Precursors, Gauge Invariance, and Quantum Error Correction in AdS/CFT

TL;DR

The paper investigates how bulk locality emerges in AdS/CFT by analyzing precursor non-uniqueness arising from boundary gauge freedom and its relation to quantum error correction. It advances an improved toy model that preserves bulk dynamics and demonstrates that a pure-gauge freedom in the smearing function can localize a bulk operator's boundary representation within a single Rindler wedge. It then shows, via entanglement-based mapping, that the same localization can be achieved using CFT entanglement, provided the bulk point lies in the corresponding entanglement wedge, and contrasts this with localization attempts in the wrong wedge which fail hermiticity and are UV-divergent. The work clarifies a deep connection between boundary gauge invariance and QEC in holography, highlighting entanglement wedge conditions as essential for consistent bulk reconstruction and suggesting avenues for extending these ideas to more complex boundary regions.

Abstract

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.

Figures (6)

• Figure 1: Global AdS$_3$, showing the light-cone for a bulk point $x$, which defines a spacelike separated region on the boundary (shaded, yellow online). The corresponding non-local boundary operator is defined à la \ref{['eq:smearing']} as an integral over this region. The local CFT operators can be time-evolved to a single Cauchy slice (shaded, pink online). This is illustrated schematically for points $A$ and $B$, where we've indicated the null lines on the boundary. In our model, the boundary operators factorize along the light-cone directions, and are trivially evolved to bilocals at the $t=0$ Cauchy slice.
• Figure 2: Rindler plane in light-cone coordinates, indicating the phase changes in the mode functions \ref{['eq:boundarylimit']} when crossing the branch cuts at $x_\pm=0$. The sign choice is arbitrary, but must be consistent across all four quadrants in order to obtain a pure-gauge Poincaré mode. We refer to these quadrants throughout as the northern (N), southern (S), eastern (E), and western (W) wedges, labelled in the obvious manner.
• Figure 3: Left: $t=0$ Cauchy slice, with a bulk point $x$ displaced slightly into the eastern Rindler wedge (shaded). Right: time-evolution of local boundary operators to bilocals at $t=0$. The dashed axes show the light-cone of the bulk point; the future and past singularities in the smearing function are indicated by the dotted lines. Point $A$ falls entirely within the eastern wedge, while one leg of $B$, and both legs of $C$, must be mapped into the east using using the entanglement of the Rindler vacuum. Note that with the bulk point as shown, at most one singular leg must be mapped, but this potential divergence is exactly cancelled by a decaying exponential arising from \ref{['eq:MinkEnt']}, so the resulting expression remains well-defined.
• Figure 4: Left: $t=0$ Cauchy slice, with a bulk point $x$ displaced slightly into east as before, but reconstruction attempted in the western (wrong) wedge Rindler wedge (shaded). Right: time-evolution of local boundary operators to bilocals at $t=0$. Note that while $A$ and $B$ can be mapped without difficulty, as discussed in the previous section, there are now points like $C$ with two divergent legs, both of which must be mapped into the western wedge. This is one more exponential in momentum than we are capable of taming, and thus localization of the associated bulk point fails.
• Figure 5: Entanglement wedge for a disconnected region, shaded. If the region is sufficiently small (left), the bulk point labelled $x$ will not be included, and hence the shaded boundary region contains no information about it. However, as the boundary region is increased, the bulk geodesics that define the entanglement wedge eventually transition to a new global minimum (right), whereupon the shaded boundary region abruptly gains information about the given bulk point. Intuitively, one needs "enough" of the boundary to reconstruct the bulk.