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On Distinguishability of Black Hole Microstates

Ning Bao, Hirosi Ooguri

TL;DR

The paper analyzes how well black hole microstates in AdS/CFT can be distinguished when measurements are limited to a boundary subregion, using the Holevo information to bound accessible information. It provides exact results in AdS_3/CFT_2 showing a sharp onset of distinguishability at a critical subregion size and eventual perfect distinguishability beyond it, with a temperature-dependent ℓ_crit. In higher dimensions, it establishes general bounds that force zero distinguishability below a threshold and maximal distinguishability beyond, while leaving the intermediate regime less constrained. The findings tie the boundary region size to bulk reconstruction intuitions from entanglement wedges and bit threads and suggest ways to reduce the required region via disconnected subregions or alternative code subspaces, pointing to intriguing links between microstate structure and holographic information flow.

Abstract

We use the Holevo information to estimate distinguishability of microstates of a black hole in anti-de Sitter space by measurements one can perform on a subregion of a Cauchy surface of the dual conformal field theory. We find that microstates are not distinguishable at all until the subregion reaches a certain size and that perfect distinguishability can be achieved before the subregion covers the entire Cauchy surface. We will compare our results with expectations from the entanglement wedge reconstruction, tensor network models, and the bit threads interpretation of the Ryu-Takayanagi formula.

On Distinguishability of Black Hole Microstates

TL;DR

The paper analyzes how well black hole microstates in AdS/CFT can be distinguished when measurements are limited to a boundary subregion, using the Holevo information to bound accessible information. It provides exact results in AdS_3/CFT_2 showing a sharp onset of distinguishability at a critical subregion size and eventual perfect distinguishability beyond it, with a temperature-dependent ℓ_crit. In higher dimensions, it establishes general bounds that force zero distinguishability below a threshold and maximal distinguishability beyond, while leaving the intermediate regime less constrained. The findings tie the boundary region size to bulk reconstruction intuitions from entanglement wedges and bit threads and suggest ways to reduce the required region via disconnected subregions or alternative code subspaces, pointing to intriguing links between microstate structure and holographic information flow.

Abstract

We use the Holevo information to estimate distinguishability of microstates of a black hole in anti-de Sitter space by measurements one can perform on a subregion of a Cauchy surface of the dual conformal field theory. We find that microstates are not distinguishable at all until the subregion reaches a certain size and that perfect distinguishability can be achieved before the subregion covers the entire Cauchy surface. We will compare our results with expectations from the entanglement wedge reconstruction, tensor network models, and the bit threads interpretation of the Ryu-Takayanagi formula.

Paper Structure

This paper contains 5 sections, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Holevo Information
  3. Holevo Information in AdS$_3$/CFT$_2$
  4. Holevo Information in AdS$_{d+1}$/CFT$_d$
  5. Discussion

Figures (3)

  • Figure 1: The Holevo information $\chi(A)$ in AdS$_3$/CFT$_2$, as a function of the radius $\ell_A$ of the subregion $A$. Note that $\chi(A)$ is identically equal to zero until $A$ covers one half of the Cauchy surface of CFT. Above this point, $\chi(A)$ increases monotonically until it achieves the maximum value, which is the Bekenstein-Hawking entropy $S_{{\rm BH}}$ of the black hole, at $\ell_{{\rm crit}}$ defined in section 2.
  • Figure 2: The Ryu-Takayanagi surfaces for region $A$ for the thermal ensemble $\rho = \sum_i p_i \rho_i$.
  • Figure 3: Two possible Ryu-Takayanagi configurations for $A_1 \cup A_2$. The total area of the surfaces on the left is $S(\rho_{A_1\cup A_2})=S(\rho_{\bar{A}_2})+S(\rho_{A_1\cup \bar{A}_2 \cup A_2})$, while that on the right is $S(\rho_{A_1 \cup A_2})=S(\rho_{A_1})+S(\rho_{A_2})$.