Explicit reconstruction of the entanglement wedge

TL;DR

This paper develops an explicit mode sum method for reconstructing the entanglement wedge in AdS/CFT, transforming a complex wave equation problem into a manageable matrix inversion task, advancing understanding of bulk-boundary encoding.

Contribution

It introduces a novel explicit reconstruction procedure for the entanglement wedge using mode sums, extending previous approaches to include boundary subregions.

Findings

Reconstruction reduces to matrix inversion, simplifying the problem.

Method applicable to scalar fields in AdS/CFT.

Provides a practical way to explicitly reconstruct bulk regions.

Abstract

The problem of how the boundary encodes the bulk in AdS/CFT is still a subject of study today. One of the major issues that needs more elucidation is the problem of subregion duality; what information of the bulk a given boundary subregion encodes. Although the proof given by Dong, Harlow, and Wall states that the entanglement wedge of the bulk should be encoded in boundary subregions, no explicit procedure for reconstructing the entanglement wedge was given so far. In this paper, mode sum approach to obtaining smearing functions for a single bulk scalar is generalised to include bulk reconstruction in the entanglement wedge of boundary subregions. It is generally expectated that solutions to the wave equation on a complicated coordinate patch are needed, but this hard problem has been transferred to a less hard but tractable problem of matrix inversion.