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

Thomas D. Galley, Lluis Masanes

Published 2016-10-16Version 1

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 show that, if we keep the first group of postulates, all the alternatives to the second group are in correspondence with a class of representations of the unitary group. Some features of these alternative 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 show that pure-state bit symmetry and no restriction of effects single out the Born rule in finite dimensions.