Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 3 / a / iii / Solution

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.