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

Motivated by bulk reconstruction of smeared boundary operators, we study the Krylov complexity of local and nonlocal primary CFT d operators from the local bulk-to-bulk propagator of a minimally coupled massive scalar field in Rindler − AdS d + 1 space. We derive analytic and numerical evidence on how the degree of nonlocality in the dual CFT d 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 AdS d + 1 space. Our results also show that the evolution of the growth rate of Krylov operator complexity in the CFT d 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.