content_image/371120.jpg content_image/371119.jpg content_image/371118.jpg content_image/371117.jpg content_image/371116.jpg
## C. Fidelity with a projective measurement 

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 

<img src='content_image/371120.jpg'>

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’: 

<img src='content_image/371119.jpg'>

The proposition operator (7) about the state of the system, just after its preparation, is 

<img src='content_image/371118.jpg'>

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: 

<img src='content_image/371117.jpg'>

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, 

<img src='content_image/371116.jpg'>

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 

