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Renormalization Group is the principle behind the Holographic Entropy Cone

Bartlomiej Czech, Sirui Shuai

TL;DR

The work reveals that holographic entropy inequalities encode a renormalization group principle: IR regions’ entanglement wedges must reach deeper into the bulk than UV regions, with inequality saturation marking equal depth. It uses contraction map techniques, majorization tests, and superbalance to show how Inclusion and Saturation arise, and it refines these results via extensive analyses of SSA, MMI, and higher-party dihedral/toric/projective-plane inequalities. A key insight is that saturation constrains erasure-correcting code access structures, linking bulk depth to code-theoretic properties of holography. The findings strengthen the operational meaning of the holographic entropy cone and suggest that RG flow is an inherent, geometry-enforced feature of holographic entanglement, with potential generalizations to time-dependent settings and quantum extremal surfaces.

Abstract

We show that every holographic entropy inequality can be recast in the form: `some entanglement wedges reach deeper in the bulk than some other entanglement wedges.' When the inequality is saturated, the two sets of wedges reach equally deep. Because bulk depth geometrizes CFT scales, the inequalities enforce and protect the holographic Renormalization Group.

Renormalization Group is the principle behind the Holographic Entropy Cone

TL;DR

The work reveals that holographic entropy inequalities encode a renormalization group principle: IR regions’ entanglement wedges must reach deeper into the bulk than UV regions, with inequality saturation marking equal depth. It uses contraction map techniques, majorization tests, and superbalance to show how Inclusion and Saturation arise, and it refines these results via extensive analyses of SSA, MMI, and higher-party dihedral/toric/projective-plane inequalities. A key insight is that saturation constrains erasure-correcting code access structures, linking bulk depth to code-theoretic properties of holography. The findings strengthen the operational meaning of the holographic entropy cone and suggest that RG flow is an inherent, geometry-enforced feature of holographic entanglement, with potential generalizations to time-dependent settings and quantum extremal surfaces.

Abstract

We show that every holographic entropy inequality can be recast in the form: `some entanglement wedges reach deeper in the bulk than some other entanglement wedges.' When the inequality is saturated, the two sets of wedges reach equally deep. Because bulk depth geometrizes CFT scales, the inequalities enforce and protect the holographic Renormalization Group.
Paper Structure (48 sections, 71 equations, 3 figures, 7 tables)

This paper contains 48 sections, 71 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Overview of results
  4. A further refinement
  5. Inequalities and holographic RG
  6. Inclusion
  7. Majorization test
  8. Superbalance and different notions of depth
  9. Saturation
  10. Examples
  11. Strong subadditivity
  12. Monogamy of mutual information
  13. Equal depth is not sufficient for saturation
  14. The five-party dihedral inequality
  15. A refinement
  16. ...and 33 more sections

Figures (3)

  • Figure 1: A spatial geometry, in which equations (\ref{['mmiresult1']}) and (\ref{['mmiresult2']}) and their analogues for $E(ACO)$ and $E(BCO)$ hold yet the monogamy of mutual information is not saturated. The example assumes that $x < 1$ and the two large cuts are much larger than 1.
  • Figure 2: Bulk regions, which become empty when inequality (\ref{['eq:5party']}) is saturated; (\ref{['eq:5partyresultO']}) is shown in blue and (\ref{['eq:5partyresultE']}) is shown in red. These regions lie deeper in the bulk (closer to the purifier $O$ / $E$) than $\cup_i E(X_i)$ does. Thus, if the inequality is saturated then the more 'infrared' regions fail to exceed the depth reached by the 'ultraviolet' regions.
  • Figure 3: Left: If $S(A) + S(B) = S(AB)$ then region (\ref{['w001']}) is empty. Middle: If region (\ref{['w001']}) is nonempty then $S(A) + S(B) > S(AB)$. Right: The converse of (\ref{['eq:saresult']}) is not true.