New insights on mutual information in the island approach to the Page curve

Abstract

In this article, we have presented one of the very important observations, regarding the behavior of mutual information of two different sets of subsystems in the after Page time scenario. This provides us with some deep insights about the conservation of geometrical entanglement. In our earlier works, we have shown how the saturation of mutual information between two specific subsystems plays a vital role in obtaining the correct Page curve for the eternal black hole. In those works, we have shown that the mutual information between $B_+$ and $B_-$, that is, $I(B_{+}: B_{-})$, vanishes at scrambling time, which leads to the correct Page curve. That means that at scrambling time, there is no correlation between $B_+$ and $B_-$. Remarkably, it is observed that at this particular value of observer's time, the mutual information between $\mathcal{I}$ and $R$, that is, $I(\mathcal{I}:R)$, becomes infinity. This indicates that the regions $\mathcal{I}$ and $R$ become maximally entangled. This provides us with a notion of conservation of geometric correlation between different regions on the Cauchy slice. In this work, we have also provided a way to calculate the tripartite mutual information of regions $\mathcal{I}$,$R_+$ and $R_-$, that is, $I(\mathcal{I}:R_+:R_-)$ on the Cauchy slice using the earlier results involving the bipartite regions. This is a new result which was missing in the earlier literature.