A new characterization of the holographic entropy cone

TL;DR

This work develops a novel majorization-based criterion to test whether covariant holographic entropies (HRT) obey the same inequalities as RT entropies, introducing light-cone configurations and null reductions to probe the holographic entropy cone. By analyzing bulk perturbations under the null energy condition, the authors connect majorization of area-perturbation contributions to the validity of holographic inequalities, and provide strong evidence that the HRT cone coincides with the RT cone, via extensive computational checks on known primitive sHIQs. The paper also introduces a tripartite-form perspective to organize sHIQs, explores the behavior of null reductions, and posits conjectures that would translate majorization safety into a complete characterization of the RT/HRT holographic entropy cone. If borne out, these results offer a fast, combinatorial method to discover and verify holographic entropy inequalities and deepen the understanding of entanglement structure in holography.

Abstract

Entanglement entropies computed using the holographic Ryu-Takayanagi formula are known to obey an infinite set of linear inequalities, which define the so-called RT entropy cone. The general structure of this cone, or equivalently the set of all valid inequalities, is unknown. It is also unknown whether those same inequalities are also obeyed by entropies computed using the covariant Hubeny-Rangamani-Takayanagi formula, although significant evidence has accumulated that they are. Using Markov states, we develop a test of this conjecture in a heretofore unexplored regime. The test reduces to checking that a given inequality obeys a certain majorization property, which is easy to evaluate. We find that the RT inequalities pass this test and, surprisingly, only RT inequalities do so. Our results not only provide strong new evidence that the HRT and RT cones coincide, but also offer a completely new characterization of that cone.

Figures (2)

• Figure 1: Left: LCC with central region $A$ for three boundary regions $A,B$ and $C$ in 1+1 dimensional Minkowski space. In any CFT, this configuration saturates SSA, ${\sf I}(B:C|A)=0$, implying that the state reduced on these three regions is a Markov state. Right: LCC in 2+1 dimensional Minkowski space for five boundary regions.
• Figure 2: LCC with three regions $A$, $B$ and $C$ as in the left figure \ref{['fig:light-cone-config']}, on $\mathbb{R}\times S^1$, with the light-cone $L$ (dashed) extended until it closes at the point $q$. The interior of the cylinder is global AdS$_3$, and the future light cone in the bulk is $\mathcal{L}$, with its null generators likewise converging at $q$. As an example we consider MMI: $\mathsf{S}(AB) + \mathsf{S}(AB)+ \mathsf{S}(AC) \geq \mathsf{S}(A) + \mathsf{S}(B) + \mathsf{S}(C) + \mathsf{S}(ABC)$. Since $A$ is the region containing the tip of the light-cone, studying MMI in this configuration is equivalent to studying its null reduction on $A$, which is SSA: $\mathsf{S}(AB) + \mathsf{S}(AC) \geq \mathsf{S}(A) + \mathsf{S}(ABC)$. We draw in black the four RT surfaces contributing for each term; the configuration saturates the inequality since every bulk null generator (thin blue lines) intersects every RT surface exactly once.

Theorems & Definitions (15)

• Lemma 1
• proof
• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Theorem 1: Karamata
• Conjecture 1
• Conjecture 2