Precursors, gauge invariance, and quantum error correction in AdS/CFT

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.

Tensor chain and constraints in tensor networks

“…Based on sub system duality, operators in the bulk can be reconstructed by the operators supported on a sub system of the boundary [10][11][12][13]. In other words, there are subspaces of the Hilbert space in the bulk which can still be reconstructed even if an amount of information on the boundary is erased [14][15][16][17]. Great progress has also been made in the realization of QEC by virtue of tensor networks [15,16,[18][19][20][21].…”

A toy model of black hole complementarity

“…It was proposed in [24] that this region is dual to bulk region called the "entanglement wedge", and we refer the reader to [25][26][27] for some recent work on this proposal. The Reeh-Schlieder theorem…”

“…This is because we may write the action of an operator on the vacuum at finite N as 27) where A 0 a |0 is the state that we would have obtained at infinite N, and the remaining terms are perturbative corrections. Now, we see that since all the terms multiplying the powers of latter cannot happen, we conclude that perturbatively the vacuum remains a separating vector.…”

Precursors and BRST symmetry

“…The piece quadratic in the α ′ s in (4.7) is exactly the ambiguity needed to localize the precursor in the CFT to leading order in N , as was shown in detail in [9]. One can now also see that one generically needs a four-parameter ambiguity if we want to be able to set K 2 in (4.5) to zero in certain regions.…”