A Unified Sheaf-Theoretic Account Of Non-Locality and Contextuality

A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the inter-relationships among the various concepts and results. We use the mathematical language of sheaves and monads to give a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed. We say that an empirical model is extendable if it can be extended consistently to all sets of measurements, regardless of compatibility. A hidden-variable model is factorizable if, for each value of the hidden variable, it factors as a product of distributions on the basic measurements. We prove that an empirical model is extendable if and only if there is a factorizable hidden-variable model which realizes it. From this we are able to prove generalized versions of well-known No-Go theorems. At the conceptual level, our equivalence result says that the existence of incompatible measurements is the essential ingredient in non-local and contextual behavior in quantum mechanics.