Hall's exact uncertainty relation and the geometry of Krylov-space operator dynamics
Phys. Rev. A 114, 042218 – Published 23 October, 2026
DOI: https://doi.org/10.1103/dnd4-f5rz
Abstract
Krylov representations of operator dynamics encode two inequivalent motions: probability transport along the discrete Krylov chain and motion of the normalized amplitude vector on the associated sphere. We lift Hall's exact uncertainty relation to Liouville space and show that, for the fixed Krylov-basis measurement, its classical Fisher information locally attains the quantum Fisher information. The amplitude vector therefore follows an exact constant-speed spherical trajectory. We construct its moving frame, identifying the first Lanczos coefficient with the speed and the higher coefficients with successive bending. Geodesic curvature controls the leading departure from Mandelstam-Tamm saturation. Separately, we compare cumulative and pointwise bounds on Krylov-index transport, which depends on the full Lanczos sequence.
