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arXiv:2601.09987

Combinatorial properties of holographic entropy inequalities

Guglielmo Grimaldi, Matthew Headrick, Veronika E. Hubeny, Pavel Shteyner

hep-thmath.COquant-phSubmitted Jan 15, 2026• Announced Jan 16, 2026• Revised Oct 16, 2026 (v2)

v1 · Jan 15, 2026 v2 · Oct 16, 2026

Abstract

A holographic entropy inequality (HEI) is a linear inequality obeyed by Ryu-Takayanagi holographic entanglement entropies, or equivalently by the minimum cut function on weighted graphs. We establish a new combinatorial framework for studying HEIs, and use it to prove several properties they share, including two majorization-related properties as well as a necessary and sufficient condition for an inequality to be an HEI. We thereby resolve all the conjectures presented in [arXiv:2508.21823], proving two of them and disproving the other two. In particular, we show that the null reduction of any superbalanced HEI passes the majorization test defined in [arXiv:2508.21823], thereby providing strong new evidence that all HEIs are obeyed in time-dependent holographic states.

Comments: 35 pages, 1 appendix, 2 figures. Watch a 5-minute video abstract here: https://youtu.be/vUtqPSxG1Io v2: clarified that our results apply to contraction-provable HEIs; new appendix on changing the purifier; other minor corrections, updates, and improvements to presentation

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A holographic entropy inequality (HEI) is a linear inequality obeyed by Ryu-Takayanagi holographic entanglement entropies, or equivalently by the minimum cut function on weighted graphs. We establish a new combinatorial framework for studying HEIs, and use it to prove several properties they share, including two majorization-related properties as well as a necessary and sufficient condition for an inequality to be an HEI. We thereby resolve all the conjectures presented in [arXiv:2508.21823], proving two of them and disproving the other two. In particular, we show that the null reduction of any superbalanced HEI passes the majorization test defined in [arXiv:2508.21823], thereby providing strong new evidence that all HEIs are obeyed in time-dependent holographic states.

Quantities computed by minimal cuts, such as entanglement entropies achievable by the Ryu-Takayanagi proposal in the AdS/CFT correspondence, are constrained by linear inequalities. We prove a previously conjectured property of all such constraints: Any

k

systems on the "greater-than" side of the inequality whose overlap is nonempty are subsumed in some

k

systems on its "less-than" side (accounting for multiplicity). This finding adds evidence that the same inequalities also constrain the entropies under time-dependent conditions because it preempts a large class of potential counterexamples. We prove several other properties of holographic entropy inequalities and comment on their relation to quantum erasure correction and the Renormalization Group.

We present a graphical method for proving holographic entanglement entropy inequalities (HEIs) in general multipartite systems. By introducing a geometric representation of the entanglement structure, we develop a systematic approach that enables one to visualize and verify the validity of HEIs for any number of subsystems

n

. Several theorems are established to formalize this method, and explicit examples are provided for systems with

n = 4

to

7

entangled regions.

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 such 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.

We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions

A

,

B

,

C

obey the inequality $I(A\ensuremath{\mathbin:}B\ensuremath{\cup}C)\ensuremath{\ge}I(A\ensuremath{\mathbin:}B)+I(A\ensuremath{\mathbin:}C)$, provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are characteristic of measures of quantum entanglement. This suggests that correlations in holographic theories arise primarily from entanglement rather than classical correlations. We also show that the Ryu-Takayanagi formula is consistent with all known general inequalities obeyed by the entanglement entropy, including an infinite set recently discovered by Cadney et al.; this constitutes strong evidence in favor of its validity.

To elucidate the meaning of holographic entropy inequalities (beyond subadditivity) which characterize the entanglement structure of geometric states in holography, arXiv:2309.06296 proposed the "tripartite form" for these inequalities, consisting of tripartite information and conditional tripartite information terms with unit coefficients. While this provides a compact and useful packaging of the inequalities, it is not a priori guaranteed that all inequalities can be recast in this form. Here we conjecture that they can, and present substantial evidence, by proving that the two known infinite families of holographic entropy inequalities found in arXiv:2309.15145 can indeed be written in the tripartite form. This is significant because such recasting is particularly nontrivial for these families. Apart from providing the explicit tripartite form expressions for every member of these two infinite families, we detail how we arrived at them, in the process deriving several useful identities which may serve as stepping stones to formulate further repackaging of the corresponding information quantities. We also illustrate the power of the tripartite form by proving a number of structural properties satisfied by any holographic entropy inequality.

The Ryu-Takayanagi formula implies many general properties of entanglement entropies in holographic theories. We review the known properties, such as continuity, strong subadditivity, and monogamy of mutual information, and fill in gaps in some of the previously-published proofs. We also add a few new properties, including: properties of the map from boundary regions to bulk regions implied by the RT formula, such as monotonicity; conditions under which subadditivity-type inequalities are saturated; and an inequality concerning reflection-symmetric states. We attempt to draw lessons from these properties about the structure of the reduced density matrix in holographic theories.