Contextuality and the Fundamental Theorems in Quantum Mechanics

Contextuality is a key feature of quantum mechanics, as was first properly brought to light by the Kochen-Specker theorem. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the KS theorem, also Wigner’s theorem, Gleason’s theorem and Bell’s theorem relate fundamentally to contextuality. The language of presheaves serves as a useful tool in this and allows to give reformulations of the theorems with a topological-geometric flavour. Moreover, we show in a mathematically exact way how much of the structure of a quantum system is encoded by contextuality, and also what is missing.