Bulk-Boundary Duality, Gauge Invariance, and Quantum Error Corrections

Recently, Almheiri, Dong, and Harlow have argued that the localization of bulk information in a boundary dual should be understood in terms of quantum error correction. We show that this structure appears naturally when the gauge invariance of the boundary theory is incorporated. This provides a new understanding of the nonuniqueness of the bulk fields (precursors). It suggests a close connection between gauge invariance and the emergence of spacetime.

Figure 1

Focusing on a single time slice of anti–de Sitter (AdS) space, bulk microcausality requires the operator $\mathrm{\Phi }(0)$ to commute with all spacelike-separated local operators. By the holographic dictionary, the boundary limits of such bulk operators are local CFT operators $\mathcal{O}(x)$, which must thus commute with $\mathrm{\Phi }(0)$.

Figure 2

A bulk operator (the black dot) can be represented as a CFT precursor consisting of bilocals (the red arcs ending with $\mathsf{x}$’s) stretching over the entire boundary. A different construction represents the bulk operators in the right bulk Rindler wedge as a precursor with support on only the right half of the boundary. Thus, bilocals localized on the left half of the CFT, as well as bilocals stretching from the left to the right half, can be removed from the precursor. This is due to the additional freedom that comes with considering the action of the precursors only on gauge-invariant states.