We prove the Quantum Null Energy Condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the Quantum Focussing Conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest.Our proof applies to free and superrenormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point p and null vector k a define a null plane N (a Rindler horizon). Given any codimension-2 surface Σ that contains p and lies on N , one can consider the von Neumann entropy S out of the quantum state restricted to one side of Σ. A second variation S out can be defined by deforming Σ along N , in a small neighborhood of p with area A. The QNEC states that T kk (p) ≥ 2π lim A→0 S out /A.

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-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: T kk ≥ 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.