Entanglement Wedge Reconstruction using the Petz Map

TL;DR

The paper addresses entanglement wedge reconstruction in AdS/CFT by recasting it as an operator-algebra quantum error correction problem. It proves that, for fixed finite-dimensional code spaces, the Petz map suffices as a recovery channel without the need for twirling, with nonperturbatively small reconstruction error. A general theorem is developed that extends Petz-map recovery to subsystem and operator-algebra QEC, providing a concrete bound on reconstruction accuracy. The work clarifies the applicability limits related to code-space size and outlines practical challenges for computing the Petz map in holographic settings. Overall, it offers a more tractable, nonperturbative tool for bulk reconstruction and deepens the connection between quantum information and holography.

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.

Figures (2)

• Figure 1: An operator $\phi_{a}$, acting on the entanglement wedge $a$ of $A = A_1 \cup A_2$, can be reconstructed on the boundary region $A$ by the map $\mathcal{D}^\dagger_{A}: M_{a} \rightarrow M_{A}$. The solid interior curves represent the RT surface of $A$ and the entire shaded region forms the entanglement wedge $a$ (restricted to a single timeslice). The darker gray areas are the entanglement wedges of $A_1$ and $A_2$ individually, and also together form the causal wedge of $A$. Since the operator $\phi_a$ is not in the causal wedge of $A$, we cannot reconstruct it simply by using the bulk and boundary equations of motion; the more sophisticated machinery of quantum error correction is required. Moreover, $\phi_a$ can only be reconstructed on $A = A_1 \cup A_2$; neither $A_1$ nor $A_2$ alone contains any information about $\phi_a$.
• Figure 2: In the Heisenberg picture, $\mathcal{M}_a\stackrel{i}{\hookrightarrow}\mathcal{B}(\mathcal{H}_\text{code})$ and $\mathcal{M}_A\stackrel{i}{\hookrightarrow}\mathcal{B}(\mathcal{H}_{CFT})$ are von Neumann subalgebras acting on the code space $\mathcal{H}_\text{code}$ and CFT Hilbert space $\mathcal{H}_{CFT}$ respectively. The Heisenberg channel $\mathcal{J}^\dagger = J^\dagger(\cdot)J$ maps boundary observables to their projection in the code space. The task of entanglement wedge reconstruction is to find a Heisenberg decoding channel $\mathcal{D}^\dagger : \mathcal{M}_a \to \mathcal{M}_A$ that maps bulk observables $\phi_a$ in $\mathcal{M}_a$ to boundary observables $\Phi_A$ in $\mathcal{M}_A$. When projected into the code space using $\mathcal{J}^\dagger$, the boundary observable $\Phi_A$ should reproduce the original bulk observable $\phi_a$. In the Schrödinger picture, the directions of all channels are reversed. The channel $\mathcal{J}$ now maps bulk states to the corresponding boundary states. The Heisenberg channels $i_a$ and $i_A$, which embed the von Neumann subalgebras $\mathcal{M}_a$ and $\mathcal{M}_A$ into the larger algebras of observables $\mathcal{B}(\mathcal{H}_\text{code})$ and $\mathcal{B}(\mathcal{H}_{CFT} )$, are the adjoints of the restriction maps onto $S(\mathcal{M}_a)$ and $S(\mathcal{M}_A)$ respectively. Finally, the decoding channel $\mathcal{D}: S(\mathcal{M}_A) \to S(\mathcal{M}_a)$ satisfies $\mathcal{D}[\mathcal{J}(\cdot)_A] \approx (\cdot)_a$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2: Barnum-Knill barnum2002reversing
• Proposition 2.1
• proof