Local, non-classical model of Bell correlations

We show how the Bell correlations can be modelled locally by relaxing the joint probability relation for independent variables $P(a,b)=P(a)P(b)$ outside classical settings, with complex/quaternion generators for the measurement outcomes encoding a deviation from this (classical) requirement. The consideration is motivated by complementarity. We analyse the pair correlations (photons or spin 1/2 particles) based on the limitations put on complementary observables when they are modelled as classical, non-commuting operators. As complementarity comes with both single particle properties and correlations insufficiently captured by classical theory together with `non-commutativity', we argue that a complete model should allow complementary observables to have new local dependences. That is, the existing information should not be restricted to be classical (simultaneously observable). Given this new correlation model, the central question is if information theory ought to be extended accordingly in quantum physics.