content_image/337687.jpg content_image/337686.jpg
in which η and ν are respectively the detection efficiency and the mean number of dark counts. On the other side, the Wigner representation [18] of the POVM element ˆ Π$_{on}$ = ˆ 1 − ˆ Π$_{off}$ is given by: 

<img src='content_image/337687.jpg'>

where L$_{m}$ ( x ) = e $^{x}$∂m x ( x $^{m}$e − $^{x}$) /m ! are the Laguerre polynomials. 

When we read the result ’on’, the non-classicality of the measurement performed by the APD can be measured by the negativity of the Wigner representation of its pre-measurement state. To avoid some mathematical difficulties, we choose for the negativity N $_{on}$( η, ν ) = W $_{on}$(0 , 0), since we have Tr { ˆ Π$_{on}$ } > 0. The evolution of this signature of non-classicality, under the noise influence, is pictured on Fig. 4. 

<img src='content_image/337686.jpg'>

Thus, this signature stays negative for a mean number of dark counts such that ν < − ln (1 − η/ 2). We can easily check this for a given APD, by simply measuring its detection efficiency η and its dark noise ν . Moreover, we can compare such a signature of non- classicality for different ”on/off” photon detectors, by simply plotting their points ( η, ν ) on Fig. 4. 

