Scanning tunneling macroscopy, black holes and AdS/CFT bulk locality

We establish resolution bounds on reconstructing a bulk field from boundary data on a timelike hypersurface. If the bulk only supports propagating modes, reconstruction is complete. If the bulk also supports evanescent modes, local reconstruction is not achievable unless one has exponential precision in knowledge of the boundary data. Without exponential precision, for a Minkowski bulk, one can reconstruct a spatially coarse-grained bulk field, but only out to a depth set by the coarse-graining scale. For an asymptotically AdS bulk, reconstruction is limited to a spatial coarse-graining proper distance set by the AdS scale. AdS black holes admit evanescent modes. We study the resolution bound in the large AdS black hole background and provide a dual CFT interpretation. Our results demonstrate that, if there is a black hole of any size in the bulk, then sub-AdS bulk locality is no longer well-encoded in boundary data in terms of local CFT operators. Specifically, in order to probe the bulk on sub-AdS scales using only boundary data in terms of local operators, one must either have such data to exponential precision or make further assumptions about the bulk state.