Geometric secret sharing in a model of Hawking radiation

TL;DR

This work presents a three-dimensional AdS construction in which black hole microstates are realized on an EOW brane with a holographic dual, producing an inception geometry behind the horizon that can be purified through ER=EPR via a wormhole to external radiation. It introduces an extended RT prescription allowing extremal surfaces to traverse the brane into inception, reproducing the Page curve (in line with island ideas) and revealing a geometric secret-sharing structure in Hawking radiation. By splitting radiation into subsystems and analyzing their entanglement wedges in the full real+inception geometry, the paper shows that individual subsystems reconstruct only parts of the interior (partial islands) while the full interior is accessible only when all radiative subsystems are combined, effectively encoding a quantum secret-sharing mechanism. The results illuminate information retrieval from Hawking radiation in a holographic setting, provide a geometric realization of ER=EPR, and highlight the role of multiboundary wormholes and eyelands in interior reconstruction, with potential extensions to more realistic (Lorentzian) evaporation dynamics.

Abstract

We consider a black hole in three dimensional AdS space entangled with an auxiliary radiation system. We model the microstates of the black hole in terms of a field theory living on an end of the world brane behind the horizon, and allow this field theory to itself have a holographic dual geometry. This geometry is also a black hole since entanglement of the microstates with the radiation leaves them in a mixed state. This "inception black hole" can be purified by entanglement through a wormhole with an auxiliary system which is naturally identified with the external radiation, giving a realization of the ER=EPR scenario. In this context, we propose an extension of the Ryu-Takayanagi (RT) formula, in which extremal surfaces computing entanglement entropy are allowed to pass through the brane into its dual geometry. This new rule reproduces the Page curve for evaporating black holes, consistently with the recently proposed "island formula". We then separate the radiation system into pieces. Our extended RT rule shows that the entanglement wedge of the union of radiation subsystems covers the black hole interior at late times, but the union of entanglement wedges of the subsystems may not. This result points to a secret sharing scheme in Hawking radiation wherein reconstruction of certain regions in the interior is impossible with any subsystem of the radiation, but possible with all of it.

Figures (22)

• Figure 1: ( a) The time-reflection-symmetric slice of the eternal BTZ black hole is a wormhole between two asymptotic regions, $A$ and $B$ . The dashed circle is the bifurcate horizon. This geometry is dual to the thermofield double state. ( b) The two-sided wormhole with one side truncated and replaced with an EOW brane (red circle) at a finite distance from the bifurcate horizon. In our model the EOW brane has an internal state structure which matches that of a 2d CFT. ( c) The EOW brane can be replaced with its holographic dual geometry. When the brane CFT is in a thermal state above the Hawking-Page transition, this geometry is a black hole within the inception disk (ID, gray) which is glued to the original geometry at the location of the brane (dashed red circle). The entropy associated with the brane is proportional to the area of the black hole horizon in the inception disk. This "inception horizon"(dashed blue circle) is an extremal surface homologous to the asymptotic region $A$. As such, it competes with the usual bifurcate horizon (dashed black circle) when we use the RT formula to search for the minimal surface which computes the entropy of $A$. In other words, the homology constraint in the RT formula can be satisfied by pulling curves through the circle where the inception disk meets the real geometry.
• Figure 2: Visualization of the glued Euclidean geometry. The $\varphi$ direction is suppressed and the partial circles are the Euclidean time circles $\tau$ and $\tau'$.
• Figure 3: Illustration of a healthy brane trajectory in Euclidean (left) and Lorentzian (right). The two figures share the time reflection symmetric slice (green dotted).
• Figure 4: The maximally extended Penrose diagram of the spacetime after inception. The real black hole spacetime (white) terminates at the EOW brane (red dashed line), then holographic inception creates the region behind (gray). The inception region also contains a black hole, due to the entanglement in the state \ref{['eq:HHstate']}. Left: since we glue convex-to-convex, we need to imagine the diagram as a folded piece of paper. Right: the causal structure is better visualized when we unfold the diagram. Note that a conventional convex-to-concave gluing would not lead to a long wormhole: it would require us to delete the other side of the brane in the inception geometry.
• Figure 5: The spatial slice of the glued geometry when we purify the inception black hole with a two boundary wormhole. Left: since we glue convex surfaces to convex surfaces, the original boundary and the purifying systems are naturally on the same side. Right: in order to increase clarity, we "unfold" the diagram on the left when we depict its various features.