Recovering the QNEC from the ANEC

TL;DR

This work delivers a mathematically rigorous proof of the quantum null energy condition (QNEC) within algebraic QFT by encoding purifications of a given state through relative modular flow and half-sided modular inclusions. It derives a variational formula for the shape derivative of relative entropy, expressed as a minimization over purifications with finite averaged null energy, and shows that this yields QNEC and convexity of the relative entropy as a function of entangling-surface shape. A central technical advance is the detailed analysis of the flowed state's averaged null energy (ANE), proving an exact affine form P_s = R + (P_ψ - R) e^{-2πs} for entire states and extending to general states by continuity, which ties the ANE directly to the derivative of relative entropy. The paper also establishes strong superadditivity of relative entropy from QNEC, discusses implications for recovery-map bounds, and outlines several promising future directions, including holographic duals and extensions to curved spacetimes.

Abstract

We study relative entropy in QFT, comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy of possible purifications. This variational expression can be used to easily prove the quantum null energy condition. We formulate Wall's conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

Figures (4)

• Figure 1: Relative entropy of $\psi_s$ for various $s$ as a function of $b$. The input relative entropy function, shown in black, is a cartoon. It has the additional property that $\partial S(b), \partial \bar{S}(-b) \rightarrow 0$ as $b \rightarrow \infty$ which is the case if the state approaches vacuum in that limit (this may not actually be the case in QFT since the wiggly cut functions, representing deformations from the Rindler cut $R$, might have bounded support.) Positive $s$ are the green curves and negative $s$ is the red curve. The relative modular flow is defined with respect to the cut at the point $b=c$.
• Figure 2: Relative entropy of $\widehat{\psi}$ as a function of $b$, where the natural self-dual cone is with respect to the algebra $\mathcal{A}_{A_c}$ (at the the origin of the $b$-axis). The dashed curves show the input relative entropy and the complement relative entropy (only in the left figure). If the complementary relative entropy for the input state $\psi$ is not finite then we have not been able to discount the possibility of a bounded jump discontinuity in the derivative of relative entropy at $b=c$ which is shown in the right figure.
• Figure 3: Domains of holomorphy for the various vectors $\Gamma_{I,II,III,IV}$ discussed in Lemma \ref{['lemma4']}. Shown are the regions in the imaginary plane upon which the complex tube regions are based with $-\infty< {\rm Re} (\eta,s) < \infty$. We have also used different colors to show where the resulting function $g_\epsilon(s,\eta)$ defined via Lemma \ref{['lemma5']} (see \ref{['gall']} and \ref{['geps']}) is bounded (blue) or not uniformly bounded (pink). The function is still bounded on compact subsets. The star marks a particularly well behaved point for $g_\epsilon$ since the function is real and monotonic as a function of $\epsilon$. If one identifies the top and bottom via $s \equiv s + i$ then the green lines become branch cuts. The right figure is for $q_\epsilon(\eta,s)$ defined in the proof of Lemma \ref{['lemma7']}, see \ref{['qeps']}. As $\epsilon \rightarrow 0$ the left and right functions are related in such away that they must become a periodic function under $s \rightarrow s + i$.
• Figure 4: Substrips of $T_I$ parameterized by $\kappa$ and $h$ and defined via $\eta = h+\kappa(s+i/2)$ with $s\in S(-1/2,0)$. The function $F^{(h,\kappa)}(s)$ on this strip is uniformly bounded by $|| c'\left| \Omega \right> ||$ on the edges so it is uniformly bounded in the bulk.

Theorems & Definitions (25)

• Definition 1: Half-sided modular inclusion
• Lemma 1
• Lemma 2
• Corollary 1: to Lemma \ref{['lemma2']}
• proof
• Lemma 3
• Theorem 1: Wall's conjecture
• proof
• Theorem 2: The Quantum Null Energy Condition (lite)
• proof