Fluctuations in the Entropy of Hawking Radiation






宮地真路 特任助教(名古屋大学)







Recent study revealed that the inclusion of Euclidean wormhole into the gravitational path integral renders the entropy of Hawking radiation consistent with unitarity, deriving the Page curve of the Hawking radiation. On the other hand, since the gravitational path integral with Euclidean wormhole computes quantities of ensemble average of theories, it is possible that the entropy of Hawking radiation of each gravity theory fluctuate wildly around the ensemble average. In this talk we show that such fluctuation is as small as the dimension of the system, ensuring the answer from the ensemble average is typical. We use the gravitational path integral to compute the fluctuations of the Hawking radiation entropy around the Page curve, in a two-dimensional model introduced by Penington \emph{et al}. Before the Page time, we find that $\delta S = e^{-S}/\sqrt{2}$, where $S$ is the black hole entropy. This result agrees with the Haar-averaged entropy fluctuations of a bipartite system, which we also compute at leading order. After the Page time, we find that $\delta S = \sqrt{2}e^{-S}/\pi$. This is not symmetric under exchange of subsystem sizes and so does not agree with the Haar average for a subsystem of fixed Hilbert space dimension. We show that the discrepancy can be attributed to an additive $\sqrt{2}/\pi$ fluctuation in the number of black hole states in a given energy band. As a by- product, our result gives a refinement on the known upper bound on the subsystem entropy fluctuation in Haar random pure state.