Impossibility of mixed-state purification in any alternative to the Born Rule

Using the existing classification of all alternatives to the Measurement Postulates of Quantum Mechanics we study the properties of multi-partite systems in these alternative theories. We prove that in all these theories the Purification Principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This implies that simple operational processes, like mixing two states, cannot be described in an agent-free universe. This appears like a crucial clue for the problem of deriving the Born Rule within "unitary quantum mechanics" or the many-worlds interpretation. We also prove that in all such modifications the task of state tomography with local measurements is impossible, and present a simple toy theory displaying all these exotic non-quantum phenomena. This toy model shows that, contrarily to previous claims, it is possible to modify the Born rule without violating the No-Signalling Principle. Finally, we argue that the quantum measurement postulates are the most non-classical amongst all alternatives.