[27] is characterized by the following POVM element [28]: in which η is the detection efficiency linked to quantum efficiencies of photodiodes used in such a device. The Wigner representation of this POVM element is then given by: with the variance σ 2 $_{x}$( η ) = 1 − η 2 η . The Gaussian character of this measurement is obvious as pictured on Fig. 6, in which we can distinguish the evolution of the variance σ 2 $_{x}$( η ) for different values of the detection efficiency η . However, the retrodiction of pre-measurement states needs some mathematical precau- tions since the trace of POVM elements (29) diverges. After taking a judicious equivalent for the Wigner representation (30), we obtain for the pre-measurement state retrodicted from 17 Figure 6: (color online) (a) Wigner representation of the POVM element ˆ Π$_{HD}$ ( x$_{I}$ ; φ ) for x$_{I}$ = 1 and η = 0 . 75 in the phase-space ( x$_{φ}$, p$_{φ}$ ). (b) Evolution of this Wigner representation with the detection efficiency η in the phase-space (x,p). ˆ Π$_{HD}$ ( x$_{I}$ ; φ ) = 1 √ π (1 − η ) exp [ − ( x$_{I}$ − √ η ˆ x$_{φ}$ ) 2 1 − η ] , (29) W $_{HD}$( x$_{φ}$, p$_{φ}$ ) = 1 √ 8 π $^{3}$η σ2 $_{x}$( η ) exp [ − ( x$_{φ}$ − x$_{I}$ / √ η ) 2 2 σ 2 $_{x}$( η ) ] , (30) 1 − η