In many cases of practical interest, rather than considering a single quantum system, we need to study a large number or collection of systems called an ensemble. Furthermore, rather than being in a single state, members of the ensemble can be found in one of two or more different quantum states. There is a given probability that a member of the ensemble is found in each of these states. We make this more concrete with a simple example.

Consider a two-dimensional Hilbert space with basis vectors {|x〉, |y〉}. We prepare a large number N of systems, where each member of the system can be in one of two state vectors


These states are normalized, so |α|2 + |β|2 = |γ|2+|δ|2 = 1, and the usual rules of quantum mechanics apply. For a system in state |a〉, if a measurement is made, then there is a probability |α|2 of finding |x〉 while there is a probability |β|2 of finding |y〉, and similarly for state |b〉.

Now suppose that we prepare na of these systems in state |a〉 and nb of the systems in state |b〉. Since we have N total systems, then


If we divide by N ,


This relation tells us that if we randomly select a member of the ensemble, the probability that it is found in state ...

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