The effects of closed timelike curves (CTCs) in quantum dynamics, and its consequences for information processing have recently become the subject of a heated debate. Deutsch introduced a formalism for treating CTCs in a quantum computational framework. He postulated a consistency condition on the chronology-violating systems which led to a nonlinear evolution on the systems that come to interact with the CTC. This has been shown to allow tasks which are impossible in ordinary linear quantum evolution, such as computational speed-ups over (linear) quantum computers, and perfectly distinguishing non-orthogonal quantum states.
Bennett and co-authors have argued, on the other hand, that nonlinear evolution allows no such exotic effects. They argued that all proofs of exotic effects due to nonlinear evolutions suffer from a fallacy they called the " linearity trap". Here we review the argument of Bennett and co-authors and show that there is no inconsistency in assuming linearity at the level of a classical ensemble, even at the presence of nonlinear quantum evolution. In fact, this is required for the very existence of empirically verifiable nonlinear evolution. The arguments for exotic quantum effects are thus seen to be based on the necessity for a fundamental distinction between proper and improper mixtures in the presence of nonlinear evolutions. We show how this leads to an operationally well-defined version of the measurement problem that we call the "preparation problem".