Holographic Wormhole Correspondence

• Holographic correspondence with wormhole geometries is a duality linking QFTs on multiple boundaries with bulk spacetimes exhibiting nontrivial wormhole topologies and entanglement features.
• The framework employs both Lorentzian and Euclidean methods to derive two-point functions and analyze causal connectivity, clarifying the role of off-diagonal correlators and direct interactions.
• Extensions to multipartite systems and traversable wormholes offer insights into quantum information measures, UV/IR factorization, and constraints on quantum channel capacities.

Holographic correspondence with wormhole geometries refers to the use of gauge/gravity duality frameworks (such as AdS/CFT) to relate quantum field theories (QFTs) with distinct or multiple boundary components to bulk spacetimes featuring nontrivial wormhole topologies. These constructions provide deep insights into the interplay between quantum information (entanglement, correlations, channel capacity) and classical or semiclassical spacetime geometry, with central implications for the structure of quantum gravity, the nature of black holes, and the emergence of spacetime from entanglement.

1. Holographic Two-Point Functions in Wormhole Geometries

In Lorentzian wormhole backgrounds constructed from solutions to higher-curvature gravity (such as 5d Einstein-Gauss-Bonnet gravity), the holographic two-point functions of dual QFT operators $O^\pm$ are derived from the on-shell gravitational action following a Lorentzian-adapted version of the GKPW prescription. The scalar field in the bulk is expanded as

$$\phi(y, x) = \int dy' \left[K^+(y, x|y')\phi_0^+(y') + K^-(y, x|y')\phi_0^-(y')\right]$$

with $K^\pm$ being bulk-to-boundary propagators localized at the two asymptotic boundaries at $x\to x^\pm$. The retarded Green’s function receives both diagonal ($\langle O^+O^+\rangle$, $\langle O^-O^-\rangle$) and off-diagonal ($\langle O^+O^-\rangle$) terms:

$$S[\phi_0] = -\frac{1}{2}\int dy\, dy'\; \phi_0(y) \Delta(y, y')\phi_0(y')$$

and

$$\langle \psi_f | T \{ O(y)O(y') \} | \psi_i \rangle = -i\Delta(y, y')$$

where $\Delta$ is directly related to the normal derivative of the bulk-to-boundary propagator at the asymptotic boundary. The Lorentzian setting requires careful control over normalizable modes and the choice of boundary conditions at the wormhole “throat”, linking the initial/final state ambiguity to physical contour prescriptions in frequency space.

In such wormhole backgrounds, off-diagonal (cross-boundary) correlation functions are manifest and generically nonvanishing—a direct signal of new physical couplings between degrees of freedom associated to each asymptotic region (Arias et al., 2010). The existence and magnitude of these cross correlators encode both entanglement and direct interaction effects in the dual QFT pair.

2. Bulk Geometry, Causal Structure, and Coupling/Entanglement Diagnosis

The wormhole solutions in higher-dimensional gravity are characterized by the absence of horizons and smooth geometric interpolation between two (or more) distinct AdS regions. For instance, the metric

$$ds^2 = R^2\left[-\cosh^2 \rho\, dt^2 + d\rho^2 + \cosh^2\rho\, d\Sigma^2\right]$$

admits causal geodesics that traverse the wormhole, linking both boundaries at $\rho \to \pm\infty$. The scalar field boundary data $\phi_0^\pm$ on each boundary act as independent sources for dual QFT operators $O^\pm$.

A geometric criterion is established: Causal connectivity of boundaries implies direct coupling in the dual field theory. If the two asymptotic AdS boundaries are causally disconnected (as in the eternal AdS black hole), cross boundary correlators signal pure entanglement—the dual state is a thermofield double. If, by contrast, causal geodesics directly join the boundaries (as in the described wormhole), then the dual CFTs are coupled, with entanglement and explicit interaction both present. This distinction is reflected in the factorization (or lack thereof) of the generating functional; Euclidean continuation, which singles out unique bulk solutions and removes normalizable mode ambiguities, aids in identifying these structures (Arias et al., 2010).

3. Multipartite and Multiboundary Wormhole Generalizations

Euclidean multiboundary AdS${}_3$ wormholes, constructed via path integrals over higher-genus Riemann surfaces or by explicit geometric quotienting, are dual to multipartite entangled states in tensor products of CFT Hilbert spaces (Balasubramanian et al., 2014). The classical geometries support $n$ disconnected boundaries connected by a common bulk region (the wormhole), with topology and moduli spaces (horizon lengths, twist parameters) determining the entanglement structure of the dual state.

Holographically, the precise structure of the dual CFT state can be written as

$$|\Sigma\rangle = \sum_{i_1\ldots i_n} A_{i_1\ldots i_n}|i_1\rangle_1\cdots|i_n\rangle_n$$

with the coefficients $A_{i_1\ldots i_n}$ controlled by the CFT $n$-point functions and conformal mappings dictated by the moduli of the Riemann surface. The system supports both bipartite and genuinely multipartite entanglement, with operationally extractable entanglement (distillable entanglement $E_D$) scaling as at least $(n+1)/2$-partite for $n$ boundaries in symmetric wormholes. There are sharp transitions in entanglement patterns as system moduli vary—phase transitions in the minimal surfaces computing holographic entanglement entropies mirror transitions between multipartite and bipartite entanglement dominance.

