  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 $T_2^J$.
