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 capital 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 capital sigma : [left angle bracket]T sub()kkright angle bracket] > or = S"/2[pi][radical]h. In a CFT, conformal transformations of this inequality give results which apply when capital 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.