Physical measurements are generally considered to produce real number valued results, but these can always be re-expressed as collections of Yes or No answers (to questions about whether or not the observed value is within various narrower and narrower intervals for example).

And a physical theory is a procedure for predicting the probabilities of positive answers to some such questions from knowledge of others – typically in the context of some particular experimental setup or system.

For example the system might consist of an electron emitted from a cathode or electron gun at a particular location into an evacuated environment which includes a barrier with two slits between the cathode and a phosphorescent screen. In this case questions with known answers include the position and time of emission and what we want to predict are probabilities of seeing a flash in various regions on the screen (which might be combined into the form of a probability density function).

In a classical theory, any range of values of momentum, position and time determines an experimentally testable question and in principle the electron could have arbitrarily precise position and momentum at any particular time, and the lattice of all such questions is just like that of all propositions in classical logic. The most complete specification of the state at each time is just a point in phase space (ie a precise value of position and momentum) and incompletely specified states can be identified as statistical mixtures of these pure states corresponding to probability density functions on the phase space.

But in quantum theory the position and momentum cannot both be measured with arbitrary precision at the same time and the lattice of all experimentally testable propositions does not have all the completeness properties of a classical logic. The most complete specification of the state in this theory corresponds to a ray (or unit vector) in a Hilbert Space (of “wave functions”) but such a pure state may still not have precise values for all observables and a pure state with precise values for one observable (often called an eigenstate for that observable) may correspond to a linear superposition of eigenstates with different eigenvalues for complementary observables. Even less completely specified states can be identified as statistical mixtures of these pure states corresponding to weighted sums or integrals of pure state projection operators to yield trace class operators called density matrices.

When QM is used to predict probabilities for outcomes at the end of an experiment where the system under study is essentially destroyed (as with the electron being absorbed into the phosphor of the screen) the question of what happens during and after the measurement does not arise. But if we place a non-destructive detector within the experiment then we need to ask how to analyse and predict what happens to the combined system. It appears in that case that if we ignore the quantum nature of the detector then the result for the electron looks as if, at the time when the intermediate measurement is made, the electron’s state suddenly jumps to the eigenstate for the value that happens to be observed. And explaining how this is consistent with a quantum description of the combined system of electron and detector is what is known as the “measurement problem”

See Quora questions:

What is the measurement problem of quantum mechanics?