Verifiable nonlinear quantum evolution implies failure of density matrices to represent proper mixtures

In a recent Letter, Bennett and coworkers [1] argue that proofs of exotic quantum effects using closed timelike curves (CTC's) based on the work of Deutsch [2], or other nonlinear quantum dynamics, suffer from a fallacy that they call the "linearity trap," by which one cannot assume that a classical mixture of input states will lead to the same mixture of corresponding output states. We show that this prescription is inconsistent with the assumption that one can verify the postulated nonlinear evolution. Specifically, if some agent can empirically verify the deterministic nonlinear action of a physical device for some set of pure-state inputs, then one cannot generally use density matrices to represent the input for a second agent who is ignorant of the actual pure state prepared. Our argument is epistemological and makes no appeal to the detailed physics of CTC's.