Entropy, Extremality, Euclidean Variations, and the Equations of Motion

TL;DR

By formulating entanglement entropy in Euclidean gravity through the replica trick, the authors show that entropy is obtained by extremizing a generalized entropy functional $A_{ ext{gen}}$, and they extend this to quantum corrections via $S_{ ext{gen}}=ig\uparrowigackslashig floor$ to all orders in $G_N$, defining quantum extremality and modular extremality. They develop a bulk dual to modular Hamiltonians and propose a linear mapping of surfaces that underpins all-order bulk reconstruction in the entanglement wedge. The work unifies variational principles with entanglement first laws to derive integrated equations of motion and to describe relative entropies in a gravitational context, including mixtures and nonlocal modular Hamiltonians. Overall, the paper provides a comprehensive framework tying extremality, quantum corrections, and modular structure into a coherent holographic entanglement program applicable to general higher-derivative gravities.

Abstract

We study the Euclidean gravitational path integral computing the Renyi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton's constant $G_N$, providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in $G_N$. We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.