Islands and Uhlmann phase: Explicit recovery of classical information from evaporating black holes

TL;DR

The paper proposes a concrete protocol to explicitly recover the interior's classical information of an evaporating black hole by measuring the Uhlmann phase of Hawking radiation, leveraging the island formula and replica wormholes. It shows that the Uhlmann phase curvature equals the symplectic form of the island plus radiation, enabling reconstruction of the island phase space and an invertible map linking initial matter configurations to late-time interior data. The approach yields a decryption procedure for classical information and allows extraction of Poisson brackets for island observables without full knowledge of the dynamical map. This work tightens the link between holographic entanglement structure and classical information recovery in quantum gravity, with implications for how information escapes black holes in a semiclassical regime.

Abstract

Recent work has established a route towards the semiclassical validity of the Page curve, and so provided evidence that information escapes an evaporating black hole. However, a protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. In this paper, we describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation. The process of black hole formation and evaporation provides an invertible map between this phase space and the space of initial matter configurations. Thus, all classical information is explicitly recovered. We assume in this paper that replica wormholes contribute to the gravitational path integral.

Figures (12)

• Figure 2.1: Spacetime is divided into asymptotic and gravitational regions, joined by a cutoff surface. Initially, the system is set up with a matter configuration that collapses to form a black hole. Then, the black hole evaporates, and all of the Hawking radiation is collected by the asymptotic region. (Note that we will continue to use this color scheme, in which the asymptotic region is red and the gravitational region is blue, throughout the paper.)
• Figure 2.2: The manifold and fields for the partition function $Z(\lambda_2,\lambda_1)$ must obey the boundary conditions $\lambda_1$ in the past $\mathcal{N}_-$, and the complex conjugate and time reversal of $\lambda_2$ in the future $\mathcal{N}_+$.
• Figure 2.3: When $\lambda_1=\lambda_2$, all the fields are invariant under time reflection and complex conjugation. This symmetry fixes a surface $\Sigma$ which divides $\mathcal{M}$ into a past piece $\mathcal{M}_-$ and a future piece $\mathcal{M}_+$.
• Figure 2.4: (a) Before the Page time, there is no island, and the entropy of the Hawking radiation is just given by the bulk entropy on the surface $R$. (b) After the Page time, enough entanglement has been built up by the Hawking process for an island to appear inside the black hole. Accounting for this in the island formula reproduces the Page curve.
• Figure 2.5: The partition function $Z_{(n)}[\lambda_n,\dots,\lambda_1]$ should be computed on manifolds $\widetilde{\mathcal{M}}_n$ with the boundary conditions shown (the case shown is for $n=3$). The $n$ gravitational regions are disconnected in (a), and this is what we would do in an ordinary QFT. However, since this is a theory of gravity, we should allow all topologies in the gravitational regions, including those which connect various copies with wormholes, for example like in (b).

Theorems & Definitions (2)

• Lemma
• proof