and a PNRD of n photons is given by in which we recognize the predictive probability (1) of having the result ’on’ when the light was prepared in the photon-number state | n 〉 : Thus, the behavior of the fidelity F $_{on}$( n, η, ν ) is the same as that of the predictive probability (28). Its evolution with the photon number n and the detection efficiency η is pictured on Fig. 5 (b). For a given efficiency, the large photon-number states are the most probable preparations leading to the result ’on’, with or without dark noise. We can also see the effect of this one on Fig. 5 (b), which is mainly a non-zero probability for a totally inefficient detector ( η = 0). This probability is the same for all the photon-number states. We understand this since the APD has actually no interaction with the signal field. The non-Gaussian character of a measurement is essential in many quantum information protocols such as entanglement purification [25]. This property can be measured by the non-Gaussian character of its pre-measurement state . We can compute for instance its ”non-Gaussianty” (non-G) [26] defined by the relative Von Neumann entropy between this pre-measurement state and its reference gaussian state, which is the gaussian state with the same covariance matrix. The non-G is therefore equal to zero only for gaussian states. Then, there is an interesting result based on Hudson-Piquet’s theorem [24]. It states that any pure state, characterized by a non-negative Wigner representation [18], is a gaussian state with a gaussian distribution for its Wigner representation. We have here an important link between the Gaussian character of a measurement and its projectivity : When a mea- surement is projective with a non-negative Wigner representation for its pre-measurement state, then this measurement is a Gaussian measurement . In order to illustrate such a Gaussian character, we determine the pre-measurement states retrodicted from results of a measuring device widely used in quantum optics : the Homodyne Detection (HD). Indeed, the inefficient HD of the value x$_{I}$ for the quadrature observable ˆ x$_{φ}$ 16 D. Non-Gaussian character of a measurement F $_{on}$( n, η, ν ) = Pr ( n | on) ∼ D → + ∞ 1 D Pr (on | n ) , (27) Pr (on | n ) = 1 − e − ν (1 − η ) n . (28)