The Exact Uncertainty Relation and Geometric Speed Limits in Krylov Space

TL;DR

The paper introduces a geometric reformulation of Hall's exact uncertainty relation in Krylov space, showing that operator evolution occurs on a unit Krylov sphere at a constant speed set by the first Lanczos coefficient $b_1$. This yields a universal, linear quantum speed limit $\\ell_K(t)=b_1 t$ that holds for all Hamiltonians, while higher Lanczos coefficients govern curvature and torsion of the trajectory. The authors derive a Krylov light cone in index space, discuss the distinction between front propagation and geometric displacement, and show that integrability emerges in the linear Krylov flow despite potentially chaotic operator growth. The framework provides a basis-invariant speed law for operator dynamics and clarifies the interplay between chaos and integrability in Krylov dynamics. The results have broad implications for quantum control, information scrambling, and the geometric understanding of operator growth.

Abstract

We show that Hall's exact uncertainty relation acquires a simple geometric form in the Krylov basis generated by the Liouvillian. In this canonical operator frame, the uncertainty equality implies that the operator amplitude vector evolves on the unit Krylov sphere with constant speed fixed solely by the first Lanczos coefficient. This yields an exact linear bound on geometric operator evolution, independent of higher Lanczos coefficients and valid for arbitrary Hamiltonians, integrable or chaotic. Our results provide the first unified geometric interpretation of exact quantum speed limits and operator growth, identifying the first Lanczos coefficient as the intrinsic speed scale of quantum dynamics.