Holographic Proof of the Quantum Null Energy Condition

TL;DR

This work provides a holographic proof of the Quantum Null Energy Condition (QNEC) at leading order in large-$N$ for CFTs and their relevant deformations with Einstein gravity duals. The authors relate the boundary QNEC to a bulk causality condition via the entanglement wedge and the area of extremal surfaces, using Fefferman–Graham near-boundary expansions to extract the finite, state-dependent contribution $S''$ to the entropy. A key technical step is constructing a bulk vector $s^{\mu}$ tangent to the family of extremal surfaces and showing its norm satisfies $s^{\mu} s_{\mu} \ge 0$, which yields $\langle T_{kk} \rangle \ge \frac{1}{2\pi \sqrt{h}} S''$ (with a refined $d=2$ form including $S'$). The results extend known free-theory proofs and, together with the holographic framework, support the universality of QNEC in quantum field theory, while also outlining several potential extensions to higher-curvature gravity, curved backgrounds, and connections to the Quantum Focussing Conjecture.

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

Figures (2)

• Figure 1: Here we show the region $\mathcal{R}$ (shaded cyan) and the boundary $\Sigma$ (black border) before and after the null deformation. The arrow indicates the direction $k^i$, and $\braket{T_{kk}}$ is being evaluated at the location of the deformation. The dashed line indicates the support of the deformation.
• Figure 2: The surface $\mathcal{M}$ in the bulk (shaded green) is the union of all of the extremal surfaces anchored to the boundary that are generated as we deform the entangling surface. The null vector $k^i$ (solid arrow) on the boundary determines the deformation, and the spacelike vector $s^\mu$ (dashed arrow) tangent to $\mathcal{M}$ is the one we construct in our proof. The QNEC arises from the inequality $s^\mu s_\mu \geq 0$.