Bulk locality from modular flow

TL;DR

This work develops a modular-flow-based framework to reconstruct bulk operators inside the entanglement wedge from boundary subregion data. By exploiting the equivalence of bulk and boundary modular flows, it derives a smearing-function construction that expresses bulk fields as integrals over boundary operators with fixed modular frequency, determined by boundary correlators. In the RT-surface limit, a tractable zero-mode formula localizes reconstruction to the surface, enabling explicit RT-based expressions. The paper also discusses how 1/N interactions, backreaction, and state dependence affect this reconstruction, situating the results within the broader context of bulk locality, entanglement wedge reconstruction, and quantum error correction.

Abstract

We study the reconstruction of bulk operators in the entanglement wedge in terms of low energy operators localized in the respective boundary region. To leading order in $N$, the dual boundary operators are constructed from the modular flow of single trace operators in the boundary subregion. The appearance of modular evolved boundary operators can be understood due to the equality between bulk and boundary modular flows and explicit formulas for bulk operators can be found with a complete understanding of the action of bulk modular flow, a difficult but in principle solvable task. We also obtain an expression when the bulk operator is located on the Ryu-Takayanagi surface which only depends on the bulk to boundary correlator and does not require the explicit use of bulk modular flow. This expression generalizes the geodesic operator/OPE block dictionary to general states and boundary regions.

Figures (2)

• Figure 1: (left) We reconstruct the zero mode using the "Cauchy slice" $\Sigma_H$ with a small null segment close to the RT surface. (right) We should really take a limit of space-like slices that approaches $\Sigma_H$.
• Figure 2: $G^{2-int}_0(\frac{1-y}{2},y+z (1-y))$ as a function of $z\in (0,1)$ for $\Delta=1$ and $y=0.1$ (which corresponds to a cross ratio of $x=0.65$). The large interval flow corresponds $G_0^{(-y,y)}(x_1,x_2)$ and the one interval flow to that of $G_0^{(y,1)}(x_1,x_2)$. We see that when $x_2$ is close to the endpoints of the interval, the flow is roughly that of the smaller interval, as we expect from locality. For $z>1/2$, the modular flow is well approximated by that of the large interval.