= Lyapunov quantum control
{c}
{title2=$\dot V=-f^2\le0$}
For $\dot\rho=-i[H_0+fH_1,\rho]$ and $\dot\rho_d=-i[H_0,\rho_d]$, the <Hilbert-Schmidt distance> gives $V=\|\rho-\rho_d\|^2/2$. The choice $f=\operatorname{Tr}(\rho_d[-iH_1,\rho])$ yields $\dot V=-f^2$. Decrease alone does not prove convergence: orthogonal pure eigenstates of a diagonal $H_0$ give zero input for a transverse Pauli control even though their distance remains nonzero. Such a law can be simulated to produce an <open-loop control>.
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