Black Hole Interiors Mapped Using Only Boundary Information, a Major Physics First

Scientists are increasingly focused on understanding the relationship between quantum entanglement and the emergence of spacetime geometry. Niloofar Vardian from Sharif University of Technology, alongside collaborators, demonstrate a novel method for reconstructing the area operator, a key quantity characterising spacetime geometry, directly from boundary quantum dynamics. Their research, detailed in a new paper, utilises Krylov complexity and the operator-algebra error-correction structure of AdS/CFT to extract information about the entanglement wedge without relying on traditional bulk geometric calculations. This significant advancement allows researchers to diagnose phenomena such as island formation and the Page transition in black holes solely through boundary evolution, offering a potentially transformative approach to probing the interiors of black holes and furthering our understanding of quantum gravity.

Reconstructing spacetime geometry via modular Krylov complexity and boundary dynamics offers a novel approach to quantum gravity

Scientists have demonstrated a groundbreaking method to reconstruct the area operator of a quantum extremal surface directly from boundary dynamics, circumventing the need for any reference to bulk geometry. This achievement merges the operator-algebra quantum error-correction structure of the AdS/CFT correspondence with the principles of modular Krylov complexity.
By utilising Lanczos coefficients derived from boundary dynamics, researchers have successfully extracted the spectrum of the Hamiltonian restricted to the algebra of the entanglement wedge, isolating its central contribution and identifying it as the area operator. The resulting construction is entirely boundary-based and remains applicable even to superpositions of semiclassical geometries.

This work establishes modular Krylov complexity as a concrete and computable probe of emergent spacetime geometry, offering a novel pathway to investigate black hole interiors solely from boundary quantum dynamics. The research bypasses traditional bulk extremization techniques by diagnosing island formation and the Page transition in evaporating black holes using only boundary modular evolution.

Specifically, the team employed modular Krylov complexity to analyse the boundary theory, revealing how the area operator, a key component in holographic calculations, can be determined without relying on the usual gravitational calculations within the bulk spacetime. The core of this breakthrough lies in the application of Lanczos reconstruction to the modular Hamiltonian associated with the entanglement wedge algebra.

This process allows for the isolation of the central contribution to the Hamiltonian’s spectrum, which is then directly identified with the area operator. This method not only avoids the complexities of bulk geometry but also extends to scenarios involving superpositions of semiclassical states, broadening its applicability to a wider range of holographic configurations.

Furthermore, the study demonstrates the power of this approach by successfully diagnosing island formation and the Page transition in evaporating black holes, relying exclusively on boundary modular dynamics. These findings establish modular Krylov complexity as a powerful tool for extracting geometric information from quantum dynamics, potentially unlocking new avenues for studying black hole interiors in time-dependent settings where a complete bulk description is unavailable. This purely boundary-based probe of quantum extremal surfaces and entanglement islands represents a significant step forward in understanding the emergence of spacetime from quantum information.

Entanglement wedge reconstruction and operator algebra decomposition for area operator identification offer promising avenues for holographic computations

Lanczos reconstruction, employing coefficients derived from boundary dynamics, underpins this work’s methodology. Specifically, the study extracts the spectrum of the Hamiltonian restricted to the algebra of the entanglement wedge, isolating its central contribution to identify the area operator. This construction remains purely boundary-based and is applicable to superpositions of semiclassical geometries, representing a significant methodological advancement.

The research utilises operator-algebra error-correction, combining it with Krylov complexity to bypass reliance on bulk geometry. Initially, the Hilbert space is decomposed into H = HA ⊗H A, establishing a framework for analysing the code subspace Hcode as a direct sum of subspaces, Hcode = ⊕α(HAα 1 ⊗H Aα 1).

The algebra A, a von-Neumann algebra within the code subspace, is then decomposed to reveal its structure and commutant A′, facilitating the identification of superselection sectors labelled by operators ZA. Unitaries UA and U A are introduced to map bulk code subspace bases to encoded states, |α, ij⟩code = UAU A |α, i⟩Aα 1 |α, j⟩ Aα 1 |χα⟩Aα 2 Aα 2, encoding logical states with redundant degrees of freedom represented by |χα⟩.

