The preparation problem in nonlinear extensions of quantum theory

Abstract: Nonlinear modifications to the laws of quantum mechanics have been proposed as a possible way to consistently describe information processing in the presence of closed timelike curves. These have recently generated controversy due to possible exotic information-theoretic effects, including breaking quantum cryptography and radically speeding up both classical and quantum computers. The physical interpretation of such theories, however, is still unclear. We consider a large class of operationally-defined theories that contain "nonlinear boxes" and show that operational verifiability without superluminal signaling implies a split in the equivalence classes of preparation procedures. We conclude that any theory satisfying the above requirements is (a) inconsistent unless it contains distinct representations for the two different kinds of preparations and (b) incomplete unless it also contains a rule for uniquely distinguishing them at the operational level. We refer to this as the preparation problem for nonlinear theories. In addition to its foundational implications, this work shows that, in the presence of nonlinear quantum evolution, the security of quantum cryptography and the existence of other exotic effects remain open questions.