The quantum dynamics of two coupled qubits

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.