= Hilbert-Schmidt Lyapunov feedback identity
{c}
{title2=$\dot V=-2fC,\quad C=\operatorname{Tr}(\rho_d[-iH_1,\rho])$}
For $V=\|\rho-\rho_d\|_{\rm HS}^2$, with both states evolving under the same drift and only $\rho$ receiving $fH_1$, 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 $\dot V=-2kC^2\leq0$. This is nonincrease, not by itself convergence to the target: additional invariant-set and reachability conditions are needed.
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