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The standard postulates of quantum theory can be divided into two groups: thefirst one characterizes the structure and dynamics of pure states, while thesecond one specifies the structure of measurements and the correspondingprobabilities. In this work we keep the first group of postulates andcharacterize all alternatives to the second group that give rise tofinite-dimensional sets of mixed states. We prove a correspondence between allthese alternatives and a class of representations of the unitary group. Somefeatures of these probabilistic theories are identical to quantum theory, butthere are important differences in others. For example, some theories havethree perfectly distinguishable states in a two-dimensional Hilbert space.Others have exotic properties such as lack of "bit symmetry", the violation of"no simultaneous encoding" (a property similar to information causality) andthe existence of maximal measurements without phase groups. We also analyzewhich of these properties single out the Born rule.

Classification of all alternatives to the Born rule in terms of informational properties