  The standard postulates of quantum theory can be divided into two groups: the
first one characterizes the structure and dynamics of pure states, while the
second one specifies the structure of measurements and the corresponding
probabilities. In this work we keep the first group of postulates and
characterize all alternatives to the second group that give rise to
finite-dimensional sets of mixed states. We prove a correspondence between all
these alternatives and a class of representations of the unitary group. Some
features of these probabilistic theories are identical to quantum theory, but
there are important differences in others. For example, some theories have
three 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) and
the existence of maximal measurements without phase groups. We also analyze
which of these properties single out the Born rule.
