Booklet Wormhole: Holographic GHZ Duality

• Booklet wormhole is a holographic dual of multipartite GHZ states, built by gluing identical AdS–Schwarzschild black holes along a common topological junction.
• The construction reproduces the entropy relation S^(m) = (m-1) S^(2) using replica-topological methods that challenge standard manifold structures.
• It reveals observer-dependent interior dynamics and tripartite traversability, offering new insights into quantum constraints and non-local entanglement in holography.

A booklet wormhole is a proposed holographic spacetime dual of a multipartite GHZ-type entangled state. It generalizes the familiar two-sided AdS black hole, which is dual to a thermofield double, to a multi-boundary geometry with a single multi-way “junction” connecting several asymptotic regions, or “pages.” In the formulation proposed for thermal GHZ states, the booklet wormhole is built from $n$ identical AdS–Schwarzschild black holes glued along a common interior interface, and it reproduces both the GHZ entropy pattern and the Euclidean partition function of the boundary theory. Its defining novelty is that the junction is topological and enforces intrinsically quantum, non-local constraints on matter fields and observables; as a result, the bulk is not an ordinary smooth manifold in the standard sense (Jiang et al., 25 Aug 2025, Jiang et al., 12 Mar 2026).

1. GHZ dual and state structure

The boundary state underlying the construction is the $n$-partite GHZ state and its thermal generalization. For finite-dimensional subsystems, the canonical forms are

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$

and

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$

The thermal version used in the holographic construction is

$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$

Each single subsystem has reduced density matrix

$$\rho = \frac{1}{Z}\sum_{i=0}^{D-1} e^{-\beta E_i} \ket{i}\bra{i},$$

so every individual boundary CFT is thermal, exactly as in the thermofield-double construction. The distinction is global: the full state is not bipartite TFD entanglement but GHZ-type multipartite entanglement distributed across all pages (Jiang et al., 25 Aug 2025).

This state has a characteristic entropy pattern. For the thermal GHZ family, the multipartite entropy obeys

$S^{(n)} = (n-1)\, S^{(2)}.$

This is precisely the relation the bulk geometry is designed to reproduce. The 2026 analysis further emphasizes a “preference state” $\ket{\tilde g^{(3)}}$, singled out by minimal complexity within the maximally entangled family, with the thermal GHZ state obtained by imaginary-time evolution,

$$\ket{\text{GHZ}} = e^{-\beta H_1/2}\ket{\tilde g^{(3)}}.$$

This suggests that the booklet wormhole is not only an entropy-matching construction but also a candidate geometric representative of a distinguished multipartite entanglement class (Jiang et al., 12 Mar 2026).

2. Bulk construction and the booklet junction

Each page of the booklet is a standard AdS–Schwarzschild exterior. The Lorentzian metric on a page is

$$ds^2 = - f(r)\, dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_{d-1}^2 ,$$

with

$n$0

The geometry is constructed by cutting each two-sided black hole along a timelike reflection-symmetric slice that passes through the bifurcation surface of the Killing horizon, then gluing the resulting pieces at a common codimension-1 interface. The pages are therefore identical outside the horizon, while their interiors meet at a single multi-way junction (Jiang et al., 25 Aug 2025).

The gluing is governed by generalized $n$1-way junction conditions. If $n$2 is the common interface, then the induced metrics satisfy

$n$3

and the extrinsic curvatures satisfy

$n$4

In the GHZ construction, the junction is chosen so that the cut is reflection-symmetric and the extrinsic curvature vanishes, giving a tensionless junction with $n$5. The nontriviality is therefore not localized matter but topology: many identical black-hole “pages” are joined along a common spine (Jiang et al., 25 Aug 2025).

The later analysis makes explicit that this common interface is not well described by an ordinary manifold structure. Outside the horizons, each page looks like an ordinary single-sided AdS black hole. Inside, each infalling observer sees something that locally looks like a standard two-sided wormhole. Globally, however, the tangent space effectively “branches” into multiple directions at the junction. The bulk is therefore treated as a generalized geometric object rather than a conventional manifold (Jiang et al., 12 Mar 2026).

3. Entropy matching and Euclidean path integrals

The central holographic claim is an exact match between the entropy structure of the thermal GHZ state and the entropy structure of the booklet wormhole. In the three-page case, the single-subsystem entropy is set by the horizon area: $n$6 The corresponding tripartite entropy is obtained from a minimal tripartition surface consisting of two interfaces, so

$n$7

This is exactly the GHZ relation $n$8 (Jiang et al., 25 Aug 2025).

For general $n$9, the holographic result is

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$0

matching the GHZ formula

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$1

The nontrivial point is that, for $$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$2, a naive RT/HRT argument would give the wrong answer. The resolution is replica-topological. When the minimal entropy surface coincides with a multipartite junction, the topology of the minimal surface becomes ambiguous. The replica path integral must therefore include different topological saddle-type configurations, obtained by thickening the junction and attaching pages in different ways. Those alternative topologies change the area term at order $$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$3, and this dominates the entropy calculation even when the corresponding change in the classical action is arbitrarily small (Jiang et al., 25 Aug 2025).

The Euclidean formulation makes the duality explicit. On the boundary, the Euclidean path integral over $$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$4 cylinders of lengths $$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$5 prepares the unnormalized state

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$6

which becomes the thermal GHZ state when $$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$7. The Euclidean booklet wormhole is built by cutting each Euclidean black hole along constant-$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$8 slices and gluing all initial slices together and all final slices together. The action of the $$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr),$$9-th page is

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$0

so the total Euclidean action is

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$1

Hence the gravitational partition function equals the thermal partition function,

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$2

This equality is the precise Euclidean statement of the duality (Jiang et al., 25 Aug 2025).

