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We consider the case of a $\sqrt{\mathrm{SWAP}}$ quantum gate and itsoptimized entangling action, via continuous dynamical decoupling, in thepresence of dephasing noise. We illustrate the procedure in the specific casewhere only the two-qubit operation is controlled and no single-qubit operationsare included in the description. To compare the optimized dynamics in thepresence of noise with the ideal case, we use the standard fidelity measure.Then we discuss the importance of using optimized gates in the quantumoperational-probabilistic theory. Because of their importance for the explicitconstruction of the completely positive maps representing the operations, wederive optimized Kraus operators in this specific case, focusing on theentanglement operation. We then show how to visualize the time evolution ofeach Kraus operator as a curve in a three-dimensional Euclidean space. Finally,we connect this formalism with the operational framework of quantum mechanicsby describing a possible set of measurements that could be performed to obtainthe Kraus operators.

Visualizing Kraus operators for dephasing noise during application of the $\sqrt{\mathrm{\mathrm{SWAP}}}$ quantum gate