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

TL;DR

This paper investigates how Krylov complexity of operators in holographic CFTs relates to bulk properties like radial momentum, revealing connections between boundary non-locality, near-horizon physics, and bulk reconstruction.

Contribution

It provides analytic and numerical insights into how boundary operator complexity evolution encodes bulk radial momentum and non-locality effects in holographic duals.

Findings

Krylov complexity growth mirrors bulk radial momentum.

Near-horizon limit matches boundary and bulk scalar observables.

Non-locality in boundary operators influences complexity evolution.

Abstract

Motivated by bulk reconstruction of smeared boundary operators, we study the Krylov complexity of local and non-local 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 non-locality 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.