Liouville-space costate equations for quantum optimal control (${\dot{\rho }}_v=K{\rho }_v,{\dot{A}}_v=-K^{†}{A}_v$)

For a real quantum optimal control objective $J=⟨⟨A|{\rho }_v(T)⟩⟩-\mathrm{Re}\int ⟨⟨A_v|{\dot{\rho }}_v-K[f]{\rho }_v⟩⟩dt-{\sum }_m{c}_m\int f_m^{2}dt/2$, first variation gives the state equation, the adjoint equation, $A_v(T)=A$, and $c_m{f}_m=\mathrm{Re}⟨⟨A_v|{\partial }_{f_m}K|{\rho }_v⟩⟩$. The adjoint is taken for the Hilbert-Schmidt inner product. With dissipation it is essential to use $K^{†}$, not silently the same generator as the forward state. These are necessary stationarity equations; they do not establish global optimality.