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Nonperturbative gravity corrections to bulk reconstruction

Elliott Gesteau, Monica Jinwoo Kang

TL;DR

The paper develops an operator-algebraic framework to incorporate nonperturbative gravitational corrections into bulk reconstruction, accommodating either finite or infinite boundary Hilbert spaces. By combining conditional expectations, modular theory, Araki–relative entropy, and the CKLT privacy/correctability duality, it links bulk and boundary relative entropies under a modified-JLMS condition to achieve state-independent approximate recovery inside the reconstruction wedge, with a cb-norm bound $\|\mathcal{E}\circ\mathcal{R}-\mathrm{Id}\|_{cb} \leq 2(2\varepsilon)^{1/4}$. It then extends the analysis to state-dependent recovery beyond the wedge via $\alpha$-bits and the universal twirled Petz map, highlighting the different guarantees (cb-norm vs operator norm) and showing how nonperturbative $G_N$ corrections shape the reconstruction landscape. The discussion connects these results to quantum islands, the information paradox, and gravitational path integrals, suggesting a robust, rigorously formulated bridge between holography, AQFT, and semiclassical gravity with potential applications to evaporating black holes and island prescriptions.

Abstract

We introduce a new algebraic framework for understanding nonperturbative gravitational aspects of bulk reconstruction with a finite or infinite-dimensional boundary Hilbert space. We use relative entropy equivalence between bulk and boundary with an inclusion of nonperturbative gravitational errors, which give rise to approximate recovery. We utilize the privacy/correctability correspondence to prove that the reconstruction wedge, the intersection of all entanglement wedges in pure and mixed states, manifestly satisfies bulk reconstruction. We explicitly demonstrate that local operators in the reconstruction wedge of a given boundary region can be recovered in a state-independent way for arbitrarily large code subspaces, up to nonperturbative errors in $G_N$. We further discuss state-dependent recovery beyond the reconstruction wedge and the use of the twirled Petz map as a universal recovery channel. We discuss our setup in the context of quantum islands and the information paradox.

Nonperturbative gravity corrections to bulk reconstruction

TL;DR

The paper develops an operator-algebraic framework to incorporate nonperturbative gravitational corrections into bulk reconstruction, accommodating either finite or infinite boundary Hilbert spaces. By combining conditional expectations, modular theory, Araki–relative entropy, and the CKLT privacy/correctability duality, it links bulk and boundary relative entropies under a modified-JLMS condition to achieve state-independent approximate recovery inside the reconstruction wedge, with a cb-norm bound . It then extends the analysis to state-dependent recovery beyond the wedge via -bits and the universal twirled Petz map, highlighting the different guarantees (cb-norm vs operator norm) and showing how nonperturbative corrections shape the reconstruction landscape. The discussion connects these results to quantum islands, the information paradox, and gravitational path integrals, suggesting a robust, rigorously formulated bridge between holography, AQFT, and semiclassical gravity with potential applications to evaporating black holes and island prescriptions.

Abstract

We introduce a new algebraic framework for understanding nonperturbative gravitational aspects of bulk reconstruction with a finite or infinite-dimensional boundary Hilbert space. We use relative entropy equivalence between bulk and boundary with an inclusion of nonperturbative gravitational errors, which give rise to approximate recovery. We utilize the privacy/correctability correspondence to prove that the reconstruction wedge, the intersection of all entanglement wedges in pure and mixed states, manifestly satisfies bulk reconstruction. We explicitly demonstrate that local operators in the reconstruction wedge of a given boundary region can be recovered in a state-independent way for arbitrarily large code subspaces, up to nonperturbative errors in . We further discuss state-dependent recovery beyond the reconstruction wedge and the use of the twirled Petz map as a universal recovery channel. We discuss our setup in the context of quantum islands and the information paradox.
Paper Structure (23 sections, 11 theorems, 80 equations, 1 table)

This paper contains 23 sections, 11 theorems, 80 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: algebraic perspective
  3. Conditional expectations
  4. Tomita--Takesaki theory and modular flow
  5. The Connes cocycle and Araki's relative entropy
  6. The Petz map from Tomita--Takesaki theory
  7. Exact infinite-dimensional entanglement wedge reconstruction
  8. Entanglement wedge reconstruction in the code subspace formalism
  9. Link to conditional expectations and Takesaki's theorem
  10. Limitations of the exact approach
  11. Private and correctable algebras and complementary recovery
  12. The completely bounded norm, privacy, and correctability
  13. A duality between privacy and correctability
  14. State-independent approximate recovery in the reconstruction wedge
  15. Setup
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathcal{H}_{code}$ and $\mathcal{H}_{phys}$ be two Hilbert spaces, $V:\mathcal{H}_{code}\rightarrow \mathcal{H}_{phys}$ be an isometry, and $\mathcal{H}^\ast_{code}$ be any finite-dimensional Hilbert space of dimension smaller or equal to the one of $\mathcal{H}_{code}$. Let $M_{A}$ be a von N be the reconstruction wedge von Neumann algebra on $\mathcal{H}_{code}$, and suppose that $M_{a^\pr

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: Jones-vNalg, p.94
  • Definition 2.8
  • Definition 2.9
  • ...and 18 more