Hadamard Products of Product Operators and the Design of   Gradient-Diffusion Experiments for Simulating Decoherence by NMR Spectroscopy
  An extension of the product operator formalism of NMR is introduced, which uses the Hadamard matrix product to describe many simple spin 1/2 relaxation processes. The utility of this formalism is illustrated by deriving NMR gradient-diffusion experiments to simulate several decoherence models of interest in quantum information processing, along with their Lindblad and Kraus representations. Gradient-diffusion experiments are also described for several more complex forms of decoherence, including the well-known collective isotropic model. Finally, it is shown that the Hadamard formalism gives a concise representation of decoherence with arbitrary correlations among the fluctuating fields at the different spins involved, and that this can be applied to both decoherence (T2) as well as nonadiabatic relaxation (T1) processes. 
Timothy F. Havel, Yehuda Sharf, Lorenza Viola, David G. Cory