IV. HOW TO STUDY AN APPARATUS ? The POVM elements should be understood as propositions about the measured system. These propositions are actually labeled by the responses of the measuring device, and this one appears as a physical implementation of tests checking these propositions. However, the POVM elements only allow us to study the behavior of an appartus, by giving the predictive probabilities (1) of having a certain response. Thus, how could we go beyond these simple conditional probabilities in order to study the measurement apparatus ? The POVM elements are not sufficient for studying such effects in the general case. Indeed, the effects of a generalized measurement on a state are given by the following trans- formation (see for example [14]): where ˆ M$_{n,µ}$ are the measurement operators, also called Kraus operators [15], describing completely the measurement process corresponding to the result ’n’. The probability P n of having this result is then given by the POVM element: The normalization of expression (8) by the probability P n is the manifestation of the so- called projection rule , leading to a non-linear transformation of the initial state contrary to an unread measurement. When we perform the measurement on only one part B of a bipartite entangled resource ˆ ρ$_{AB}$ , the state of the other part A - conditioned on the result of this measurement - depends on the POVM element corresponding to the expected result ’n’: In quantum optics, such protocols have been recently implemented for preparing more ”ex- otic” non-classical states from gaussian resources, such as Fock states [16] or Schr¨ odinger’s cat states [17]. 8 ˆ ρ [ n ] A , cond = 1 P n Tr$_{B}$ { ˆ ρ$_{AB}$ ˆ 1$_{A}$ ⊗ ˆ Π$_{n}$ } . (10) ˆ Π$_{n}$ = ∑ µ ˆ M † n,µ ˆ M$_{n,µ}$. (9) ˆ ρ → ˆ ρ [ n ] cond = ∑ µ ˆ M$_{n,µ}$ ˆ ρ ˆ M † n,µ P n , (8) A. Measurement effects on a system