only PNRD which is the most faithful to the measurement performed by the APD corre- sponds to the photon number n = 0. In terms of retrodictive probabilities (22), this means that the most probable state in which the system was prepared, before the measurement giving the result ’off’, is the vacuum state. For a low efficiency η → 0, all the photon-number states become equally probable. We can understand this since the measurement is not at all projective ( π$_{off}$ ≃ 0) for such efficiencies. Figure 5: (color online) Fidelities between the measurements performed by an APD and PNRDs : (a) Evolution of the fidelity, between the measurement corresponding to the result ’off’ and projective measurements of n photons, with the quantum efficiency η . (b) Evolution of the predic- tive probability Pr (on | n ) with the quantum efficiency η and the photon-number n , for two mean numbers of dark counts ν . For the measurement corresponding to the result ’on’, the situation is more delicate since the POVM element corresponding to this result leads to divergences in the retrodiction of the pre-measurement state. Indeed, we can only determine an asymptotic equivalent for this pre-measurement state: where D is in fact the dimension of the Hilbert space. The fidelity between this measurement ˆ ρ [on] $_{retr}$( η, ν ) ∼ D → + ∞ 1 D D ∑ n =0 [ 1 − e − ν (1 − η ) $^{n}$] | n 〉〈 n | , (26) 15