A Semiclassical Diagnostic for Spacetime Emergence

Abstract: Recent developments have shown that some semiclassical spacetimes cannot emerge from a traditional application of the rules of holography, prompting proposals for restoring their emergence with "observer rules". In this paper, we propose a general semiclassical diagnostic of such failures of emergence, and of the extent to which observer rules can fix them. Our diagnostic is the presence of certain "evanescent" quantum extremal surfaces, which are distinguished by an upper bound on their area rather than their generalized entropy. In particular, the generalized entropy of an evanescent QES may be large: even though its area term must be small, its bulk entanglement term is unconstrained. This feature is explained by an operational distinction between classical and quantum connectivity in semiclassical gravity, or equivalently between the two summands of the generalized entropy.

• The paper introduces a diagnostic for spacetime emergence by identifying evanescent QES as indicators of holographic error correction breakdown.
• It employs tensor network models, specifically slice-normal networks, to formalize the interplay between generalized entropy and emergent geometry.
• The study demonstrates that excising regions beyond low-entropy QES restores a reliable emergent bulk description in diverse geometrical settings.

Semiclassical Diagnostics of Spacetime Emergence: The Role of Evanescent Quantum Extremal Surfaces

Introduction

The holographic principle postulates that gravitational effective field theory (EFT) in a bulk region can emerge from the dynamics of a lower-dimensional non-gravitational theory. However, recent advances in AdS/CFT and quantum information theory demonstrate that the emergence of semiclassical spacetime is subject to nontrivial limitations. This paper addresses these limitations by introducing a geometric and tensor-network-based diagnostic: the presence of evanescent quantum extremal surfaces (QES) signals a failure of spacetime emergence according to strict quantum error correction criteria.

The authors analyze the interplay between the generalized entropy formula, the area term, and bulk quantum entanglement. They employ tensor network models—specifically, slice-normal networks inspired by Python’s lunch geometry—to formalize the information-theoretic properties underlying the bulk-boundary map. Their diagnostic sheds light on both failure and partial restoration of spacetime emergence in situations such as closed cosmologies, fully evaporated black holes, and more.

Criteria for Emergence and Evanescent QES

In the conventional holographic context, the bulk-to-boundary map $V$ should approximately preserve inner products between subexponentially complex states up to errors of order $e^{-N^2}$, with $N^2$ the number of fundamental degrees of freedom. However, particular smooth semiclassical geometries (e.g., closed universes entangled with AdS regions or fully evaporated black holes) violate this criterion (2605.06780).

The core proposal of the paper is that such violations can be diagnosed by the existence of evanescent QES, defined as follows: a QES $X$ is evanescent if, for all code subspace states $\psi$,

$$\inf_\psi S_\mathrm{gen}[X] \leq \mathcal{O}(\log(1/G)),$$

where $S_\mathrm{gen}$ is the generalized entropy and $G$ is Newton's constant. Crucially, the bound depends only on the area term, not the full generalized entropy. This operational distinction reflects the fact that degrees of freedom associated with the area term are inaccessible from EFT, whereas the bulk entanglement entropy is not.

Figure 1: The evanescent QES (green cross) homologous to the fundamental description for AS$^2$ cosmology (left) and the fully evaporated black hole (right).

The presence of an evanescent QES homologous to the fundamental boundary implies that the semiclassical spacetime region beyond this surface does not emerge in the boundary description: subexponentially complex overlaps of code subspace states become easily distinguishable (e.g., using swap tests) within the bulk EFT, violating the fundamental holographic error correction structure.

Slice-Normal Tensor Networks and $\chi$-Entropies

To formalize the diagnostic, the authors introduce slice-normal tensor networks, where non-isometric tensor contractions (postselections) correspond to QES of minimal area. The contracted legs—the $e^{-N^2}$0-states—encode the "area" contribution to the generalized entropy. The key result is that for reliable emergence, all $e^{-N^2}$1-entropies must satisfy

$e^{-N^2}$2

with $e^{-N^2}$3 the entropy carried by each $e^{-N^2}$4-state.

Figure 2: A slice geometry and corresponding slice-normal tensor network. Surfaces $e^{-N^2}$5 and $e^{-N^2}$6 of locally minimal area yield random unitaries joined by $e^{-N^2}$7-states.

If any $e^{-N^2}$8-state fails this entropy bound, the network signalizes a breakdown in the emergent description: no code subspace states are reliably mapped to the boundary. The failure is both robust and code-independent (state-independent).

Excision Protocol and Spacetime Diagnostics

Whenever an evanescent QES is present, one can excise the non-emergent region by modifying the code subspace to exclude degrees of freedom lying beyond the offending $e^{-N^2}$9-state in the tensor network. This restabilizes the code's error-correcting properties and restricts the emergent region to the domain of dependence of the non-excised spacetime—a geometric protocol with direct operational meaning.

