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