# How to transform a wigner function to represent loss of mode information (coarse graining)?

I have a highly multi-mode gaussian wigner function representing an optical field:

$$W\left(\{p\},\{q\}\right)=\mathrm{Exp}\left(-\sum_{j=0}^{f}(b_{j}q^{2}_{j}+a_{j}p^{2}_{j})\right).$$

However the detector I am modeling can only distinguish "groupings" of the modes labelled by by $j$ (i.e. there are only a few distinguishable modes but thousands of physical modes). In a sense many physical modes are "coarse grained" into a detection mode.

Normally I would just use a POVM for the detection function containing a sum of all the modes in question, but for somewhat complicated reasons (that are not relevant to the question) I can not do this. Instead I'm trying to figure out how to perform such a (non-unitary) transformation to the mode variables themselves.

Does anyone know how to apply such a transformation to the wigner function?