By I. T. Todorov, D. Ter Haar

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Conversely, the same array of detectors shown in Fig. 1 could have been used to measure the positions of particles from some other kind of preparation, involving a diﬀerent target or even an entirely diﬀerent preparation apparatus. Having distinguished preparation from measurement, we need to be more precise about just what is being prepared. At ﬁrst, one might say that it is the particle (more generally, the object of the subsequent measurement) that is prepared. While this is true in an obvious and trivial sense, it fails to characterize the speciﬁc result of the preparation.

Commuting sets of operators So far we have discussed only the properties of single operators. The next two theorems deal with two or more operators together. Theorem 5. If A and B are self-adjoint operators, each of which possesses a complete set of eigenvectors, and if AB = BA, then there exists a complete set of vectors which are eigenvectors of both A and B. Proof. Let {|an } and {|bm } be the complete sets of eigenvectors of A and B, respectively: A|an = an |an , B|bm = bm |bm . We may expand any eigenvector of A in terms of the set of eigenvectors of B: |an = cm |bm , m where the coeﬃcients cm depend on the particular vector |an .

The average value of a dynamical variable R, represented by the operator R, in the virtual ensemble of events that may result from a preparation procedure for the state, represented by the operator ρ, is R = Tr(ρR) . 1) Here Tr denotes the trace. The state operator is also referred to as the statistical operator, and sometimes as the density matrix, although the latter term should be restricted to its matrix form in coordinate representation. 1 Basic Theoretical Concepts 47 these will be developed later.