We define this fidelity as the overlap between the pre-measurement state ˆ ρ [ n ] retr retrodicted from a certain result ’n’ and a target state | ψ$_{tar}$ 〉 , in which we would like checking the system before its interaction with the apparatus. Such a fidelity [23] can be written as With the retrodictive approach, we have an interesting interpretation for this overlap. This is nothing else than the retrodictive probability (2) of preparing the system in the target state | ψ$_{tar}$ 〉 , before the measurement process giving the result ’n’: The proposition operator (7) about the state of the system, just after its preparation, is We note that the marginal probability P tar = Pr ( ψ$_{tar}$ ) of preparing the system in this target state is 1 /D in this particular case. It ensures a maximum mixture of pure states constituting a basis of the Hilbert space. This basis is composed of the target state | ψ$_{tar}$ 〉 and ( D − 1) other pure states. Thus, for having an exhaustive description of the system, we need at least one other proposition: It corresponds to the preparation of the system in the state ˆ ρ$_{tar}$ = ˆ Θ$_{tar}$ / Tr { ˆ Θ$_{tar}$ } with the probability P tar = ( D − 1) /D . When the measurement giving the result ’n’ is sufficiently faithful F $_{n}$( ψ$_{tar}$ ) ≃ 1, the most probable state in which the system was prepared before its interaction with the apparatus is this target state | ψ$_{tar}$ 〉 . We now illustrate this notion for the measurements performed by an APD. For the mea- surement corresponding to the result ’off’, the fidelity with an ideal photon number-resolved detection (PNRD) of n photons is given by, The evolution of this fidelity with the detection efficiency η is plotted for different photon numbers n on Fig. 5 (a). As we can expect it for the response ’off’ of an efficient APD, the 14 F off ( n, η ) = Pr ( n | off) = η (1 − η ) $^{n}$. (25) ˆ Θ$_{tar}$ = ˆ 1 − ˆ Θ$_{tar}$ . (24) ˆ Θ$_{tar}$ = | ψ$_{tar}$ 〉〈 ψ$_{tar}$ | . (23) F $_{n}$( ψ$_{tar}$ ) = Pr ( ψ$_{tar}$ | n ) = Tr { ˆ ρ [ n ] $_{retr}$ˆ Θ$_{tar}$ } . (22) F $_{n}$( ψ$_{tar}$ ) = 〈 ψ$_{tar}$ | ˆ ρ [ n ] retr | ψ$_{tar}$ 〉 . (21) C. Fidelity with a projective measurement