We consider the transformations of quantum states obtainable by a process of the following sort. Combine the given input state with a specially prepared initial state of an auxiliary system. Apply a unitary transformation to the combined system. Measure the state of the auxiliary subsystem. If (and only if) it is in a specified final state, consider the process successful, and take the resulting state of the original (principal) system as the result of the process.
We review known information about exact realization of transformations by such a process. Then we present results about approximate realization of finite partial transformations. We consider primarily the issue of approximation to within a specified positive epsilon, but we also address the question of arbitrarily close approximation.