In the second lesson, we described "generalized" measurements that do not obey the restrictive criteria of von Neumann's ideal projective measurement. These measurements, which provide more or less partial information on the state of a quantum system, often correspond to situations closer to real experiments than projective measurements. An important special case of a generalized measure is defined by a set of positive Hermitian operators forming a "POVM" (Positive Operator Valued Measure). The link between generalized measures, POVMs and projective measures was recalled, and a number of interesting examples were presented.
As we saw in the first lesson, a simple model of a measurement process is realized by coupling a quantum system S to a set of N independent measurement spins or "qubits" (constituting an angular momentum J = N/2). We have shown that the partial acquisition of information resulting from the coupling of S with a single qubit is a POVM, and have described how the accumulation of POVM measurements resulting from coupling with a set of qubits is transformed into a projective measurement. We also showed that the acquisition of information about S resulting from the POVM measure is akin to a Bayesian inference process in probability theory. We concluded the lesson by considering a curious example of measurement, in which it seems that information is obtained "without the measured system having interacted with the device". The paradox arises, as in other such cases, from the undue use of classical concepts to describe a quantum situation.
