On the Orthocomplementation of State-Property-Systems of Contextual Systems

Abstract

We describe a model in which the maximal change of state of the system due to interaction with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the limit N = 2 the system reduces to a model for the spin measurements on a quantum spin-1 2 particle. This model fits in the hidden measurement approach to quantum mechanics in which quantum probabilities are explained as due to an uncontrollable fluctuation in the measurement process. In the limit N → ∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate situations the change of state due to measurement is neither 'maximal' (i.e. quantum) nor 'zero' (i.e. classical). We show that two of the axioms used in Piron's representation theorem for quantum mechanics are violated, namely the covering law and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability of a state transition in general not only depends on the angular distance between the two states but also on the measurement context which induces the state transition. Therefore our models also shed new light on Gleason's theorem and suggest that transition probability maybe is not a secondary concept which can be derived from the structure on the set of states and properties, but instead should be regarded as a primitive concept by its own right for which the measurement context is crucial.