Large Breakdowns of Entanglement Wedge Reconstruction

TL;DR

This work shows that the reconstruction wedge (RW) — the boundary-accessible bulk region consistent across a code subspace — can be parametrically smaller than the entanglement wedge (EW) even when backreaction is small. By constructing simple dustball code subspaces in AdS/CFT, the authors demonstrate macroscopic separations between RW and EW, including cases where the quantum extremal surface diverges macroscopically from the Ryu-Takayanagi surface. They illustrate that subspace-dependent bulk reconstruction is essential, independent of horizons, and that the position of the quantum extremal surface can be significantly different from the RT surface. The results provide a tractable, horizon-free setting to study state-dependent reconstruction and offer insight into the geometry of generalized entropy and potential implications for black hole interiors in holography.

Abstract

We show that the bulk region reconstructable from a given boundary subregion --- which we term the reconstruction wedge --- can be much smaller than the entanglement wedge even when backreaction is small. We find arbitrarily large separations between the reconstruction and entanglement wedges in near-vacuum states for regions close to an entanglement phase transition, and for more general regions in states with large energy (but very low energy density). Our examples also illustrate situations for which the quantum extremal surface is macroscopically different from the Ryu-Takayanagi surface.

Figures (4)

• Figure 1: In blue is the reconstruction wedge of $A$ (blue boundary region). In red is the reconstruction wedge of $\bar{A}$ (red boundary region). Both $A$ and $\bar{A}$ are defined by dividing the boundary into four equal pieces and taking the union of two disconnected components. The quantum extremal surface of each connected component is depicted as thin dashed gray lines. (a) Neither reconstruction wedge includes the white region (the "middle") in the two-dimensional code subspace corresponding to the state Qubit 1 on top of the vacuum. (b) Adding a Bell pair between the entanglement wedge of one of $A$'s connected components and the middle causes the reconstruction wedge of $A$ to include the middle.
• Figure 2: In blue is the reconstruction wedge of $A$ (blue boundary region). In red is the reconstruction wedge of $\bar{A}$ (red boundary region). $A$ can reconstruct Qubit $1$ through Qubit $k$ provided there are at least $k$ Bell pairs entangling the middle and the entanglement wedge of one of the connected components of $A$.
• Figure 3: The Penrose diagram of the Oppenheimer-Snyder collapse. In the range $r < R$, the metric is hyperbolic FRW. For $r > R$, it is BTZ. The horizons do not reach the $t = 0$ surface.
• Figure 4: In blue is the entanglement wedge of $A$ (blue boundary region). In red is the entanglement wedge of $\bar{A}$ (red boundary region). $A$ is defined by dividing the boundary into $2n+1$ equal pieces and taking the union of every-other connected component. The quantum extremal surface of each connected component is depicted as thin dashed gray lines. The dustball is the shaded circle in the middle. (a) Neither $A$'s nor $\bar{A}$'s entanglement wedge includes the dustball when it is in the maximally-mixed state in our code subspace. Therefore neither's reconstruction wedge includes the dustball. (b) $A$'s entanglement wedge includes the dustball when it is in a pure state. Therefore the entanglement wedge is much larger than the reconstruction wedge, in this state.