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

## TL;DR

This paper explores how entanglement wedge nesting in holographic conformal field theories implies the quantum null energy condition, especially on curved backgrounds with curvature-squared couplings, extending previous holographic proofs.

## Contribution

It establishes necessary and sufficient conditions for entanglement wedge nesting to imply the quantum null energy condition in dimensions up to five, including effects of curvature-squared couplings.

## Key findings

- Entanglement wedge nesting implies the quantum null energy condition in certain holographic setups.
- The quantum focusing conjecture leads to the quantum null energy condition in the non-gravitational limit.
- Conditions are extended to CFTs on arbitrary curved backgrounds with curvature-squared couplings.

## 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.