# Demonstration of the Hayden-Preskill protocol via mutual information

## TL;DR

This paper constructs a spin-1/2 particle system to mimic black holes and demonstrates how rapid mixing enables the Hayden-Preskill protocol to allow information to escape from the black hole, confirming the original conjecture.

## Contribution

The paper explicitly demonstrates the Hayden-Preskill protocol using a spin system and shows the critical role of rapid mixing for information retrieval.

## Key findings

- Rapid mixing leads to quick information escape from the black hole.
- Insufficient mixing results in slow information escape.
- Mutual information between outside particles and the black hole indicates information transfer.

## Abstract

We construct the Hayden-Preskill protocol by using a system of spin-1/2 particles and demonstrate information flows of this system which can mimic black holes. We first define an analogous black hole $A$ as a collection of such particles. Second, we take the particles from inside to outside the black hole to define an analogous system of Hawking radiation $B$ as outside particles. When the black hole and the radiation have the maximum entanglement at the Page time, we take an entangled pair system $C$ and $D$. The particles of $C$ fall into the black hole while their counterparts of $D$ remain outside. If we assume rapid mixing of the particle states in the black hole $A \cup C$, can the information of $C$ rapidly escape from the black hole like a mirror? We numerically show that if we turn on the rapid mixing in the black hole, the original information of $C$ rapidly escapes from the black hole to outside in the form of the mutual information between $B$ and $D$. On the other hand, if the mixing between $A$ and $C$ is not enough, the information escapes slowly. Hence, we explicitly demonstrate the original conjecture of Hayden and Preskill. We emphasize that enough mixing is an essential condition to make the Hayden-Preskill protocol functionally work.