Coarse graining pure states in AdS/CFT

TL;DR

The authors show that Euclidean multi-boundary wormholes in AdS/CFT can encode a coarse-grained description of a pure-state black hole without ensemble averaging. They formulate a replica-based coarse-graining map that, when applied to the region outside a time-symmetric apparent horizon, reproduces the horizon entropy as S̄ = Area(γ)/4G and links gravitational wormholes to GHZ-like entanglement patterns in the CFT. By analyzing B-states and thin-shell geometries, and by introducing massive probes, they derive explicit CFT duals, characterize the coarse-grained density matrix ρ̄, and demonstrate Page-like behavior in toy evaporation models with an island-like transition. The work also discusses ensemble interpretations as Gaussian statistics over UV coefficients and connects to random tensor-network pictures and ETH-inspired operator structure. Overall, the paper provides a concrete higher-dimensional framework integrating wormholes, coarse graining, and horizon thermodynamics in AdS/CFT, with explicit CFT constructions and replica calculations.

Abstract

We construct new Euclidean wormhole solutions in AdS(d+1) and discuss their role in UV-complete theories, without ensemble averaging. The geometries are interpreted as overlaps of GHZ-like entangled states, which arise naturally from coarse graining the density matrix of a pure state in the dual CFT. In several examples, including thin-shell collapsing black holes and pure black holes with an end-of-the-world brane behind the horizon, the coarse-graining map is found explicitly in CFT terms, and used to define a coarse-grained entropy that is equal to one quarter the area of a time-symmetric apparent horizon. Wormholes are used to derive the coarse-graining map and to study statistical properties of the quantum state. This reproduces aspects of the West Coast model of 2D gravity and the large-c ensemble of 3D gravity, including a Page curve, in a higher-dimensional context with generic matter fields.

Figures (7)

• Figure 1: An on-shell Euclidean AdS$_{d+1}$ wormhole with six disconnected boundaries. Each boundary (shown in black) is $\mathbb{R}^d$ with the flat metric or $S^d$ with the round metric. The red lines are matter sources with planar or spherical symmetry. We consider examples where these matter sources are thin shells of pressureless fluid. There are similar wormholes sourced by end-of-world branes; in that case the spacetime ends at the red lines, and only replica-symmetric 'fractional' wormholes with $k < 2$ are on shell. The $i_m$'s label the particular matter configuration, or the flavor of the EOW brane.
• Figure 2: The Euclidean spacetime associated to the GHZ-like quantum state $|\Psi_k\rangle$ is a 'windmill' geometry. In the CFT, $|\Psi_k\rangle$ is defined on the $t=0$ slice of $k$ copies of the CFT, which is an $(S^{d-1})^k$ that cannot be extended smoothly into a bulk Cauchy slice. States of this type are not required to satisfy holographic entanglement inequalities. Any pair of boundaries can be connected by a spatial slice, so the quantum states on all $k$ boundaries are correlated by the gravitational constraints. Gluing this to another windmill for $\langle \Psi_k|$ produces the wormhole geometry in figure \ref{['fig:introWormhole']}.
• Figure 3: Penrose diagram of a black hole with a time-symmetric apparent horizon, $\gamma$, at $t=0$ (which is also extremal). The corresponding Euclidean geometry is shown in eq. \ref{['psi1overlap']} and the $t=0$ spatial slice is in eq. \ref{['spatialslice']}. For thin shells and $B$-states, the apparent horizon coincides with the event horizon (gray), but sending in additional matter from the boundary leads to the figure shown.
• Figure 4: Euclidean $B$-state black hole. The physical region, to the right in the figure, is a portion of the eternal black hole bounded by the asymptotic AdS boundary (black) and the EOW brane (red). Positive tension branes lead to solutions that cover at most half the thermal circle at the boundary.
• Figure 5: Euclidean thin-shell black hole, obtained by gluing a piece of global AdS (left) to a piece of the eternal black hole (right) across a thin shell (red). Each point on the diagram is $S^{d-1}$, except for the dashed line, which is the center of Euclidean global AdS.