The geometry of contextuality
Contextuality is one of the key features of quantum mechanics. It asserts that physical quantities cannot be assigned definite values if these are to satisfy certain functional relations as they do in classical physics. As a result the phase space picture with its underlying geometry is lost in quantum mechanics, leading to a number of interpretational problems in standard quantum theory.
On the other hand contextuality has been proposed as a candidate for a potential better performance of quantum computers with recent results stressing the importance of geometrical notions, in particular, (group) cohomology.
Motivated by a number of conceptual problems in the Copenhagen interpretation, the topos approach to quantum theory has developed a new perspective on quantum theory – building on contextuality explicitly. Generalizing to arbitrary topoi in constructing representations for physical theories one finds a reformulation of quantum mechanics in the topos of presheaves over the context category (the partial order of commutative subalgebras of some von Neumann algebra).
Within this generalization quantum theory resembles classical physics in many ways. In particular, it suggests the study of the geometric properties of the quantum state space in the form of the spectral presheaf and its connections to contextuality.