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

Eric Mintun, Joseph Polchinski, Vladimir Rosenhaus

TL;DR

The paper addresses how bulk locality in a holographic boundary theory emerges and how bulk information can be localized in different boundary regions. It shows that gauge invariance in the boundary naturally yields a quantum error correction–like structure for precursors, without requiring a separate code subspace. Through a discrete three-site model and a continuum free-field construction, it demonstrates gauge-invariant readout from subsets of boundary data and a gauge-driven equivalence between different precursor representations (Poincaré and Rindler). The findings illuminate the deep connection between gauge symmetry and spacetime emergence, and they point to a broader role for gauge principles in quantum information theory applied to holography.

Abstract

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 non-uniqueness of the bulk fields (precursors). It suggests a close connection between gauge invariance and the emergence of spacetime.

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

TL;DR

The paper addresses how bulk locality in a holographic boundary theory emerges and how bulk information can be localized in different boundary regions. It shows that gauge invariance in the boundary naturally yields a quantum error correction–like structure for precursors, without requiring a separate code subspace. Through a discrete three-site model and a continuum free-field construction, it demonstrates gauge-invariant readout from subsets of boundary data and a gauge-driven equivalence between different precursor representations (Poincaré and Rindler). The findings illuminate the deep connection between gauge symmetry and spacetime emergence, and they point to a broader role for gauge principles in quantum information theory applied to holography.

Abstract

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 non-uniqueness of the bulk fields (precursors). It suggests a close connection between gauge invariance and the emergence of spacetime.

Paper Structure

This paper contains 5 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A discrete model
  3. Free-field precursors
  4. Conclusion
  5. Appendix

Figures (2)

  • Figure 1: Focusing on a single time slice of AdS, bulk microcausality requires the operator $\Phi(0)$ 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 $\Phi(0)$.
  • Figure 2: A bulk operator (black dot) can be represented as a CFT precursor consisting of bilocals (red arcs ending on 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.