Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality

In this Letter we prove a simple theorem in quantum information theory, which implies that bulk operators in the anti–de Sitter/conformal field theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion $A$, provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a quantum error correcting code.

Figure 1

Factorizing the bulk and boundary on a time slice. The entanglement wedge ${\mathcal{E}}_A$ is shaded. For simplicity we have shown a connected boundary region $A$, although this might not be the case.