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Teleportation=Translation: Continuous recovery of black hole information

Jeongwon Ho

TL;DR

The paper addresses how to recover black hole information via a continuous, unitary flow that links discrete algebraic teleportation to geometric modular flow.It achieves this by lifting Type III algebras to a Haagerup–Kosaki crossed-product semifinite setting, constructing a canonical L^p interpolation, and defining a self-adjoint generator ̃G for the teleportation flow.A key result is the exact operator identity ̃G = 2P, where P is the modular Hamiltonian difference, proven through half-sided modular inclusion theorems, modular perturbation theory, and Nelson’s analytic-vector framework, with robustness from non-commutative L^p geometry.The work provides an analytic mechanism to study emergent geometry from quantum information, offers a correlation-function test in holographic models, and outlines how back-reaction could be incorporated in this algebraic-geometric framework.

Abstract

The \textit{Teleportation=Translation} conjecture posits that the recovery of information from a black hole is dual to a geometric translation in the emergent spacetime. In this paper, we establish this equivalence by constructing a continuous family of unitaries that bridges the discrete algebraic teleportation protocol and modular flow. We resolve the obstruction of dynamic idempotency, inherent in Type III von Neumann algebras, by employing the Haagerup-Kosaki crossed-product construction. This lift to the semifinite envelope yields a canonical, dynamically consistent path whose unique self-adjoint generator $\tilde{G}$ is proven to be exactly twice the modular Hamiltonian difference, $\tilde{G}=2(K_{\tilde{\mathcal{M}}}-K_{\tilde{\mathcal{N}}})$. We establish this identity as a closed operator equivalence using Nelson's analytic vector theorem and quantify its structural robustness via Kosaki's non-commutative $L^p$ theory. Our results provide a concrete analytic mechanism for probing emergent geometry from quantum information, offering a kinematic framework naturally extendable to include gravitational back-reaction.

Teleportation=Translation: Continuous recovery of black hole information

TL;DR

The paper addresses how to recover black hole information via a continuous, unitary flow that links discrete algebraic teleportation to geometric modular flow.It achieves this by lifting Type III algebras to a Haagerup–Kosaki crossed-product semifinite setting, constructing a canonical L^p interpolation, and defining a self-adjoint generator ̃G for the teleportation flow.A key result is the exact operator identity ̃G = 2P, where P is the modular Hamiltonian difference, proven through half-sided modular inclusion theorems, modular perturbation theory, and Nelson’s analytic-vector framework, with robustness from non-commutative L^p geometry.The work provides an analytic mechanism to study emergent geometry from quantum information, offers a correlation-function test in holographic models, and outlines how back-reaction could be incorporated in this algebraic-geometric framework.

Abstract

The \textit{Teleportation=Translation} conjecture posits that the recovery of information from a black hole is dual to a geometric translation in the emergent spacetime. In this paper, we establish this equivalence by constructing a continuous family of unitaries that bridges the discrete algebraic teleportation protocol and modular flow. We resolve the obstruction of dynamic idempotency, inherent in Type III von Neumann algebras, by employing the Haagerup-Kosaki crossed-product construction. This lift to the semifinite envelope yields a canonical, dynamically consistent path whose unique self-adjoint generator is proven to be exactly twice the modular Hamiltonian difference, . We establish this identity as a closed operator equivalence using Nelson's analytic vector theorem and quantify its structural robustness via Kosaki's non-commutative theory. Our results provide a concrete analytic mechanism for probing emergent geometry from quantum information, offering a kinematic framework naturally extendable to include gravitational back-reaction.

Paper Structure

This paper contains 18 sections, 10 theorems, 35 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Discrete Protocol: Canonical Shift
  3. Preliminaries: Tomita–Takesaki Theory
  4. The Canonical Shift Protocol
  5. From a Discrete Protocol to a Continuous Path
  6. The Physical Requirement: Dynamical Idempotency
  7. Path A: Pedagogical (CP-Map) Interpolation
  8. Path B: The Canonical Haagerup–Kosaki Interpolation
  9. The Unitary Path and its Generator
  10. ALGEBRAIC VERIFICATION: $\tilde{G} = 2P$
  11. Verification in Half-Sided Modular Inclusions
  12. Rigorous Derivation via Modular Perturbation Theory
  13. Stability Quantification via Non-Commutative $L^p$ Spaces
  14. Correlation-Function Test of the Conjecture
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 3.2

(Idempotent CP maps are conditional expectations) Let $\mathcal{M}$ be a von Neumann algebra and $\omega$ a faithful normal state on $\mathcal{M}$. Let $E: \mathcal{M} \to \mathcal{M}$ be a normal, unital, completely positive (CP) map satisfying $E^2 = E$ and $\omega \circ E = \omega$. Then $\mathca

Theorems & Definitions (26)

  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 3.5: Canonical Interpolation Path
  • Theorem 3.6: Properties of the Canonical Path
  • proof
  • Remark 3.7: Modular Covariance and Core Preservation
  • ...and 16 more