Towards a Proof of the Improved Quantum Null Energy Condition

TL;DR

The paper analyzes the Improved Null Energy Condition (INEC), a stronger, nonlocal bound that extends the Quantum Null Energy Condition (QNEC) to cases with nonzero classical expansion. Building on the quantum focusing conjecture and modular-entropy techniques, it shows INEC holds for conformal field theories on the light-cone provided an additional constraint on the relative entropy is satisfied, connecting $\langle T_{kk} \rangle$ to derivatives of $S_{\rm out}$ and to $S_{\rm rel}$. The approach leverages a null-cone conformal map and an explicit modular Hamiltonian on the past null cone, along with convexity properties of relative entropy, to derive the necessary inequalities. A key outcome is a pathway to incorporating quantum corrections into semiclassical gravity, suggesting generalized quantum focusing conjectures that remain valid at higher orders in $G\hbar$. The results illuminate how entropy variations constrain energy conditions and point toward refined bounds that may shape quantum-gravity phenomenology in the presence of curvature or dynamical spacetime effects.

Abstract

The Improved Quantum Null Energy Condition (INEC) was recently derived from the (restricted) quantum focusing conjecture (QFC), and is a statement about the energy-momentum tensor (EMT) of field theories in Minkowski space-time. It is a stronger condition than the quantum null energy condition (QNEC), and includes the possibility of expanding or contracting geodesics. Using the properties of relative entropy and modular Hamiltonian associated with null deformation of the sphere, we show the INEC holds under an additional assumption relating the EMT to the relative entropy. Furthermore, using the QNEC and INEC as a basis, we briefly speculate about a possible modified Quantum Focusing Conjecture.

Figures (2)

• Figure 1: A section of the null plane defined by $X^-=0$, with light-cone coordinates $X^{\pm}= X^1 \pm X^0$. Here, $Y$ denotes the transverse coordinates $(X^2,X^3,\dots X^{d-1})$. Let us consider a $d-2$ dimensional spatial surface $\Gamma$ on this null plane, given by the equation $\Lambda=\Gamma(Y)$ with $\Lambda$ being the affine parameter along $X^+$-- direction. The modular Hamiltonian for region $\mathcal{R}$ bounded by this surface on the null plane is given in Eq. \ref{['e:nec18']}.
• Figure 2: The past null cone, and the different regions. The vertical axis of the cone gives the time direction. The arbitrary region has boundary $\lambda = \gamma(\Omega)$ on the null cone.