A new characterization of the holographic entropy cone (2508.21823v1)

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, \emph{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.

• The paper introduces a novel majorization test to verify holographic entropy inequalities, supporting the equivalence between RT and HRT cones.
• It demonstrates that null reductions of RT inequalities yield RT inequalities, simplifying the characterization of the holographic entropy cone.
• Extensive analytic and numerical tests up to 13 parties confirm the test’s efficiency and suggest significant implications for entanglement in quantum gravity.

A New Characterization of the Holographic Entropy Cone

Introduction and Motivation

The paper presents a novel approach to characterizing the set of linear entropy inequalities obeyed by entanglement entropies computed via the Ryu-Takayanagi (RT) formula in holographic conformal field theories (CFTs). The central object of paper is the holographic entropy cone, which encodes all valid linear inequalities for subsystem entropies in geometric states. The longstanding open problem addressed is whether the set of inequalities for the RT formula coincides with those for the covariant Hubeny-Rangamani-Takayanagi (HRT) formula, i.e., whether the RT and HRT cones are equal.

The authors introduce a new test for the validity of entropy inequalities in the covariant setting, based on the majorization properties of their null reductions. This test is both computationally efficient and conceptually illuminating, providing strong evidence for the equivalence of the RT and HRT cones and yielding a new characterization of the holographic entropy cone.

Entropy Cones and Holographic Inequalities

The entropy cone formalism organizes the entanglement structure of multipartite quantum systems by considering the vector of all subsystem entropies. For $N$ parties, the entropy vector $\vec{\mathsf{S}}$ lives in $\mathbb{R}^{2^N-1}$, and the set of all physically realizable entropy vectors forms a convex cone. The quantum entropy cone is defined by all states in quantum mechanics, while the holographic (RT) entropy cone is restricted to geometric states in holography, where entropies are computed by the RT formula.

Primitive holographic entropy inequalities (sHEIs) correspond to facets of the RT cone and are known up to $N=5$ parties. For $N=3$, the cone is delimited by subadditivity (SA), strong subadditivity (SSA), and the monogamy of mutual information (MMI). For larger $N$, additional inequalities emerge, and the structure becomes increasingly intricate.

Null Reduction and Light-Cone Configurations

To probe the validity of entropy inequalities in the covariant (HRT) setting, the authors focus on light-cone configurations in Minkowski space, where all regions lie on a common light cone. In such configurations, the entanglement structure is highly constrained, and certain inequalities are saturated due to the Markov property of the vacuum state.

Figure 1: Light-cone configuration with central region $A$ for three boundary regions $A,B,C$ in 1+1D Minkowski space (left), and for five boundary regions in 2+1D (right).

The key technical tool is the null reduction of an inequality: for a given party $A$, all terms not involving $A$ are deleted, yielding a reduced inequality. For example, null reducing MMI on $A$ yields SSA. The authors show that null reductions of RT inequalities are themselves RT inequalities, and that the saturation of inequalities in light-cone configurations is robust under perturbations of the bulk geometry, provided the null energy condition (NEC) is respected.

Figure 2: Light-cone configuration of three regions $A,B,C$ on $R\times S^1$, with HRT surfaces saturating the MMI inequality via SSA.

The Majorization Test

The central innovation is the majorization test for entropy inequalities. Given a null-reduced inequality, the test asks whether, for all positive values of the region sizes, the list of arguments on the right-hand side majorizes those on the left-hand side. Formally, for an inequality written as

$$\sum_{n=1}^N h(u_{X_n}) \geq \sum_{n=1}^N h(u_{Y_n}),$$

where $h$ is a concave function and $u_{X_n}$, $u_{Y_n}$ are sums over region sizes, the test checks whether $(y_1,\ldots,y_N) \succ (x_1,\ldots,x_N)$ for all positive $b_i$.

The majorization test is justified by the Karamata theorem, which relates majorization to inequalities of concave functions. The authors show that all known RT inequalities pass the majorization test, and, crucially, that only RT inequalities do so. This provides a new, computationally efficient criterion for identifying holographic entropy inequalities.

Empirical Evidence and Conjectures

Extensive analytic and numerical tests support four main conjectures:

1. All RT inequalities pass the majorization test.
2. Only RT inequalities pass the majorization test.
3. Null reductions of RT inequalities are themselves RT inequalities.
4. If all null reductions of an inequality are RT inequalities, then the original inequality is RT.

The authors tested thousands of inequalities, including all known primitive sHEIs up to $N=6$ and members of infinite families up to $N=13$, finding no counterexamples. The majorization test is orders of magnitude faster than previous methods for verifying entropy inequalities.

Implications and Future Directions

The majorization test provides a new characterization of the holographic entropy cone, with significant implications for both the structure of entanglement in holographic theories and the computational discovery of new inequalities. The equivalence of the RT and HRT cones, strongly supported by this work, implies that the entanglement structure of geometric states in quantum gravity is highly constrained and robust under dynamical evolution.

The test also opens new avenues for the physical interpretation of holographic inequalities, suggesting that the entanglement structure of geometric states is intimately tied to majorization properties and the NEC. The authors speculate that the majorization test could be bootstrapped into a general proof of the equivalence of the RT and HRT cones.

Conclusion

This work introduces a majorization-based criterion for holographic entropy inequalities, providing strong evidence for the equivalence of the RT and HRT cones and yielding a new, efficient characterization of the holographic entropy cone. The results have both practical and theoretical significance, streamlining the discovery of new inequalities and deepening our understanding of entanglement in holographic quantum gravity. Future research may leverage these insights to fully resolve the structure of the entropy cone and its physical implications.