Convex sets

Our object is to understand the geometry of the set of all possible states of a quantum system that can occur in nature. This is a very general question, especially since we are not trying to define ‘state’ or ‘system’ very precisely. Indeed we will not even discuss whether the state is a property of a thing, or of the preparation of a thing, or of a belief about a thing. Nevertheless we can ask what kind of restrictions are needed on a set if it is going to serve as a space of states in the first place. There is a restriction that arises naturally both in quantum mechanics and in classical statistics: the set must be a convex set. The idea is that a convex set is a set such that one can form ‘mixtures’ of any pair of points in the set. This is, as we will see, how probability enters (although we are not trying to define ‘probability’ either). From a geometrical point of view a mixture of two states can be defined as a point on the segment of the straight line between the two points that represent the states that we want to mix. We insist that given two points belonging to the set of states, the straight line segment between them must belong to the set too. This is certainly not true of any set.