  We investigate the difference between classical and quantum dynamics of
coupled magnetic dipoles. We prove that in general the dynamics of the
classical interaction Hamiltonian differs from the corresponding quantum model,
regardless of the initial state. The difference appears as non
positive-definite diffusion terms in the quantum evolution equation of an
appropriate positive phase-space probability density. Thus, it is not possible
to express the dynamics in terms of a convolution of a positive transition
probability function and the initial condition as can be done in the classical
case. We conclude that the dynamics is a quantum element of NMR quantum
information processing. There are two limits where our quantum evolution
coincide with the classical one: the short time limit before spin-spin
interaction sets in and the long time limit when phase diffusion is
incorporated.
