  In any given experimental scenario, the rules of quantum theory provide
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
descriptions have been known for this set, even in the simplest scenarios.
Here, we obtain the analytical description of a complete set of quantum
statistics in terms of extremal points. This is made possible by finding all
bipartite quantum states and pairs of binary measurements which can be
self-tested, i.e. identified from statistics only. Our description provides a
direct insight into the properties and limitations of quantum theory. These are
not expressed in terms of Hilbert spaces, but rather directly in terms of
measurement observation statistics.