4. Symmetries, special Killing fields, and quantum junction conditions

The booklet wormhole is constrained by the symmetry of the GHZ state. For the thermofield double one has

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$3

but for the thermal GHZ state the invariance is richer: $$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$4 equivalently under combinations such as

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$5

In the bulk, these become multi-boost symmetries generated by

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$6

The crucial observation is that distinct Killing vector fields can coincide in the entanglement wedge of a single page. The authors argue that this cannot happen on a connected smooth manifold without degeneracy. The bulk dual therefore requires special Killing fields that standard manifolds cannot realize (Jiang et al., 12 Mar 2026).

This symmetry analysis leads directly to the matter-field junction conditions. Classical local matching conditions of Israel type are not sufficient. Instead, the consistent junction conditions are intrinsically quantum and non-local. For three infalling observers—Alice, Bob, and Charlie—each can define a conserved momentum operator in the interior, but the multi-boost symmetry implies that these observables are not independent. One obtains the Dirac-type constraint

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$7

More generally, for any conserved charge $$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$8,

$$|\text{GHZ}_n\rangle = \frac{1}{\sqrt D}\sum_{i=0}^{D-1} |i\rangle^{\otimes n}.$$9

These act as first-class constraints on the physical Hilbert space and define the junction as a quantum gauge theory rather than a classical matching surface (Jiang et al., 12 Mar 2026).

A direct consequence is observer dependence in the interior. If excitations originate on page 1, then in the symmetric subspace of pages 2 and 3,

$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$0

If Alice sees a localized momentum eigenstate, Bob and Charlie see a correlated non-local state spread over their interiors. The information is global, but each observer accesses only a gauge-reduced part of it. This makes the booklet wormhole a concrete realization of observer-dependent bulk physics (Jiang et al., 12 Mar 2026).

5. Traversability and multipartite information flow

For ordinary two-sided wormholes, traversability is achieved by a double-trace deformation coupling the two boundaries,

$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$1

or, in discrete form,

$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$2

Detailed AdS$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$3 constructions show that such a coupling produces negative averaged null energy, opens the wormhole, and gives a bulk picture of teleportation through a traversable throat (Bak et al., 2018).

For booklet wormholes, the same mechanism does not work at the bipartite level. In the thermal GHZ state, bipartite correlators between two pages are diagonal and do not depend on insertion times, so a standard two-page double-trace deformation does not generate the phase structure required for traversability. The appropriate deformation is tripartite: $$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$4 The relevant diagnostic is then a commutator involving operators on three pages, such as

$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$5

Within the $$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$6-symmetric subspace of pages 2 and 3, the analysis reduces to an effective two-sided problem for a composite operator, and an imaginary part appears in the relevant correlator when the tripartite coupling is sufficiently large. This is the booklet analogue of traversable-wormhole signaling (Jiang et al., 12 Mar 2026).

The information flow is correspondingly multipartite. A localized wave packet injected from one page is pure and localized in that observer’s frame, but it generally evolves into a non-local mixed state on each remaining page. The bulk state must satisfy constraints such as

$$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$7

so the information that was initially localized on one page is transferred into entanglement between the other pages. In this sense, signals traversing a booklet wormhole do not simply emerge as local excitations on another boundary; they emerge as correlations distributed across multiple pages (Jiang et al., 12 Mar 2026).

6. Conceptual status, comparisons, and open questions

The booklet wormhole is not merely a multi-boundary wormhole in the usual sense. Compared with standard smooth multi-boundary AdS constructions, it introduces a single multi-way junction gluing more than two spacetimes at one interface. The interface is topological, carries no localized energy in the GHZ construction, and can be moved in imaginary time without changing the state so long as the total Euclidean length is fixed. This makes the junction’s position a gauge freedom rather than an observable (Jiang et al., 12 Mar 2026).

It also differs from multi-mouth traversable wormholes in asymptotically flat four-dimensional gravity. Those constructions connect several asymptotic regions through a common interior throat and are traversable between any pair of mouths, but they are built from quantum backreaction in asymptotically flat spacetime and have fundamental-group distinctions such as $$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$8 versus $$\ket{\beta\text{-GHZ}_n} \coloneqq \frac{1}{\sqrt Z} \sum_{i=0}^{D-1} e^{-\frac{\beta E_i}{2}} \ket{i}^{\otimes n}, \qquad Z = \sum_{i=0}^{D-1} e^{-\beta E_i}.$$9 for different multi-mouth topologies (Emparan et al., 2020). The booklet wormhole, by contrast, is specifically a holographic AdS construction tailored to GHZ-type multipartite entanglement (Jiang et al., 25 Aug 2025).

The principal conceptual consequence is that conventional holographic entropy inequalities are not universal once multi-way junctions and replica-topological ambiguities are admitted. The booklet wormhole circumvents the usual inequalities because different topologies are inevitably included in the gravitational path integral, even in the large-$$\rho = \frac{1}{Z}\sum_{i=0}^{D-1} e^{-\beta E_i} \ket{i}\bra{i},$$0 limit. The authors describe this as the first explicit holographic duality with a non-perturbative quantum effect (Jiang et al., 25 Aug 2025).

Several open problems remain. The 2026 analysis raises the problem of formulating fully dynamical matter fields at the junction, clarifying the role of observer-dependent interior states, and understanding more precisely how quantum reference frames organize the constrained Hilbert space. It also leaves open the detailed dynamics of tripartite traversability and the extent to which booklet wormholes can illuminate the information paradox, bulk locality, and the limits of manifold-based holographic reconstruction (Jiang et al., 12 Mar 2026).