The Exact Uncertainty Relation and Geometric Speed Limits in Krylov Space

TL;DR

This paper presents a geometric interpretation of quantum speed limits using Krylov space, showing that operator evolution occurs at a constant speed determined by the first Lanczos coefficient, applicable to all Hamiltonians.

Contribution

It introduces a unified geometric framework for quantum speed limits based on Krylov space, highlighting the role of the first Lanczos coefficient as the intrinsic speed scale.

Findings

Operator amplitude vector evolves on the unit Krylov sphere at constant speed.

Provides an exact linear bound on operator evolution independent of higher Lanczos coefficients.

First Lanczos coefficient identified as the intrinsic speed scale of quantum dynamics.

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.