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Using the existing classification of all alternatives to the measurementpostulates of quantum theory we study the properties of bi-partite systems inthese alternative theories. We prove that in all these theories thepurification principle is violated, meaning that some mixed states are not thereduction of a pure state in a larger system. This allows us to derive themeasurement postulates of quantum theory from the structure of pure states andreversible dynamics, and the requirement that the purification principle holds.The violation of the purification principle implies that there is someirreducible classicality in these theories, which appears like an importantclue for the problem of deriving the Born rule within the many-worldsinterpretation. We also prove that in all such modifications the task of statetomography with local measurements is impossible, and present a simple toytheory displaying all these exotic non-quantum phenomena. This toy model showsthat, contrarily to previous claims, it is possible to modify the Born rulewithout violating the no-signalling principle. Finally, we argue that thequantum measurement postulates are the most non-classical amongst allalternatives.

Any modification of the Born rule leads to a violation of the purification and local tomography principles