Bulk Locality and Quantum Error Correction in AdS/CFT

TL;DR

The paper proposes that bulk locality in AdS/CFT emerges from quantum error correction in the boundary CFT, with bulk operators acting as logical operations on a code subspace and AdS-Rindler reconstruction realized via operator algebra quantum error correction. It develops a framework of subregion-subregion duality, contrasting causal and entanglement wedges, and shows how bulk correlation and backreaction constrain recoverability, connecting these ideas to quantum secret sharing and the holographic entropy bound. The authors bridge global and wedge-based bulk reconstructions, introduce precise coding-theory statements, and discuss potential concrete realizations through tensor networks such as MERA. This coding-theory perspective clarifies the limits of bulk locality in holography and provides a tractable route to understanding the emergence of spacetime in AdS/CFT.

Abstract

We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.

Figures (10)

• Figure 1: $AdS_3$ reconstruction globally, and in an AdS-Rindler wedge.
• Figure 2: Coordinates for the AdS-Rindler wedge for $AdS_3$, shaded in blue. In this case we have $-\infty<x<\infty$.
• Figure 3: Three examples of $AdS_3$-Rindler reconstruction. Shown here is a top-down view of a bulk Cauchy slice whose boundary is $\Sigma$. On the left, the blue shaded region is the intersection of this Cauchy slice with the causal wedge for a CFT region $A$ that is the complement of a small boundary interval around the boundary point $Y$. In the center we have the point $x$ lying in the causal wedge of two different CFT regions, $A$ and $B$. $A$ borders the blue and green regions, while $B$ borders the green and yellow regions. The black circle segments are $\chi_A$, $\chi_B$, and $\chi_{A\cap B}$. On the right we have split $\Sigma$ into a union of three disjoint intervals, $A$, $B$, and $C$, and the circle segments are $\chi_A$, $\chi_B$, and $\chi_C$.
• Figure 4: Correcting for erasures in AdS/CFT. Bulk quantum information at point in the center is protected in the CFT against the erasure of the boundary of any one of the green regions, but bulk information at the point near the boundary is completely lost by an erasure of the boundary of the red region.
• Figure 5: Potentially troublesome bulk correlation. Here $\phi(x)$ is an operator that acts within the code subspace $\mathcal{H}_{\mathcal{C}}$, and which we thus expect can be represented as an operator on $A$. $\phi(y)$ we similarly expect to reconstructed on $\overline{A}$, but there is nonvanishing correlation between them in the ground state $|\Omega\rangle$. Using inequality \ref{['cirac']}, this correlation puts a lower bound on the accuracy with which we can view AdS/CFT as quantum error correction in the conventional sense of section \ref{['errcorsec']}.

Theorems & Definitions (2)

• Theorem
• proof