Krylov operator complexity in holographic CFTs: Smeared boundary reconstruction and the dual proper radial momentum

Abstract

Motivated by bulk reconstruction of smeared boundary operators, we study the Krylov complexity of local and nonlocal primary CFTd operators from the local bulk-to-bulk propagator of a minimally coupled massive scalar field in Rindler-AdSd+1 space. We derive analytic and numerical evidence on how the degree of nonlocality in the dual CFTd observable affects the evolution of Krylov complexity and the Lanczos coefficients. Curiously, the near-horizon limit matches with the same observable for conformally coupled probe scalar fields inserted at the asymptotic boundary of AdSd+1 space. Our results also show that the evolution of the growth rate of Krylov operator complexity in the CFTd takes the same form as to the proper radial momentum of a probe particle inside the bulk to a good approximation. The exact equality only occurs when the probe particle is inserted in the asymptotic boundary or in the horizon limit. Our results capture a prosperous interplay between Krylov complexity in the CFT, thermal ensembles at finite bulk locations and their role in the holographic dictionary.