Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 50 / 1 / c

The Lyapunov quantum control law requires an exact instantaneous expectation value. Cyclicity writes it aswhere $A(t)$ is a known Hermitian observable. A measurement on one system produces a random outcome, not its exact ensemble expectation, and changes the system state. Repeated measurements for state estimation or continuous weak measurements introduce noise and backaction absent from the assumed closed-system equation. Thus the law cannot be implemented unchanged as exact measurement-based quantum feedback while retaining the dynamics and guarantee in part (a). A conditional-state feedback scheme must include the measurement evolution and rederive its stability properties.

If the initial density operator and Hamiltonian model are known, we can instead solve the deterministic equations in a computer, calculate $f(t)$ along the simulated trajectory, and apply that precomputed waveform in the laboratory as open-loop control. Ensemble measurements can help calibrate such a waveform, but they do not supply nondisturbing instantaneous feedback from an individual evolving system.