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