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For $\dot{\rho }=-i[H_0+f{H}_1,\rho ]$ and ${\dot{\rho }}_d=-i[H_0,{\rho }_d]$, the Hilbert-Schmidt distance gives $V=\Vert \rho -{\rho }_d{\Vert }^2/2$. The choice $f=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])$ 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.

For actual final density operator ${\rho }_f(t_F)$ and target ${\rho }_d$, minimizeThis Hilbert-Schmidt distance is nonnegative and is zero exactly at the target, for pure or mixed states. The subscript $f$ denotes evolution under the chosen control; $t_F$ is fixed. A simple overlap $\mathrm{Tr}({\rho }_d{\rho }_f)$ need not have its maximum at a general mixed target, whereas this distance has the required equality condition. For purely Quantum Hamiltonian dynamics, the target must lie on the initial state's unitary orbit of a density operator to attain zero error; otherwise the constrained optimization finds the closest accessible state instead.

Three standard objectives are state preparation, observable optimization and process implementation. All optimizations are over admissible control inputs and a specified final time $T$.

For state preparation, steer an initial density operator to a target ${\rho }_d$. An exact goal is ${\rho }_f(T)={\rho }_d$, or one may minimize the squared Hilbert-Schmidt distanceFor a pure target ${\rho }_d=|{\psi }_d⟩⟨{\psi }_d|$, maximizing $⟨{\psi }_d|{\rho }_f(T)|{\psi }_d⟩$ is a useful alternative. The target must be physically attainable: Hamiltonian engineering alone preserves the initial spectrum of the density operator.

For an observable objective, choose a Hermitian operator $O$ and maximize its expected final value:Population transfer is a special case with $O$ the projector onto the desired level or subspace; suppressing an unwanted population reverses the sign or minimizes this quantity.

For coherent process engineering, implement a target unitary operator $V$ on every input, rather than merely arranging one state transfer. If the propagator is $U_f(T)$, a phase-insensitive error isThe error vanishes precisely when $U_f(T)=e^{i\varphi }V$: equality in the Hilbert-Schmidt inner product Cauchy-Schwarz inequality bound makes the two operators proportional. For an open-system process, the analogous objective compares the implemented quantum channel with the target channel. These objectives can be supplemented by pulse-energy, amplitude, bandwidth and duration constraints in quantum optimal control.

The density operators are Hermitian, so their squared Hilbert-Schmidt distance is $2V=\mathrm{Tr}({\rho }^2)+\mathrm{Tr}({\rho }_d^2)-2\mathrm{Tr}(\rho {\rho }_d)$. Both purity of a density operator values are constant under their respective unitary time evolution: cyclicity gives $\mathrm{Tr}(\rho [H,\rho ])=0$ for every Hermitian $H$. ConsequentlyThe terms containing $H_0$ cancel because $\mathrm{Tr}([-i{H}_0,\rho ]{\rho }_d)=-\mathrm{Tr}(\rho [-i{H}_0,{\rho }_d])$. The quantity $F=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])$ is real: $[-i{H}_1,\rho ]$ is Hermitian, and the trace of the product of two Hermitian matrices equals its conjugate. The Lyapunov quantum control choice $f=F$ therefore gives $\boxed{\dot{V}=-f^2\le 0}$. For $t_2\ge t_1$, integration yields $V(t_2)=V(t_1)-{\int }_{t_1}^{t_2}f(t{)}^{2}dt\le V(t_1)$. Since the distance is $\sqrt{2V}$, it is monotonically nonincreasing. The decrease need not be strict when $f=0$.

The Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product gives $0\le J\le 1$ for $n$-dimensional unitary operators. Equality $J=0$ holds exactly when $U=e^{i\varphi }{U}_T$. When absolute gate phase matters, use the phase-sensitive Hilbert-Schmidt distance $\Vert U-U_T{\Vert }^2/(2n)=1-\mathrm{Re}\mathrm{Tr}(U_T^{†}U)/n$ instead.