Energy is Entanglement

TL;DR

<3-5 sentence high-level summary>Energy is Entanglement demonstrates that for theories with gravity duals the diagonal second variation of entanglement entropy under null deformations saturates the Quantum Null Energy Condition, yielding $S''_{vv}=2\pi\langle T_{vv}\rangle$. The authors prove this at leading order in $1/N$ via holographic arguments and argue that subleading corrections do not alter the result, while conjecturing saturation for all interacting theories. They extend the analysis to non-null deformations under finiteness and locality assumptions and connect the saturation to gravitational dynamics through the generalized entropy, showing that Einstein's equations emerge from entropy considerations. The work tightly ties energy, entanglement, and gravity together through the holographic entanglement entropy framework and bulk modular energy, with implications for ANEC, QFC, and curvature-driven GR equations.

Abstract

We compute the local second variation of the von Neumann entropy of a region in theories with a gravity dual. For null variations our formula says that the diagonal part of the Quantum Null Energy Condition is saturated in every state, thus providing an equivalence between energy and entropy. We prove that the formula holds at leading order in 1/N, and further argue that it will not be affected at higher orders. We conjecture that the QNEC is saturated in all interacting theories. We also discuss the special case of free theories, and the implications of our formula for the Averaged Null Energy Condition, Quantum Focusing Conjecture, and gravitational equations of motion. We show that the leading-order gravitational equations of motion, Einstein's equations, are equivalent to leading-order saturation of the QFC for Planck-width deformations.

Figures (2)

• Figure 1: Most of our work concerns the variations of entanglement entropy for the yellow region $\mathcal{R}$ whose boundary $\partial \mathcal{R}$ lies on the null plane $u=0$. The entangling surface is specified by the function $V(y)$.
• Figure 2: By restricting attention to $z<z_{*}$ the geometry is close to pure AdS, and we can solve for $\delta \bar{X}$ perturbatively. All of the $z<z_{*}$ data imprints itself as boundary conditions at $z=z_{*}$. We show that these boundary conditions are unimportant for our analysis, which means that a perturbative calculation is enough.