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

Sergio E. Aguilar-Gutierrez1,*, Hugo A. Camargo2,†, Viktor Jahnke2,‡, Keun-Young Kim2,3,§, and Mitsuhiro Nishida4,∥

  • *Contact author: sergio.ernesto.aguilar@gmail.com
  • †Contact author: hugo.camargo@phys.ncts.ntu.edu.tw
  • ‡Contact author: v.jahnke@unesp.br
  • §Contact author: fortoe@gist.ac.kr
  • ∥Contact author: mnishida124@gmail.com
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Phys. Rev. D 112, 126014 – Published 18 December, 2025

DOI: https://doi.org/10.1103/6bgg-vglp

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.

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Phys. Rev. D 112, 126014– Published 18 December, 2025
  • Received 2 October 2025
  • Accepted 4 December 2025
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