Relative entropy equals bulk relative entropy

TL;DR

The authors propose that the boundary modular Hamiltonian for a boundary subregion in a holographic theory is dual to a bulk operator consisting of the extremal surface area term plus the bulk modular Hamiltonian inside the entanglement wedge. They show that, to leading order in 1/G_N, the boundary relative entropy equals the bulk relative entropy, implying that the entanglement wedge is the natural bulk region dual to the boundary region and that modular flow is matched across the two descriptions. The work analyzes linear and graviton corrections, demonstrates the canonical-energy interpretation in symmetric cases, and discusses smooth bulk modular evolution and implications for entanglement-wedge reconstruction, while outlining extensions to higher-derivative gravity, non-extremal surfaces, and the nature of bulk distillable entanglement. Overall, it provides a concrete, testable bridge between boundary information-theoretic quantities and bulk gravitational dynamics, with broad implications for holographic duality and quantum gravity.

Abstract

We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular flow is dual to the bulk modular flow in the entanglement wedge, with implications for entanglement wedge reconstruction.

Figures (3)

• Figure 1: The red segment indicates a spatial region, $R$, of the boundary theory. The leading contribution to the entanglement entropy is computed by the area of an extremal surface $\cal S$ that ends at the boundary of region $R$. This surface divides the bulk into two, region $R_b$ and its complement. Region $R_b$ lives in the bulk and has one more dimension than region $R$. The leading correction to the boundary entanglement entropy is given by the bulk entanglement entropy between region $R_b$ and the rest of the bulk.
• Figure 2: In this figure we are considering the thermofield double state. (a) Acting with the bulk modular Hamiltonian $e^{- i t K_{\rm bulk,R }}$ we get a new state on the horizontal line that has a singularity at the horizon. (b) The area term introduces a kink, or a relative boost between the left and right sides. Then the state produced by the full right side Hamiltonian is non-singular, and locally equal to the vacuum state.
• Figure 3: In both figures the region $R$ is the union of the two red intervals and the Ryu-Takayanagi surface is the dotted black line, while the boundary of $R_C$ is the blue dashed line (color online). In $a)$, the shaded region denotes the defining spatial slice $R_b$ of the entanglement wedge. In $b)$, the shaded region is the defining spatial slice $R_C$ of the causal wedge. The modular flow of an operator close to the Ryu-Takayanagi surface will be approximately local, so that $\phi_1(s)$ will be almost local and, after some $s$, it will be in causal contact with $\phi_{C1}$. This flow takes the operator out of this slice to its past or to its future. Alternatively, if we consider an operator near the boundary of the causal wedge $\phi_{C2}$, it is clear that, under modular flow, $[\phi_{C2}(s),\phi_2] \not=0$.