. Following the presentation in Hubeny:2024fjn, we are representing a 2-edge by a line segment and higher $k$-edge by a rounded polygon around the corresponding vertices.
• Figure 5: An example of a KC-PMI with a partial $\beta\text{-set}$, and its transformations under different CG-maps. Panels (a), (b) and (c) show the (disconnected) correlation hypergraph ${\mathrm{H}}_{\mathcal{P}'}$ for the same ${\sf N}=3$ KC-PMI $\mathcal{P}'$, corresponding to the tensor product of two Bell pairs between parties $\{1,2\}$ and $\{3,0\}$. The dashed blue lines indicate the partition $\Theta^\circ$ associated to three different CG-maps (to ${\sf N}=2$ for (a) and (c), and to ${\sf N}=1$ for (b)). Panels (d), (e), and (f) show (respectively) the correlation hypergraph ${\mathrm{H}}_{\mathcal{P}}$ after the coarse-graining transformation, with blue vertices corresponding to the non-trivial equivalent classes. To simplify the figure, we have omitted labels for the hyperedges. Notice that in $\mathcal{P}'$, the $\beta\text{-set}$$\beta(1230)$ is partial, but it is mapped to a partial $\beta\text{-set}$ in (d), a vanishing one in (e), and a positive one in (f). Furthermore, in (f), notice that the quotient hypergraph does not have the red hyperedge, which is instead added by the weak union closure (since the two yellow hyperedges have non-empty intersection). This example also shows that an essential $\beta\text{-set}$ in the ${\sf N}$-party system, for example $\beta(12)$ in (f), can result from the coarse-graining of a partial $\beta\text{-set}$ in the ${\sf N}'$-party system, in this case $\beta(123)$ in (c).

Theorems & Definitions (40)

• Definition 1: Correlation hypergraph of a KC-PMI
• Theorem 1
• proof
• Theorem 2
• proof
• Definition 2: Weak union closure
• Theorem 3
• proof
• Definition 3: Hypergraph quotient
• Theorem 4