Quantum process tomography with informational incomplete data of two J-coupled heterogeneous spins relaxation in a time window much greater than T₁

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 magnetization state (N_j) by an initial sequence of radiofrequency pulses. From the reconstructed quantum map, we infer both longitudinal (T₁) and transversal (T₂) relaxation times, and introduce the J-coupling relaxation times ($T_1^J,T_2^J$), 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 $T_2^J.$ (paper)