Holographic proof of the quantum null energy condition

Abstract

We use holography to prove the quantum null energy condition (QNEC) at leading order in large $N$ for CFTs and relevant deformations of CFTs in Minkowski space which have Einstein gravity duals. Given any codimension-two surface $\mathrm{\ensuremath{\Sigma}}$ which is locally stationary under a null deformation in the direction $k$ at the point $p$, the QNEC is a lower bound on the energy-momentum tensor at $p$ in terms of the second variation of the entropy to one side of $\mathrm{\ensuremath{\Sigma}}$: $⟨{T}_{kk}⟩\ensuremath{\ge}{S}^{\ensuremath{'}\ensuremath{'}}/2\ensuremath{\pi}\sqrt{h}$. In a CFT, conformal transformations of this inequality give results which apply when $\mathrm{\ensuremath{\Sigma}}$ is not locally stationary. The QNEC was proven previously for free theories, and taken together with our result this provides strong evidence that the QNEC is a true statement about quantum field theory in general.