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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi>CFT</a:mi> </a:mrow> <a:mrow> <a:mi>d</a:mi> </a:mrow> </a:msub> </a:mrow> </a:math> operators from the local bulk-to-bulk propagator of a minimally coupled massive scalar field in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mrow> <c:mi>Rindler</c:mi> <c:mtext>−</c:mtext> <c:mi>AdS</c:mi> </c:mrow> <c:mrow> <c:mi>d</c:mi> <c:mo>+</c:mo> <c:mn>1</c:mn> </c:mrow> </c:msub> </c:math> space. We derive analytic and numerical evidence on how the degree of nonlocality in the dual <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:msub> <e:mrow> <e:mi>CFT</e:mi> </e:mrow> <e:mrow> <e:mi>d</e:mi> </e:mrow> </e:msub> </e:mrow> </e:math> 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>AdS</g:mi> </g:mrow> <g:mrow> <g:mi>d</g:mi> <g:mo>+</g:mo> <g:mn>1</g:mn> </g:mrow> </g:msub> </g:mrow> </g:math> space. Our results also show that the evolution of the growth rate of Krylov operator complexity in the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:msub> <i:mrow> <i:mi>CFT</i:mi> </i:mrow> <i:mrow> <i:mi>d</i:mi> </i:mrow> </i:msub> </i:mrow> </i:math> 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.