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Optimal control of two-qubit quantum systems attracts high interest due toapplications ranging from two-qubit gate generation to optimization of receiverfor transferring coherence matrices along spin chains. State preparation andmanipulation is among important tasks to study for such systems. Typicallycoherent control, e.g. a shaped laser pulse, is used to manipulate two-qubitsystems. However, the environment can also be used $\unicode{x2013}$ as anincoherent control resource. In this article, we consider optimal statemanipulation for a two-qubit system whose dynamics is governed by theGorini-Kossakowski-Sudarshan-Lindblad master equation, where coherent controlenters into the Hamiltonian and incoherent control into both the Hamiltonian(via Lamb shift) and the superoperator of dissipation. We exploit twophysically different classes of interaction with coherent control and optimizethe Hilbert-Schmidt overlap between final and target density matrices,including optimization of its steering to a given value. We find the conditionswhen zero coherent and incoherent controls satisfy the Pontryagin maximumprinciple, and in addition, when they form a stationary point of the objectivefunctional. Moreover, we find a case when this stationary point provides theglobally minimal value of the overlap. Using upper and lower bounds for theoverlap, we develop one- and two-step gradient projection methods operatingwith functional controls.

Optimal State Manipulation for a Two-Qubit System Driven by Coherent and Incoherent Controls