Gravity/Ensemble Duality

TL;DR

The work analyzes how gravity-based entropy calculations, including the Ryu-Takayanagi and island formalisms, reproduce the Page curve while Hawking’s late-time entropy growth appears to conflict with unitarity. It demonstrates the state paradox across standard, bath-coupled, and doubly holographic settings and clarifies the precise RT prescriptions and homology rules that apply in each case. A central claim is that the paradox is resolved if the gravitational path integral computes ensemble-averaged quantities over unitary theories, rather than a single theory, linking islands, Page curves, and Hawking radiation to gravity/ensemble duality. The paper also introduces a squared RT prescription for double holography and analyzes how islands and Page curves arise consistently at multiple holographic layers, highlighting ensemble interpretations as essential for resolving paradoxes. Overall, the results bolster gravity/ensemble duality as a coherent framework for reconciling unitarity with gravitational entropy calculations in black hole evaporation.

Abstract

For the first time, a gravitational calculation was recently shown to yield the Page curve for the entropy of Hawking radiation, consistent with unitary evolution. However, the calculation takes as essential input Hawking's result that the radiation entropy becomes large at late times. We call this apparent contradiction the state paradox. We exhibit its manifestations in standard and doubly-holographic settings, with and without an external bath. We clarify which version(s) of the Ryu-Takayanagi prescription apply in each setting. We show that the two possible homology rules in the presence of a braneworld generate a bulk dual of the state paradox. The paradox is resolved if the gravitational path integral computes averaged quantities in a suitable ensemble of unitary theories, a possibility supported independently by several recent developments.

Figures (12)

• Figure 1: Top: Penrose diagrams for an evaporating black hole. The light green region is the entanglement wedge of the radiation that has arrived at infinity before (left) and after (right) the Page time. Middle: State paradox. The RT prescription yields the Page curve for the entropy of the radiation, but only if the same entropy is assumed to follow Hawking's rising curve when determining the entanglement wedge. Bottom: Resolution of the state paradox by gravity/ensemble duality. The ensemble-averaged state is mixed, and its entropy follows Hawking's curve. The ensemble-averaged entropy follows the Page curve.
• Figure 2: Examples of holographic duality. Left: The solid bulk $M_d$ is dual to a holographic CFT$_{d-1}$ on $M_{d-1}$ (blue boundary). Right: In this example, $M_{d-1}$ is a manifold with boundary, so the boundary theory is a BCFT$_{d-1}$ and $M_d$ contains an end of the world brane EOW$_d$. (Despite the appearance of a BCFT this is a "singly holographic" example. In Sections \ref{['dh']} and \ref{['dhb']} we will consider a doubly holographic setting where the EOW$_d$ is a braneworld that localizes gravity and contains a holographic CFT$_d$.)
• Figure 3: RT prescription, applied in the setting shown on the right of Fig. \ref{['fig:2.1']}. The entropy of the boundary region $R_{d-1}$ is given by the generalized entropy of its entanglement wedge EW$(R_{d-1})$. $\gamma_d$ is the quantum extremal surface.
• Figure 4: Hawking radiation is absorbed by a distant Dyson sphere near the boundary. In Hawking's semiclassical analysis, the Dyson sphere entropy will grow monotonically. The quantum state on the global bulk slices shown is pure. Each global slice is the entanglement wedge of its respective boundary slices. Thus the RT prescription implies that the entropy of the global boundary vanishes, as required by CFT unitarity. However, at late times the extrapolate dictionary demands that $S(\hbox{boundary})=S(\hbox{Dyson})$. This contradiction is the state paradox.
• Figure 5: Compared to Fig. \ref{['fig:2.3']}, the Hawking radiation is collected in a localized reservoir on the Dyson sphere. The RT prescription is applied to a nearby boundary region $R_{d-1}$. The entanglement wedge EW$(R_{d-1})$ is shown in light green. After the Page time, it contains a disconnected island $I$, the black hole interior, because this choice minimizes the generalized entropy. This yields the Page curve for $S(R_{d-1})$. However, the extrapolate dictionary would yield Hawking's curve; this is the state paradox.