Contextuality and state-space geometry

We shall explore the connection between state-space geometry and the Abramsky-Brandenburger sheaf-theoretic framework for classifying no-go theorems. The classic example of such a no-go theorem is the Kochen-Specker theorem. No-go results prohibit any theory from the specified class, e.g.~non-contextual theories, from replicating the empirical predictions of quantum theory. The sheaf-theoretic framework allows such no-go results to be generalised according to a certain kind of topology relating to the compatibility of the measurements used. We show that there is a correspondence between a class of no-go results and a class of polygonal state-spaces. The latter is a family of models whose geometric realisation lies in the equatorial plane of the Bloch sphere. This shows that the geometry of the state space used to define a physical theory related in a crucial way to the type of contextuality the theory exhibits. In particular, it also yields an understanding of the quantitative violation that quantum theory yields for the chained Bell inequality.