  High fidelity quantum state transfer is an essential part of quantum
information processing. In this regard, we address the problem of maximizing
the fidelity in a quantum state transformation process satisfying the
Liouville-von Neumann equation. By introducing fidelity as the performance
index, we aim at maximizing the similarity of the final state density operator
with the one of the desired target state. Optimality conditions in the form of
a Maximum Principle of Pontryagin are given for the matrix-valued dynamic
control systems propagating the probability density function. These provide a
complete set of relations enabling the computation of the optimal control
strategy.
