Quantum process tomography with informational incomplete data of two $J$-coupled heterogeneous spins relaxation in a time window much greater than $T_1$
We reconstruct the time dependent quantum map corresponding to the relaxation process of a two-spin system in liquid-state NMR at room temperature. By means of quantum tomography techniques that handle informational incomplete data, we show how to properly post-process and normalize the measurements data for the simulation of quantum information processing, overcoming the unknown number of molecules prepared in a non-equilibrium magnetisation state ($N_j$) by an initial sequence of radiofrequency (RF) pulses. From the reconstructed quantum map, we infer both longitudinal ($T_1$) and transversal ($T_2$) relaxation times, and introduce the $J$-coupling relaxation times ($T^J_1,T^J_2$), which are relevant for quantum information processing simulations. We show that the map associated to the relaxation process cannot be assumed approximated unital and trace-preserving for times greater than $T_2^J$.