Quasi-probabilities in quantum mechanics: fundamental ideas and applications

In addition to the usual projective measurements, quantum mechanics allows for alternative ways to extract information from a quantum system. Some of these lead to a quasi-probability distribution for the observable measured which are not positively defined. In perfect analogy with the Wigner quasiprobability distribution, the presence of negative regions in the distribution can be used to spot pure quantum behaviors of the system or the dynamics. I will present a particular scheme, called quantum non-demolition, where quasi-probability distribution arises naturally. It exploits an additional quantum detector coupled to the system to be measured which allows us to gain important information about the wave-function of the system. I will discuss what are the advantages and disadvantages of this approach with some practical examples: the measure of the work done on a quantum system driven by an external field and the calculation of the derivative of a quantum operator. In the last part of the talk, I will discuss the connection with the violation of the Leggett-Garg inequalities, and how this approach identifies pure quantum effects and quantum-to-classical transition due to the interaction with an environment.