Figure 3: Excision protocol in tensor networks—degrees beyond the small $N^2$0-entropy surface (left) are excised, yielding an emergent bulk region (right).

A covariant version identifies the outermost evanescent QES homologous to the full fundamental description and restricts the emergent spacetime to its outer wedge.

Case Studies and Heuristic Consequences

The framework is explicitly illustrated in several geometrical regimes:

• AS$N^2$1 Cosmology: The non-emergence of the baby-universe region is associated with the existence of empty set QES of vanishing area.
• Fully Evaporated Black Holes: The horizon QES at the endpoint of evaporation becomes evanescent (area falls to $N^2$2), so the black hole interior is non-emergent in the boundary CFT; the information is accessible in Hawking radiation [EngGes25b].
• Iterated or Nested QES: Multiple non-emergent regions can appear, requiring multiple excision steps; inclusion of observers may only partially restore emergence.

  Figure 4: Canonical purification of the Hawking radiation entanglement wedge after the Page time produces an evanescent QES upon purification.

Bulk Interpretation: Classical vs. Quantum Connectivity

A significant claim is that the area and bulk entanglement contributions to generalized entropy are operationally distinct with respect to emergent geometry. Large bulk entropy does not counteract the non-emergence induced by a vanishing area term—the emergent region must be classically connected, not merely quantum-entangled [EngLiu23]. This separates and clarifies the physical status of "ER=EPR": quantum connectivity (entanglement) does not suffice for emergent geometry if not supported by adequate classical connectivity (area).

Observer Rules and Ob-Evanescent QES

The addition of an "observer" in the bulk can, in some cases, extend the emergent region by changing the effective homology constraint—i.e., by modifying the boundary data to which the QES are homologous. The diagnostic generalizes accordingly: with a bulk observer of entropy $N^2$3, one considers Ob-evanescent QES for which the area term is bounded above by $N^2$4.

Figure 5: Observer-modified slice-normal tensor network with new QES homologous to the observer clone and asymptotic boundary; non-emergent regions remains present if left QES is evanescent.

However, even observer rules may leave portions of spacetime non-emergent (e.g., nested QES or multiple observers), demonstrating that observer complementarity does not universally restore the bulk.

Large $N^2$5 Limit, Volatile Spacetimes, and the Algebraic Approach

The authors connect their results to the algebraic and large $N^2$6 approaches to holography, showing that the presence of evanescent QES is associated with the breakdown of code isometry at large $N^2$7—i.e., the nonexistence of a boundary state whose correlation functions converge to the bulk EFT correlators in the $N^2$8 limit [Ges25].

In quantum volatile spacetimes (e.g., with divergently large interior volume at increasing $N^2$9), the operational meaning and complexity-class of the non-emergence criteria are more subtle. Diagnosing emergence requires considering polynomially complex operators, a technical distinction from previous analyses focused on operators with $X$0 complexity.

Figure 6: Schematic depiction of excision: AS$X$1 cosmology, evaporated black hole, and canonical-purified Hawking wedge before (left) and after (right) covariant excision; emergent regions are shaded.

Implications, Open Problems, and Future Developments

The proposed diagnostic provides a concrete, operational criterion for the emergence (and excision) of spacetime geometry from quantum error correction and tensor network principles. Extension of this paradigm to de Sitter space, wormhole physics, and more general classes of holographic spacetimes is a natural avenue for further research [Zha26, Har26].

Connections to recent developments leveraging non-local "magic" for area term calculation in quantum codes [CaoChe26], as well as dynamic tensor network constructions for modeling evaporating black holes, suggest the robustness of the diagnostic beyond static settings.

Key open questions include: the precise role of complexity-theoretic class of operations in the boundary, characterization of emergent regions in the presence of multiple observers or nested QES, and the CFT signature of evanescent QES for fixed $X$2.

Conclusion

This work firmly establishes the area-term-dominated evanescent QES as a semiclassical diagnostic for (non-)emergence of spacetime, rooted in quantum error correction structure of the holographic map. The diagnostic links geometric, information-theoretic, and algebraic frameworks, demonstrating operational non-equivalence between classical and purely quantum connectivity. The approach offers powerful tools for analyzing the frontiers—and limitations—of holographic bulk reconstruction, quantum gravity, and black hole information.

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References:

• "[A Semiclassical Diagnostic for Spacetime Emergence]" (2605.06780)
• "[Observer complementarity for black holes and holography]" (Engelhardt et al., 8 Jul 2025)
• "[Further Evidence Against a Semiclassical Baby Universe in AdS/CFT]" (Engelhardt et al., 20 Apr 2025)
• "[State-dependent geometries from magic-enriched quantum codes]" (Cao et al., 13 Mar 2026)
• "[A no-go theorem for large $X$3 closed universes]" (Gesteau, 17 Sep 2025)
• "[Cryptographic Censorship]" (Engelhardt et al., 2024)
• "[Spoofing Entanglement in Holography]" (Engelhardt et al., 2024)