Decoherence and the conditions for the classical control of quantum systems

We find the conditions for one quantum system to function as a classical controller of another quantum system: the controller must be an open system and rapidly diagonalized in the diagonal basis of the controller variable that is coupled to the controlled system. This causes decoherence in the controlled system that can be made small if the rate of diagonalization is fast. We give a detailed example based on the quantum optomechanical control of a mechanical resonator. The resulting equations are structurally similar to recently proposed models for consistently combining quantum and classical stochastic dynamics.