Demonstration of the Hayden-Preskill Protocol via Mutual Information

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 ⋃ 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.