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

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.

💡 Research Summary

The paper presents a novel, purely boundary‑based method for reconstructing the area operator of a quantum extremal surface (QES) and for diagnosing entanglement‑island formation in evaporating black holes, using the framework of modular Krylov complexity. The authors begin by reviewing the operator‑algebra quantum error‑correction (OAQEC) description of AdS/CFT, emphasizing that the QES area appears as a central operator (L_A) in the code subspace. While previous works treated (L_A) as an input, this work shows how to extract it directly from boundary dynamics.

The key technical tool is modular Krylov complexity, a generalization of Krylov complexity where the generator is the modular Hamiltonian (K_A = -\log\rho_A) associated with a boundary region (A). By repeatedly acting with the modular Liouvillian on a reference operator (or state) and applying the Lanczos algorithm, one obtains a tridiagonal matrix whose entries ({a_n, b_n}) encode the projected modular Hamiltonian (K_A) in the Krylov basis. Diagonalizing this matrix yields the spectrum of (K_A). Because the area operator sits in the center of the algebra, the difference between the full modular Hamiltonian and its Krylov‑reconstructed part isolates (L_A): \