Maximizing complementary quantities by projective measurements

In this work we study the so-called quantitative complementarity quantities. We focus in the following physical situation: two qubits ($q_A$ and $q_B$) are initially in a maximally entangled state. One of them ($q_B$) interacts with a $N$-qubit system ($R$). After the interaction, projective measurements are performed in each of the qubits of $R$, in a basis that is chosen after independent optimization procedures: maximization of the visibility, the concurrence and the predictability. For a specific maximization procedure, we study in details how each of the complementary quantities behave, conditioned on the intensity of the coupling between $q_B$ and the $N$ qubits. We show that, if the coupling is sufficiently "strong", independent of the maximization procedure, the concurrence tends to decay quickly. Interestingly enough, the behavior of the concurrence in this model is similar to the entanglement dynamics of a two qubit system subjected to a thermal reservoir, despite that we consider finite $N$. However the visibility shows a different behavior: its maximization is more efficient for stronger coupling constants. Moreover, we investigate how the distinguishability, or the information stored in different parts of the system, is distributed for different couplings.