content_image/357419.jpg content_image/357420.jpg content_image/357418.jpg
[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 . 

<img src='content_image/357419.jpg'>

<img src='content_image/357420.jpg'>

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 . 

<img src='content_image/357418.jpg'>

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 

