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-2 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 $Σ$: $\langle T_{kk}\rangle \geq S"/2π\sqrt{h}$. 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.

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