Modular Krylov Complexity as a Boundary Probe of Area Operator and Entanglement Islands

TL;DR

This paper demonstrates that the area operator of quantum extremal surfaces can be reconstructed solely from boundary dynamics using modular Krylov complexity, enabling boundary-based diagnosis of black hole phenomena.

Contribution

It introduces a boundary-only method combining OAQEC and modular Krylov complexity to reconstruct the area operator and diagnose entanglement islands without bulk geometry.

Findings

Reconstruction of the area operator from boundary dynamics.

Diagnosis of island formation and Page transition boundary-only.

Establishment of modular Krylov complexity as a probe of spacetime geometry.

Abstract

We show that the area operator of a quantum extremal surface can be reconstructed directly from boundary dynamics without reference to bulk geometry. Our approach combines the operator-algebra quantum error-correction (OAQEC) structure of AdS/CFT with modular Krylov complexity. Using Lanczos coefficients of boundary modular dynamics, we extract the spectrum of the modular Hamiltonian restricted to the algebra of the entanglement wedge and isolate its central contribution, which is identified with the area operator. The construction is purely boundary-based and applies to superpositions of semiclassical geometries as well. As an application, we diagnose island formation and the Page transition in evaporating black holes using boundary modular evolution alone, bypassing any bulk extremization. More broadly, our results establish modular Krylov complexity as a concrete and computable probe of emergent spacetime geometry, providing a new route to accessing black hole interiors from boundary quantum dynamics.