  A puzzling aspect of the AdS/CFT correspondence is that a single bulk
operator can be mapped to multiple different boundary operators, or precursors.
By improving upon a recent model of Mintun, Polchinski, and Rosenhaus, we
demonstrate explicitly how this ambiguity arises in a simple model of the field
theory. In particular, we show how gauge invariance in the boundary theory
manifests as a freedom in the smearing function used in the bulk-boundary
mapping, and explicitly show how this freedom can be used to localize the
precursor in different spatial regions. We also show how the ambiguity can be
understood in terms of quantum error correction, by appealing to the
entanglement present in the CFT. The concordance of these two approaches
suggests that gauge invariance and entanglement in the boundary field theory
are intimately connected to the reconstruction of local operators in the dual
spacetime.
