= Liouville-space costate equations for quantum optimal control
{c}
{title2=$\dot\rho_v=K\rho_v,\quad\dot A_v=-K^\dagger A_v$}
For a real <quantum optimal control> objective $J=\langle\!\langle A|\rho_v(T)\rangle\!\rangle-\operatorname{Re}\int\langle\!\langle A_v|\dot\rho_v-K[f]\rho_v\rangle\!\rangle dt-\sum_m c_m\int f_m^2dt/2$, first variation gives the state equation, the adjoint equation, $A_v(T)=A$, and $c_m f_m=\operatorname{Re}\langle\!\langle A_v|\partial_{f_m}K|\rho_v\rangle\!\rangle$. The adjoint is taken for the <Hilbert-Schmidt inner product>. With dissipation it is essential to use $K^\dagger$, not silently the same generator as the forward state. These are necessary stationarity equations; they do not establish global optimality.
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