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An algebraic description of the Page transition

Haocheng Zhong

TL;DR

The paper develops an algebraic framework for the Page transition in black hole evaporation by recasting the Page curve as a phase transition in channel recoverability within approximate quantum error correction with complementary recovery. It generalizes to infinite dimensions using algebraic relative entropy, covering type III factors, and furnishes explicit relative-entropy probes for type I/II to signal the Page time. Central contributions include a reconstruction-theorem-based picture of island emergence, a detailed algebraic treatment for Type I/II factors with ΔS1/ΔS2-based probes, and a discussion of extensions to Type III factors via crossed products. The work provides a principled, algebraic mechanism linking quantum error correction, island formulas, and gravitational entropy, enabling a type-agnostic diagnostic of information recovery in black hole evaporation.

Abstract

In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.

An algebraic description of the Page transition

TL;DR

The paper develops an algebraic framework for the Page transition in black hole evaporation by recasting the Page curve as a phase transition in channel recoverability within approximate quantum error correction with complementary recovery. It generalizes to infinite dimensions using algebraic relative entropy, covering type III factors, and furnishes explicit relative-entropy probes for type I/II to signal the Page time. Central contributions include a reconstruction-theorem-based picture of island emergence, a detailed algebraic treatment for Type I/II factors with ΔS1/ΔS2-based probes, and a discussion of extensions to Type III factors via crossed products. The work provides a principled, algebraic mechanism linking quantum error correction, island formulas, and gravitational entropy, enabling a type-agnostic diagnostic of information recovery in black hole evaporation.

Abstract

In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
Paper Structure (8 sections, 9 theorems, 63 equations, 3 figures)

This paper contains 8 sections, 9 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The reconstruction theorem with complementary recovery
  4. Basics of von Neumann algebra
  5. Algebraic entropies
  6. Page transition via AQEC with complementary recovery
  7. An algebraic description in factors of type I/II
  8. Discussions

Key Result

Theorem 1.1

Given a finite-dimensional Hilbert space $\mathcal{H}$ (called the physical space) which is decomposed into two parts $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$ and a code space $\mathcal{H}_{code}$ with condition $dim\mathcal{H}_{code}\leq\left| A \right|$, the isometric encoding $V: where $s(\mathcal{H}_{(\cdot)})$ denotes the set of all density operators in $\mathcal{H}_{(\cdot)}

Figures (3)

  • Figure 1: The Penrose diagram of an evaporating black hole. A Cauchy slice (blue solid curve) is divided into three regions $\Sigma_{\text{Island}},\Sigma_{X},\Sigma_{\text{Rad}}$ by a codimension-2 surface $X$ (orange dot) and a cutoff surface (red dotted line).
  • Figure 2: A sketch of the Page curve (red line) for the entropy of Hawking radiation, which is upper bounded by the two saddle-point solutions: the vanishing island solution $S_{\text{Rad}}^{\text{no-island}}$ (blue line) and the non-vanishing solution $S_{\text{Rad}}^{\text{island}}$ (yellow line). The transition of the two dominant solutions happens at the Page time $t_P$, which is call the Page transition.
  • Figure 3: $\Delta S$ (green line) is defined as the difference between $S_{\text{Rad}}^{\text{island}}$ and $S_{\text{Rad}}$, which monotonically decreases until the Page time $t_P$, i.e. we have $\Delta S(t_2)\leq\Delta S(t_1)$ for $t_1\leq t_2\leq t_P$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 11 more