 Hilbert-Schmidt Lyapunov feedback identity

For $V=\Vert \rho -{\rho }_d{\Vert }_{\mathrm{HS}}^2$, with both states evolving under the same drift and only $\rho$ receiving $f{H}_1$, cyclicity of the trace cancels the drift terms and gives the displayed identity. The scalar $C$ is real because $-i[H_1,\rho ]$ is Hermitian. Choosing $f=kC$, $k>0$, makes $\dot{V}=-2k{C}^2\le 0$. This is nonincrease, not by itself convergence to the target: additional invariant-set and reachability conditions are needed.