Subsequently, the modular Hamiltonian associated with the entanglement wedge algebra is examined, allowing for the extraction of the area operator’s spectrum. The algebraic entropy S(ρcode, MA) is calculated using the formula S(ρcode, MA) = tr(ρcode LA) + S(ρcode, A), where LA is a linear operator related to the area of the quantum extremal surface.

Defining LA = ⊕αS(χα, MA)Iaα ⊗I aα, the algebraic Ryu-Takayanagi formula is achieved, linking the area operator to the algebraic structure of the boundary theory. This approach successfully diagnoses island formation and the Page transition in evaporating black holes using only boundary modular dynamics.

Area operator reconstruction via boundary modular evolution and Krylov complexity reveals emergent spacetime geometry

Researchers demonstrate the direct reconstruction of an area operator from boundary dynamics, circumventing the need for reference to bulk geometry. This work combines the operator-algebra error-correction structure of AdS/CFT with Krylov complexity to achieve this result. By utilising Lanczos coefficients derived from boundary dynamics, the spectrum of the Hamiltonian restricted to the entanglement wedge algebra is extracted, isolating its central contribution which is then identified as the area operator.

The construction employed is entirely boundary-based and remains applicable to superpositions of semiclassical geometries. As a specific application, the study successfully diagnoses island formation and the Page transition in evaporating black holes using boundary modular evolution alone, effectively bypassing traditional bulk extremization methods.

This approach establishes modular Krylov complexity as a concrete and computable probe of emergent spacetime geometry. Modular Krylov complexity provides a new method for accessing black hole interiors from boundary quantum dynamics. The research extracts the spectrum of the modular Hamiltonian restricted to the entanglement wedge algebra, isolating the central component identified as the area operator.

This method does not rely on bulk geometry and extends to time-dependent geometries, offering a novel approach to studying holographic states. Furthermore, the work demonstrates the ability to diagnose island formation, observing the Page transition in evaporating black holes solely through boundary modular dynamics.

These findings establish modular Krylov complexity as a powerful tool for extracting geometric information from quantum dynamics, opening avenues for investigating black hole interiors in scenarios where a complete bulk description is unavailable. The area operator reconstruction relies on a decomposition of the Hilbert space into HA ⊗ HA, where A and A represent von Neumann algebras satisfying specific relationships detailed within the study. Operators within the algebra A act on the boundary region A, while those in its commutant A act on the complement, reflecting the complementary recovery expected in AdS/CFT.

Reconstructing spacetime geometry via boundary dynamics and Krylov complexity offers a novel path toward quantum gravity

Researchers have demonstrated a method for reconstructing the area operator of a quantum extremal surface directly from boundary dynamics, circumventing the need to reference bulk geometry. This reconstruction leverages the operator-algebra error-correction structure inherent in the AdS/CFT correspondence, combined with the mathematical technique of Krylov complexity.

By analysing the Lanczos coefficients derived from boundary dynamics, the spectrum of the Hamiltonian restricted to the entanglement wedge is extracted, isolating a central contribution identified as the area operator. This purely boundary-based construction extends to superpositions of semiclassical geometries and successfully diagnoses island formation and the Page transition in evaporating black holes without requiring traditional bulk extremization procedures.

These findings establish Krylov complexity as a viable and computationally accessible tool for probing emergent spacetime geometry, offering a novel approach to investigating black hole interiors from boundary observations. The authors acknowledge that the accuracy of their method relies on the large-N limit of the holographic quantum error-correcting code, where leakage outside the code subspace is suppressed. Future research could explore connections between this boundary modular diagnostics and continuous tensor network descriptions of conformal field theory-bath dynamics, potentially providing an explicit real-time representation of information flow in such systems.