Holographic entropy inequalities pass the majorization test

Abstract: 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 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.

• The paper proves that all balanced holographic entropy inequalities pass the majorization test, confirming their extension to time-dependent settings.
• It employs contraction map techniques to reveal robust combinatorial structures that ensure validity under null reduction and superbalance conditions.
• The work connects holographic entanglement constraints with quantum error correction and RG flows, offering deep insights into gravitational dynamics.

Holographic Entropy Inequalities and the Majorization Test

Overview

The paper "Holographic entropy inequalities pass the majorization test" (2601.09989) establishes new structural properties for holographic entropy inequalities—linear entropy constraints arising from the Ryu-Takayanagi (RT) formula in AdS/CFT and tensor network models. Specifically, it proves that all such inequalities and their null reductions satisfy the majorization test, thus precluding a substantial class of possible violations in time-dependent holographic spacetimes. This result supports the claim that the inequalities extend beyond the strictly static RT context into more general dynamic (HRT) settings. The work situates its findings within the framework of contraction map proofs, linking mathematical combinatorics to underlying physical principles such as quantum error correction and holographic RG flows.

Holographic Entropy Inequalities: Structure and Properties

Holographic entropy inequalities constrain the possible patterns of entanglement entropy attainable by minimal (RT) or maximin (HRT) surfaces for multipartite boundary subsystems. Formally, these take the schematic form:

$$LHS = \sum_{i=1}^L S(X_i) \geq \sum_{j=1}^R S(Y_j) = RHS$$

with $X_i$ and $Y_j$ unions of atomic regions, and all coefficients unity by convention. Important examples include strong subadditivity (SSA) and the monogamy of mutual information.

Prior universal properties include:

• Balance: Each atomic region appears with equal net frequency on both sides, canceling UV-divergences at region boundaries.
• Superbalance: Each region pair appears with equal net frequency, canceling divergences at pairwise boundaries.

Almost all holographic inequalities are not only balanced but also superbalanced, with SSA and its convex combinations as exceptions. The structure of the inequalities can be captured succinctly using indicator matrices for inclusion of atomic regions.

Majorization Test: Statement and Proof

A major open question is whether all known holographic entropy inequalities remain valid when entropies are computed via the HRT formula (i.e., for covariant, time-dependent surfaces), rather than just by minimal surfaces on time-symmetric slices. The majorization test, introduced as a sufficient combinatorial check, posits that for any null reduction of a balanced holographic entropy inequality, a specific majorization relation among associated vectors holds:

$\vec{v} \prec \vec{z}$

That is, for all $k$, the maximal sum over $k$-tuples of LHS components does not exceed the maximal sum over $k$-tuples of RHS components, for arbitrary positive weights assigned to atomic regions not involved in the null reduction.

The main technical result is that all null reductions of all balanced holographic entropy inequalities pass the majorization test. The proof utilizes the contraction map construction, a combinatorial machinery underlying all known holographic inequalities, to explicitly demonstrate the existence of required dominance relations in the sums over region appearances. Central to the argument is the mapping of overlap structures between LHS and RHS region sets using inner products, collinearity properties, and indicator vectors.

Null Reductions and Validity Preservation

In addition, the paper proves that the null reduction of any superbalanced holographic entropy inequality is itself a valid holographic entropy inequality, with an explicit contraction map provided. In this context, null reduction means retaining only those terms involving a given atomic region, thereby producing a derived inequality among subsystems containing that region.

This property fails for non-superbalanced inequalities, as illustrated by SSA: the null reduction of SSA can yield invalid inequalities.

Corollaries and Interpretations

Several corollaries follow:

• For any $k$ LHS regions with nontrivial intersection, there exist $k$ RHS regions in the same inequality containing at least the intersection and with region multiplicities at least as large. This result, denoted "Dominance," strengthens the combinatorial structure underlying the entropy cone.
• In any maximally tight inequality (i.e., one that defines a facet of the entropy cone other than subadditivity), the LHS regions have empty total intersection.

The implication is that holographic entropy inequalities are not only constraints on linear combinations of entropies but also implement a form of coarse majorization in the cover structure of regions. The results strongly suggest that the inequalities are robust under generic dynamical evolution in the bulk, thus acting as physical constraints on the class of admissible spacetimes in holographic theories.

Connections to Quantum Error Correction and RG Flows

The paper further discusses connections to quantum erasure correction and the holographic renormalization group. When holographic entropy inequalities are saturated, structures relevant for quantum error correction emerge in the bulk, consistent with the role of minimal surfaces encoding the protectable logical subspace against boundary erasures. RG flow interpretations relate the combinatorial dominance implied by the inequalities to bulk coverage by entanglement wedges, serving as order parameters for bulk IR correlations not locally encoded in the ultraviolet.

The authors remark that while these correspondences are explicit in static cases, the majorization property provides a combinatorial foundation for extending such interpretations to dynamical (time-dependent) settings.

Contrapositives and Limitations

The paper also highlights, with synchronous work, that the converses of its main results are not valid: majorization-satisfying null reductions do not guarantee overall validity of an inequality, and the validity of all null reductions does not guarantee that an inequality is holographic.

Conclusion

This work rigorously establishes that all balanced holographic entropy inequalities, and all null reductions thereof, pass the majorization test and, when superbalanced, preserve validity under null reduction. These results strengthen the evidence that holographic entropy inequalities extend to time-dependent, dynamical settings, and illuminate deep combinatorial and physical structure in the entropy cone. They suggest fruitful directions for further research into the interplay between graph-theoretic constraints, quantum error correction frameworks, and the dynamics of gravity in AdS/CFT, especially in terms of formulating physical principles that bridge discrete contraction map properties and continuum gravitational evolution.