  This tutorial is devoted to review the modern tools of quantum mechanics,
which are suitable to describe states, measurements, and operations of
realistic, not isolated, systems in interaction with their environment, and
with any kind of measuring and processing devices. We underline the central
role of the Born rule and and illustrate how the notion of density operator
naturally emerges, together the concept of purification of a mixed state. In
reexamining the postulates of standard quantum measurement theory, we
investigate how they may formally generalized, going beyond the description in
terms of selfadjoint operators and projective measurements, and how this leads
to the introduction of generalized measurements, probability operator-valued
measures (POVM) and detection operators. We then state and prove the Naimark
theorem, which elucidates the connections between generalized and standard
measurements and illustrates how a generalized measurement may be physically
implemented. The "impossibility" of a joint measurement of two non commuting
observables is revisited and its canonical implementations as a generalized
measurement is described in some details. Finally, we address the basic
properties, usually captured by the request of unitarity, that a map
transforming quantum states into quantum states should satisfy to be physically
admissible, and introduce the notion of complete positivity (CP). We then state
and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate
the connections between the CP-maps description of quantum operations, together
with their operator-sum representation, and the customary unitary description
of quantum evolution. We also address transposition as an example of positive
map which is not completely positive, and provide some examples of generalized
measurements and quantum operations.
