Krylov complexity of modular Hamiltonian evolution

TL;DR

This work extends Krylov, or Lanczos, complexity to modular Hamiltonian evolution, linking entanglement structure to fine-grained dynamical growth. It shows that the modular Lanczos spectrum encodes the entanglement spectrum and that spread complexity attains a Page-curve–like plateau in random modular Hamiltonians, highlighting that entanglement spectrum—not entropy alone—governs complexity. In 2d CFTs, the authors derive universal modular growth with Lyapunov exponent $\lambda^{mod}_L=2\pi$ and a scrambling time set by the local modular temperature, providing concrete analytic results for both state and operator evolution. The results illuminate the interplay between entanglement, complexity, and holography, and suggest entanglement spectrum as a key diagnostic in quantum gravity contexts.

Abstract

We investigate the complexity of states and operators evolved with the modular Hamiltonian by using the Krylov basis. In the first part, we formulate the problem for states and analyse different examples, including quantum mechanics, two-dimensional conformal field theories and random modular Hamiltonians, focusing on relations with the entanglement spectrum. We find that the modular Lanczos spectrum provides a different approach to quantum entanglement, opening new avenues in many-body systems and holography. In the second part, we focus on the modular evolution of operators and states excited by local operators in two-dimensional conformal field theories. We find that, at late modular time, the spread complexity is universally governed by the modular Lyapunov exponent $λ^{mod}_L=2π$ and is proportional to the local temperature of the modular Hamiltonian. Our analysis provides explicit examples where entanglement entropy is indeed not enough, however the entanglement spectrum is, and encodes the same information as complexity.