# Recovering the QNEC from the ANEC

## TL;DR

This paper proves Wall's conjecture linking the shape derivative of relative entropy to the averaged null energy, providing a new method to derive the quantum null energy condition in quantum field theory.

## Contribution

It establishes Wall's conjecture as a theorem for operator algebras with half-sided modular inclusion, connecting relative entropy derivatives to null energy and offering new insights into quantum energy conditions.

## Key findings

- Proved Wall's conjecture relating shape derivatives of relative entropy to null energy.
- Derived a new proof of strong superadditivity of relative entropy.
- Formulated a theorem connecting operator algebra properties to quantum energy conditions.

## Abstract

We study relative entropy in QFT, comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy of possible purifications. This variational expression can be used to easily prove the quantum null energy condition. We formulate Wall's conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.