This chapter will treat of single systems, ignoring that they may consist of parts, and may interact with other systems.
The custom in quantum theory to refer to physical quantities as observables reflects the initial conviction that each physical quantity must be associated with a realizable measurement (or experimental) arrangement. The stochastic response function is intuitively thought of as giving the probabilities of measurement outcomes. Later we shall look to see how measurement too can be modelled as a quantum-mechanical process, but for now we rely on that intuition. In the notation of the preceding chapter, we readas the probability that a measurement of observable m yields a value in Borel set E. Before turning to the Hilbert space representation, I want to explain some concepts definable generally in terms of the stochastic response function, which will prepare us for questions to be raised later.
There are two concepts of purity for states. To begin, if s is a state, and m is an observable, letbe the function which assigns probabilities to outcomes of measurements of m, on the basis of state s. Then if s, t, and u are states such that for a number 0 < b ≤ 1 and all observables m, we call s a mixture of states t and u. This convex sum of the stochastic response functions can also be countable, provided only that the 'weights' are non-negative and sum to 1. We call s pure if it is not a mixture of other states. When (1) holds we also call t and u components of s. These components provide an example of the following relation: