Fluctuation-Growth Speed Limits in Two-Level Quantum Systems

Quantum speed limits (QSLs) impose fundamental constraints on how rapidly quantum systems evolve. While traditional QSLs bound the rate of change of a state or expectation value, recent work by Hamazaki (2024) extends these limits to the fluctuations of observables, setting an upper bound on how fast an observable's uncertainty can grow. We tested this fluctuation-growth bound in a two-level (spin-½) system with time-dependent observables. For a single-axis observable, the bound is tight—the fluctuation-growth rate reaches the maximal value allowed by the bound. For a two-axis observable, the bound is loose—the fluctuation-growth rate remains below the limit except at special instants. These results support the universality of Hamazaki's bound and illustrate the conditions under which a quantum observable reaches or falls below the maximal rate of fluctuation growth.