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In this work, we address the problem of maximizing fidelity in a quantumstate transformation process controlled in such a way as to keep decoherencewithin given bounds. We consider a three-level $\Lambda$-type atom subjected toMarkovian decoherence characterized by non-unital decoherence channels. Weintroduce fidelity as the performance index for the quantum statetransformation process since the goal is to maximize the similarity of thefinal state density operator with the one of the desired target state. Weformulate the quantum optimal control problem with state constraints where thelater reflect the fact that the decoherence level remains within a pre-definedbound. Optimality conditions in the form of a Maximum Principle of Pontryaginin Gamkrelidze's form are given. These provide a complete set of relationsenabling the computation of the optimal control strategy. This is a novelapproach in the context of quantum systems.

A Quantum Optimal Control Problem with State Constrained Preserving Coherence