Feedback based on state overlap need not be stabilizing
= Feedback based on state overlap need not be stabilizing
{title2=$f=\operatorname{Tr}(\rho_d\rho)\not\Rightarrow\dot V\leq0$}
Take $H_0=0$, $H_1=\sigma_y/2$, $\rho_d=(I+\sigma_z)/2$ and $\rho=(I+\sigma_x)/2$. The overlap is $f=1/2$, but $C=\operatorname{Tr}(\rho_d[-iH_1,\rho])=-1/2$, so the <Hilbert-Schmidt Lyapunov feedback identity> gives $\dot V=1/2>0$. A target overlap and the <derivative> of that overlap along the control direction are different quantities.