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 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.

💡 Research Summary

In this work the authors investigate the Krylov operator complexity (KOC) of holographic conformal field theories (CFTs) by exploiting the bulk‑to‑bulk propagator of a minimally‑coupled massive scalar field in Rindler‑AdS(_{d+1}) (RAdS). The central idea is to treat the two‑point Wightman function
(C(t,r)=\langle\phi(t,r)\phi(0,r)\rangle)
as the autocorrelation input for the Lanczos algorithm. By applying the Lanczos recursion to (C(t,r)) one obtains a set of Lanczos coefficients ({b_n(r)}) that define a tridiagonal “Krylov Hamiltonian”. The Krylov complexity is then defined as the average position of a fictitious particle hopping on this semi‑infinite chain, (K(t,r)=\sum_n n|\psi_n(t)|^2).

Bulk set‑up
The RAdS metric is
(ds^2=-(r^2-1)dt^2+\frac{dr^2}{r^2-1}+r^2 dH_{d-1}^2),
with the AdS boundary at (r\to\infty) and a horizon at (r=1). A scalar of mass (m^2=\Delta(\Delta-d)) obeys the Klein‑Gordon equation, and its exact bulk‑to‑bulk propagator is known (Eq. 2.14):
(G_\Delta(P;P’)=c_\Delta,\xi^\Delta,{}_2F_1!\big(\frac\Delta2,\frac{\Delta+1}2;\Delta+1-\frac d2;\xi^2\big)),
where (\xi) is the chordal distance depending on the radial coordinates of the two points.

Lanczos coefficients and non‑locality
Fourier‑transforming (C(t,r)) yields a real, even function of time, suitable for the Lanczos recursion. The authors find that the behavior of the coefficients ({b_n(r)}) is controlled by the radial position (r), which in the holographic dictionary corresponds to the degree of non‑locality of the dual boundary operator (via the HKLL smearing kernel).

• Near the boundary ((r\to\infty)): the propagator reduces to the standard boundary two‑point function, and the Lanczos coefficients grow linearly, (b_n\approx\alpha,n). Consequently the Krylov complexity grows exponentially, (K(t)\sim e^{\alpha t}).
• Near the horizon ((r\to1)): the propagator becomes essentially constant; the coefficients saturate, (b_n\approx\beta), leading to a linear growth of complexity, (K(t)\sim\beta t).
• For intermediate radii ((1<r<\infty)): the coefficients display a non‑linear profile, reflecting the partial smearing of the boundary operator. The authors provide extensive numerical data (Appendix E) showing how the slope and curvature of (b_n(r)) vary smoothly between the two limits.

Relation to proper radial momentum
A key result is the quantitative comparison between the time derivative of Krylov complexity, (\dot K(t,r)), and the proper radial momentum (p_r(t,r)) of a probe particle falling in the same geometry. For a free particle the conserved energy yields
(p_r(t,r)=m,\dot r/(r^2-1)).
Integrating the geodesic equations gives an exponential increase of (p_r) at early times, turning into a linear increase near the horizon. The authors compute (\dot K(t,r)) from the Lanczos data and find an excellent match: the two functions differ by less than 5 % for all (r), and become exactly equal in the two asymptotic regimes (boundary and horizon). This provides strong evidence that Krylov complexity encodes the bulk radial dynamics of infalling probes.

HKLL smearing and scale‑non‑locality map
Using the Hamilton‑Kabat‑Lifschytz‑Low (HKLL) reconstruction, the bulk field is expressed as a smeared integral over a boundary primary (O_\Delta). The smearing kernel’s width grows with decreasing (r), so that bulk operators at finite (r) correspond to increasingly non‑local boundary operators. The authors argue that the radial coordinate thus acts as a “non‑locality scale”, and the observed deformation of the Lanczos coefficients with (r) is a direct manifestation of this scale dependence.

Comparison with (T\bar T) deformations
In Appendix F the authors compare their results with those obtained for a CFT deformed by the irrelevant (T\bar T) operator, which effectively introduces a finite radial cutoff in the bulk. They show that the Lanczos coefficients in the deformed theory acquire an effective scaling dimension (\Delta_{\rm eff}=\Delta+\alpha/r^2), reproducing the same qualitative behavior (linear → saturated growth) as the finite‑(r) bulk propagator. This suggests that the Krylov‑complexity picture is robust under such irrelevant deformations.

Technical aspects
The paper includes a careful analysis of the analytic domain of the Wightman function in the complex time plane, ensuring that the Lanczos recursion converges. The “moment method” is employed to extract the coefficients analytically in the near‑boundary and near‑horizon limits, while full numerical integration is used for intermediate radii. The authors also discuss the “switch‑back effect” (a temporary decrease of complexity) and its possible holographic interpretation.

Implications and outlook
The work establishes a concrete holographic dictionary entry: the growth rate of Krylov operator complexity of a (smeared) boundary operator is dual to the proper radial momentum of a bulk probe. This extends earlier SYK‑based observations to higher‑dimensional, genuinely holographic settings and shows that the correspondence holds not only for strictly local operators but also for a controlled family of non‑local operators parametrized by the bulk radial coordinate. The authors point out several future directions, including the inclusion of interactions, back‑reaction, and the exploration of Krylov complexity in time‑dependent backgrounds or in the presence of quantum chaos diagnostics such as OTOCs.

In summary, the paper provides analytic and numerical evidence that Krylov complexity in holographic CFTs faithfully captures bulk radial dynamics, that the degree of non‑locality of the boundary operator is encoded in the radial position of the bulk field, and that this framework survives relevant deformations like (T\bar T). It thus offers a promising new tool for probing the interior of AdS spacetimes from boundary quantum information measures.