Proof of the Quantum Null Energy Condition

TL;DR

The paper proves the Quantum Null Energy Condition (QNEC) for free and superrenormalizable bosonic quantum field theories on stationary null surfaces by reducing the problem to a family of decoupled 1+1 dimensional chiral CFTs via null quantization and an auxiliary system. Using the replica trick, the authors analyze the second-order perturbation of the entanglement entropy under infinitesimal deformations of a codimension-2 surface along a null direction, showing that the second variation is nonpositive, which yields the QNEC bound $\langle T_{kk}\rangle \ge \frac{\hbar}{2\pi\mathcal{A}} S_{out}''$. The calculation relies on careful analytic continuation across replica sheets and a decomposition of the state into near-vacuum pencil and auxiliary sectors, culminating in a sign-preserving argument for $S^{(2)''}$. The results extend to $D=2$, higher spin, and interactions, with dimensional reduction and robustness arguments, highlighting a deep link between quantum information and energy conditions with potential implications for quantum gravity and holography.

Abstract

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 $S_\text{out}$ of the quantum state restricted to one side of $Σ$. A second variation $S_\text{out}^{\prime\prime}$ can be defined by deforming $Σ$ along $N$, in a small neighborhood of $p$ with area $\cal A$. The QNEC states that $\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2π} \lim_{{\cal A}\to 0}S_\text{out}^{ \prime\prime}/{\cal A}$.

Figures (3)

• Figure 1: The spatial surface $\Sigma$ splits a Cauchy surface, one side of which is shown in yellow. The generalized entropy $S_\text{gen}$ is the area of $\Sigma$ plus the von Neumann entropy $S_\text{out}$ of the yellow region. The quantum expansion $\Theta$ at one point of $\Sigma$ is the rate at which $S_\text{gen}$ changes under a small variation $d\lambda$ of $\Sigma$, per cross-sectional area $\cal A$ of the variation. The Quantum Focussing Conjecture states that the quantum expansion cannot increase under a second variation in the same direction. If the classical expansion and shear vanish (as they do for the green null surface in the figure), the Quantum Null Energy Condition is implied as a limiting case. Our proof involves quantization on the null surface; the entropy of the state on the yellow spacelike slice is related to the entropy of the null quantized state on the future (brighter green) part of the null surface.
• Figure 2: The state of the CFT on $x > \lambda$ can be defined by insertions of $\partial \Phi$ on the Euclidean plane. The red lines denote a branch cut where the state is defined.
• Figure 3: Sample plots of the imaginary part (the real part is qualitatively identical) of the naïve bracketed digamma expression in \ref{['secondParenthesis']} and the one in \ref{['eq-continuation']} obtained from analytic continuation with $z=-m-i\alpha_{ij}$ for $m=3$ and various values of $\alpha_{ij}$. The oscillating curves are \ref{['secondParenthesis']}, while the smooth curves are the result of applying the specified analytic continuation prescription to that expression, resulting in \ref{['eq-continuation']}.