An ideal measurement checks a simple proposition corresponding to a projector ˆ Π$_{n}$ = | ψ$_{n}$ 〉〈 ψ$_{n}$ | in the Hilbert space. However, in more realistic situations, a measuring device is characterized by POVM elements which are not at all projectors. We have actually an evaluation of the projectivity of a measurement with the purity π$_{n}$ of its pre-measurement state: such that 1 /D < π$_{n}$ ≤ 1. When the pre-measurement state is a pure quantum state with π$_{n}$ = 1, the measurement performed by the apparatus is projective for the response ’n’, but it may be non-ideal . Indeed, the POVM element corresponding to such a projectivity ( π$_{n}$ = 1) is given by where η$_{n}$ = Tr { ˆ Π$_{n}$ } can be viewed as the detection efficiency of the state | ψ$_{n}$ 〉 , by using the predictive approach in which the predictive probability Pr ( n | ψ$_{n}$ ) = η$_{n}$ . We clearly see on this property the complementarity between the predictive and retrodictive approaches in quantum physics. For an APD, the projectivity is only relevant for the measurement corresponding to the response ’off’ which is characterized in the ideal case ( η = 1) by the vacuum state | 0 〉 . Indeed, the pre-measurement state retrodicted from the result ’off’ is and the projectivity of this measurement is given by: 13 π$_{n}$ = Tr [ ( ˆ ρ [ n ] retr ) 2 ] , (17) ˆ Π$_{n}$ = η$_{n}$ | ψ$_{n}$ 〉〈 ψ$_{n}$ | (18) ˆ ρ [off] retr ( η, ν ) = η ∞ ∑ n =0 (1 − η ) $^{n}$| n 〉〈 n | , (19) π$_{off}$ ( η, ν ) = η 2 − η . (20) Contrary to the POVM element (15), we can see that this state and its properties do not depend on the dark noise ν . We can easily understand this since the dark counts correspond to results ’on’. Thus, when we have the result ’off’, the properties of the performed mea- surement only depends on the detection efficiency η . If we have a quasi-ideal measurement η ≃ 1, we are sure of the pre-measurement state even for a dark noise arbitrarely large, which may be interesting for conditional protocols based on such measurements. B. Projectivity