 Solution

The deterministic law cannot be transplanted unchanged into measurement-based quantum feedback. Evaluating its state-dependent scalar requires knowledge of the current quantum state. In fact $C=\mathrm{Tr}(i[H_1,{\rho }_d]\rho )$ is the expectation value of a Hermitian observable, but that expectation cannot be obtained exactly from a single system without measurement backaction. Tomography uses many independently prepared copies, not a nondisturbing readout of one evolving state.

Continuous weak measurements can supply a noisy record and a conditional state estimate, but then the state follows stochastic measurement dynamics with additional disturbance and decoherence, rather than the closed unitary equation used in part (a). A new stochastic stability analysis is required. One may instead simulate the corrected Lyapunov quantum control from known initial data and apply its waveform as open-loop control, or learn a pulse across repeated experiments as in Question 3. Those procedures do not constitute real-time state feedback on a single quantum trajectory. The separate source error in part (a) remains: even ideal knowledge of the state would not make its printed overlap law monotone.