Abstract
In any experimental setting, the rules of quantum theory provide the statistical distributions that the observed outcomes are expected to follow. The set formed by all these distributions contains the imprint of quantum theory, capturing some of its core properties. So far, only partial explicit descriptions of this set have been found for Bell-type settings, even in the simplest scenario. Here, we present the complete explicit and analytical description of a full set of quantum statistics in terms of extreme points. This is made possible by identifying all bipartite quantum states and pairs of binary measurements which can be self-tested, that is, reconstructed from empirical statistics only. Our description reveals precisely some of quantum theory’s extent and limitations.