This chapter presents the theoretical framework, based on Gleason’s theorem, allowing to describegeneralized measurements, in addition to von Neumann measurements, in terms of POVMs, probability operators and post-measurement operators. It mentions the Naimark theorem according to which a generalized measurement is a von Neumann measurement if one describes it in a Hilbert space of higher dimension. Examples of generalized measurements are given: imperfect measurements, simultaneous measurements of noncommuting operators. It presents the Zurek model that accounts for the decoherence process occurring in a measurement and shows that the quantum measurement process, including state collapse, is not a physical evolution. Finally, it studies the case of successive measurements using Bayes statistics in which the state collapse appears as an updating of the information about a system, and the fundamental property of repeatability of quantum mechanics.

This is a brief excursion into atopic that belongs to quantum information theory. We prove here that the Connes-Kirchberg problem is equivalent to a question raised by Tsirelson in connection with quantum mechanics. The question involves correlation matrices of various kinds that we discuss using operatorvalued measures in order to highlight the equivalence ofthe Tsirelson problem with the LLP of certain free product C*-algebras.