Abstract
In any experimental setting, quantum physics provides the statistical distributions that the observed outcomes are expected to follow. The set formed by all these distributions contains the imprint of quantum theory and captures some of its core properties. So far, only partial explicit descriptions of this set have been found for Bell-type settings in which entangled states can be shared and measured by independent observers. Here we obtain the complete explicit and analytical description of a full set of quantum statistics in terms of its extremal points. This is made possible by finding all bipartite quantum states and pairs of binary measurements that can be self-tested, that is, reconstructed from empirical statistics only. Our description precisely reveals some of the extent and limitations of quantum theory.