# Entanglement Wedge Reconstruction using the Petz Map

## TL;DR

This paper demonstrates that the Petz map alone suffices for entanglement wedge reconstruction in holography, simplifying previous methods and ensuring non-perturbatively small errors, with broad implications for quantum error correction.

## Contribution

It shows that the simpler Petz map can replace the twirled Petz map for entanglement wedge reconstruction, broadening its applicability and reducing complexity.

## Key findings

- Petz map achieves entanglement wedge reconstruction without twirling.
- Reconstruction errors are non-perturbatively small.
- General theorem extends Petz map use to quantum error correction.

## Abstract

At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedure over bulk and boundary modular time, and hence it can be somewhat intractable to evaluate in practice. We show that a much simpler channel, the Petz map, is sufficient for entanglement wedge reconstruction for any code space of fixed finite dimension - no twirling is required. Moreover, the error in the reconstruction will always be non-perturbatively small. From a quantum information perspective, we prove a general theorem extending the use of the Petz map as a general-purpose recovery channel to subsystem and operator algebra quantum error correction.