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The Quantum Null Energy Condition, Entanglement Wedge Nesting, and Quantum Focusing

Chris Akers, Venkatesa Chandrasekaran, Stefan Leichenauer, Adam Levine, Arvin Shahbazi Moghaddam

TL;DR

This work shows that Entanglement Wedge Nesting (EWN) and the Quantum Focusing Conjecture (QFC) yield the Quantum Null Energy Condition (QNEC) for holographic CFTs on curved backgrounds with curvature-squared bulk terms, identifying geometric conditions that must hold for $d\leq 5$. By analyzing Fefferman–Graham expansions and the generalized entropy functional, the authors connect bulk geometric inequalities to boundary energy and entropy data, deriving the ordinary QNEC in curved space and a stronger Conformal QNEC via Weyl transformations. They demonstrate that the QNEC derived from QFC and from EWN are equivalent under the same assumptions and establish scheme-independence of the bound, while discussing bulk entropy contributions and the necessity of smearing. The paper outlines future directions including relevant deformations, higher-dimensional generalizations, deeper links between EWN and QFC, and a direct Conformal QNEC derivation from QFC, highlighting the broader impact for quantum gravity bounds in curved holographic settings.

Abstract

We study the consequences of Entanglement Wedge Nesting for CFTs with holographic duals. The CFT is formulated on an arbitrary curved background, and we include the effects of curvature-squared couplings in the bulk. In this setup we find necessary and sufficient conditions for Entanglement Wedge Nesting to imply the Quantum Null Energy Condition in $d\leq 5$, extending its earlier holographic proofs. We also show that the Quantum Focusing Conjecture yields the Quantum Null Energy Condition as its nongravitational limit under these same conditions.

The Quantum Null Energy Condition, Entanglement Wedge Nesting, and Quantum Focusing

TL;DR

This work shows that Entanglement Wedge Nesting (EWN) and the Quantum Focusing Conjecture (QFC) yield the Quantum Null Energy Condition (QNEC) for holographic CFTs on curved backgrounds with curvature-squared bulk terms, identifying geometric conditions that must hold for . By analyzing Fefferman–Graham expansions and the generalized entropy functional, the authors connect bulk geometric inequalities to boundary energy and entropy data, deriving the ordinary QNEC in curved space and a stronger Conformal QNEC via Weyl transformations. They demonstrate that the QNEC derived from QFC and from EWN are equivalent under the same assumptions and establish scheme-independence of the bound, while discussing bulk entropy contributions and the necessity of smearing. The paper outlines future directions including relevant deformations, higher-dimensional generalizations, deeper links between EWN and QFC, and a direct Conformal QNEC derivation from QFC, highlighting the broader impact for quantum gravity bounds in curved holographic settings.

Abstract

We study the consequences of Entanglement Wedge Nesting for CFTs with holographic duals. The CFT is formulated on an arbitrary curved background, and we include the effects of curvature-squared couplings in the bulk. In this setup we find necessary and sufficient conditions for Entanglement Wedge Nesting to imply the Quantum Null Energy Condition in , extending its earlier holographic proofs. We also show that the Quantum Focusing Conjecture yields the Quantum Null Energy Condition as its nongravitational limit under these same conditions.

Paper Structure

This paper contains 52 sections, 116 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Relation to other work
  3. Entanglement Wedge Nesting
  4. Subregion Duality
  5. Entanglement Wedge Nesting
  6. Near-Boundary EWN
  7. Bulk Metric
  8. Extremal Surface
  9. The $z$-Expansion of EWN
  10. Variations of the Entropy
  11. State-Independent Inequalities
  12. Leading Inequality
  13. Subleading Inequality
  14. The Quantum Null Energy Condition
  15. The Conformal QNEC
  16. ...and 37 more sections

Figures (1)

  • Figure 1: Here we show the holographic setup which illustrates Entanglement Wedge Nesting. A spatial region $A_1$ on the boundary is deformed into the spatial region $A_2$ by the null vector $\delta X^i$. The extremal surfaces of $A_1$ and $A_2$ are connected by a codimension-one bulk surface $\mathcal{M}$ (shaded blue) that is nowhere timelike by EWN. Then the vectors $\delta \bar{X}^\mu$ and $s^\mu$, which lie in $\mathcal{M}$, have nonnegative norm.