4. Quantum Information Measures and Wormhole Geometry

The geometric realization of quantum information measures—including entanglement of purification $E_P$—is established by identifying $E_P$ with the area of an entanglement wedge cross-section ($E_W$) (Bao et al., 2018, Bao, 2018). In multiboundary wormholes, minimal surfaces corresponding to wormhole throats compute not only Ryu-Takayanagi (RT) entanglement entropy but also $E_W$, supporting direct map between multipartite entanglement features in the CFT and geometric features in the bulk.

This connection enables the translation of multipartite entanglement inequalities (e.g., $$E_W(A_1:A_2:\cdots:A_n) \leq E_W(A_1A_2:A_3\cdots:A_n) + E_W(\ldots)$$) into geometric inequalities involving lengths or areas of connecting surfaces (throats) in the multi-boundary wormhole. It also identifies optimal purifications—minimizing entanglement between subsystems—as associated to the state supported on the wormhole boundary components (Bao et al., 2018, Bao, 2018).

5. UV/IR Factorization and Nonlocal Couplings in Multiboundary Contexts

Euclidean AdS wormholes constructed in various bulk theories (Einstein-dilaton, Einstein–Yang–Mills, etc.) yield duals with striking UV/IR features: at short distances, cross-boundary correlators are finite (contrasting with the standard UV divergence of single-boundary correlators), indicating that the dual QFTs factorize at high energies. However, in the infrared, nonlocal couplings mediate soft interactions between the boundary sectors (Betzios et al., 2019). The precise nonlocality scale and momentum dependence can be analyzed in the dual using multi-trace deformation technology, requiring “softer than local” couplings to maintain the regularity of holographic correlators. This is exemplified in toy models as well as full bulk computations of two-point (and nonlocal, e.g., Wilson loop) correlators.

Table: Properties of Cross-Boundary Correlators in Holographic Wormholes

Regime	Diagonal (Same Boundary)	Off-diagonal (Cross-Boundary)
Short-distance/UV	Divergent	Finite
Long-distance/IR	Power-law/Exponential decay	Comparable magnitude
Holographic interpretation	Local QFT singularity	Soft, nonlocal interaction

6. Dynamical, Traversable, and Quantum Protocol Aspects

Traversable wormholes, achieved via double-trace deformations that violate the averaged null energy condition (ANEC) in the bulk, transform classically nontraversable bridges into viable quantum channels between boundaries (Ahn et al., 2020, Bao et al., 2019). The signal transmission capacity of such channels is controlled by the parameters of the deformation (e.g., coupling $h$), bulk spacetime dimension, and is tightly linked to the chaos-scrambling properties of the dual CFT (via butterfly velocity $v_B$). Quantum random walk analysis provides a precise mapping between bulk traversability and quantum channel capacity, even when the boundary state is a superposition over distinct classical geometries.

In higher dimensions, analytic formulas for ANEC violation and signal transfer bounds display precise dimensional dependence. The amount of information transmittable through the wormhole is suppressed with increasing spacetime dimension due to faster decay of ANEC violation and reduced butterfly velocity $v_B = 1/(d-1)$ (Ahn et al., 2020).

7. Implications for Complexity, Decoherence, and Chronology

Black hole geometries incorporating wormhole regions from horizon to singularity have been analyzed using the complexity–volume (CV) and complexity–action (CA) conjectures, giving a geometric basis for the scaling of computational complexity in the dual CFT state in terms of bulk spatial volumes or bulk action evaluated on Wheeler–DeWitt patches (Bousder, 2023). The analogy with Gabriel's horn (finite volume, infinite surface area) in these solutions elucidates paradoxes regarding information storage in black holes—bulk finiteness does not limit information capacity, as the boundary surface (horizon/ER bridge) can encode unlimited detail.

Decoherence processes modeled by four-boundary wormhole geometries (constructed in AdS${}_3$) illustrate how energy exchange with an environment drives the system from entangled to classical (and finally uncorrelated) states via moduli evolution, “pinching off” the wormhole throat and leading to loss of mutual information (Anegawa et al., 2020).

In time machine models based on Misner-type backgrounds, holography illustrates that strong self-interactions in the dual CFT (and therefore in the bulk geometry) prohibit propagation across chronology horizons by splitting the bulk into disconnected pieces meeting only on a measure-zero boundary set; entanglement, even if present at leading order, cannot be used to transmit information across causally disconnected regions once $1/N$ (quantum gravity) corrections are included (Emparan et al., 2021).

---

The holographic correspondence with wormhole geometries thus systematically relates nontrivial bulk topologies and causal structures to fine-grained properties of quantum entanglement, interaction, and information flow in dual boundary quantum field theories, providing a quantitative bridge connecting the structure of spacetime with quantum information and computation theoretic quantities.