Demonstration of the Hayden-Preskill protocol via mutual information

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.