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Quantum Black Holes: Entropy and Entanglement on the Horizon

Etera R. Livine, Daniel R. Terno

TL;DR

This work develops simple Loop Quantum Gravity horizon models in which a Schwarzschild horizon is built from $n$ elementary patches of spin $s$, and analyzes both entropy and entanglement of the horizon. By mapping intertwiners to a random-walk problem, it obtains the horizon's microstate count and shows the entropy follows $S^{(s)}_n \sim n\ln(2s+1) - \tfrac{3}{2}\ln n + \cdots$, with the universal logarithmic correction coefficient $-\tfrac{3}{2}$. It also computes horizon entanglement for bipartitions, relates the logarithmic term to total correlations across the horizon, and discusses an evaporation interpretation via the unentangled fraction of qubits; a semiclassical time scale recovers Hawking-like evaporation. Beyond spin-$\tfrac{1}{2}$, the universal correction persists for general spin $s$, and the study develops an area renormalisation (coarse-graining) framework revealing a square-root scaling of macroscopic area with the number of patches. Overall, the paper forges a concrete bridge between quantum information concepts and quantum geometry in LQG, providing universal predictions and a renormalisation picture for black hole horizons.

Abstract

We are interested in black holes in Loop Quantum Gravity (LQG). We study the simple model of static black holes: the horizon is made of a given number of identical elementary surfaces and these small surfaces all behaves as a spin-s system accordingly to LQG. The chosen spin-s defines the area unit or area resolution, which the observer uses to probe the space(time) geometry. For s=1/2, we are actually dealing with the qubit model, where the horizon is made of a certain number of qubits. In this context, we compute the black hole entropy and show that the factor in front of the logarithmic correction to the entropy formula is independent of the unit s. We also compute the entanglement between parts of the horizon. We show that these correlations between parts of the horizon are directly responsible for the asymptotic logarithmic corrections. This leads us to speculate on a relation between the evaporation process and the entanglement between a pair of qubits and the rest of the horizon. Finally, we introduce a concept of renormalisation of areas in LQG.

Quantum Black Holes: Entropy and Entanglement on the Horizon

TL;DR

This work develops simple Loop Quantum Gravity horizon models in which a Schwarzschild horizon is built from elementary patches of spin , and analyzes both entropy and entanglement of the horizon. By mapping intertwiners to a random-walk problem, it obtains the horizon's microstate count and shows the entropy follows , with the universal logarithmic correction coefficient . It also computes horizon entanglement for bipartitions, relates the logarithmic term to total correlations across the horizon, and discusses an evaporation interpretation via the unentangled fraction of qubits; a semiclassical time scale recovers Hawking-like evaporation. Beyond spin-, the universal correction persists for general spin , and the study develops an area renormalisation (coarse-graining) framework revealing a square-root scaling of macroscopic area with the number of patches. Overall, the paper forges a concrete bridge between quantum information concepts and quantum geometry in LQG, providing universal predictions and a renormalisation picture for black hole horizons.

Abstract

We are interested in black holes in Loop Quantum Gravity (LQG). We study the simple model of static black holes: the horizon is made of a given number of identical elementary surfaces and these small surfaces all behaves as a spin-s system accordingly to LQG. The chosen spin-s defines the area unit or area resolution, which the observer uses to probe the space(time) geometry. For s=1/2, we are actually dealing with the qubit model, where the horizon is made of a certain number of qubits. In this context, we compute the black hole entropy and show that the factor in front of the logarithmic correction to the entropy formula is independent of the unit s. We also compute the entanglement between parts of the horizon. We show that these correlations between parts of the horizon are directly responsible for the asymptotic logarithmic corrections. This leads us to speculate on a relation between the evaporation process and the entanglement between a pair of qubits and the rest of the horizon. Finally, we introduce a concept of renormalisation of areas in LQG.

Paper Structure

This paper contains 15 sections, 7 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The spin-1/2 black hole model
  3. Entropy from random walk
  4. Entanglement
  5. About evaporation
  6. Black holes at the interplay between quantum gravity and quantum information
  7. The spin-1 black hole model and generalization
  8. Random walk analogy, entropy renormalisation and universal log correction
  9. Elementary scale and irreducibility
  10. On area renormalisation in LQG
  11. Conclusions and Outlook
  12. Entanglement of the black hole state
  13. Evaluation of the degeneracies ${}^s d_j^{(n)}$ for arbitrary spin
  14. Entanglement calculations
  15. Overview of the representations of the permutation group

Key Result

Proposition 1

The degeneracy coefficients $d^{(n)}_j$ in the decomposition of the tensor product $(V^{1/2})^{\otimes 2n}$ into the irreducible spin-$j$ representations equal the number of returns ${\rm RWM}(j)$ to the spot $j$ after $2n$ iterations of the random walk of step $1/2$ with a mirror in $j=0$ and start where ${\rm RW}(j)$ is the number of returns to the spot $j$ after $2n$ iterations of the random wa

Figures (1)

  • Figure 1: Entanglement between $2k$ and $2n-2k$ qubits for $2k=2$ (left) and $2k=4$ (right). The entanglement of the two qubit block for $n=4$ is 0.706 ebits, and the entanglement of the four qubit block for $n=6$ is 0.9429 ebits.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7