# Quantum null energy condition The QNEC states $Q:=T_{a b} k^{a} k^{b}-\frac{1}{2 \pi} S^{\prime \prime} \geq 0$, where $S$ is the von Neumann entropy on one side of the null congruence. Both terms diverge, but the divergences cancel usually. The difference is independent of the renormalisation scheme or regulator. When they don't cancel, [[2017#Fu, Marolf]] proposes to use bare quantities. ## Refs - original proposal - [[2015#Bousso, Fisher, Leichenauer, Wall]] - curved spacetime: - [[FuKoellerMarolf201706]][](https://arxiv.org/pdf/1706.01572.pdf) - proofs - [[2017#Balakrishnan, Faulkner, Khandker, Wang]]: uses causality and the [[0619 Lightcone OPE|lightcone OPE]] - [[2025#Hollands, Longo]]: uses an explicit formula for shape dependence of [[0199 Relative entropy|relative entropy]] - covariant - [[2023#Kudler-Flam, Leutheusser, Rahman, Satishchandran, Speranza]] ## Saturation - see [[2018#Leichenauer, Levine, Shahbazi-Moghaddam]] - when the QNEC is saturated, energy is von Neumann entropy, hence "energy is entanglement" - this happens for theories with gravity dual (at leading order in $1/N$)