Holographic proof of the quantum null energy condition

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 Σ 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 Σ: ⟨Tₖₖ⟩≥S^''/2π√h. In a CFT, conformal transformations of this inequality give results which apply when Σ 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.