It is not necessary to understand the full structure of quantum measurement theory to understand quantum measurements in a wide range of experiments. This pedagogical approach was pointed out to me by Alexander Korotkov, and is the one he uses in his work on continuous measurements [343, 524, 638]. When we measure a single observable, and when this observable is not being changed during the measurement by any dynamics other than the measurement process, then only a small addition to Bayesian inference is required to describe quantum measurements. In fact, once we have made this addition, for infinitesimal time-steps we can include a Hamiltonian under which the measured observable changes with time, and obtain a full description of the continuous measurement of any quantum observable.

There is also another situation in which quantum measurement theory simplifies: when the system is linear, and when the observable being measured is a linear combination of the canonical coordinates. In this case, even when the observable is undergoing linear dynamics, a continuous quantum measurement reduces to a classical continuous measurement of a classical linear system, with the addition of a specified amount of white (flat-spectrum) noise. This noise is the “quantum back-action” of the measurement.