Probability Theory with Superposition Events: A Classical Generalization in the Direction of Quantum Mechanics
In finite probability theory, events are subsets of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density matrices induced by the experiments or `measurements' is the Luders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over Z_2, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over Z_2 that can model many characteristic non-classical results of QM.