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

Question

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?

This post imported from StackExchange Physics at 2016-03-28 19:42 (UTC), posted by SE-user quantum_loser

Answer 1

I think your question might be too broad... Unlike density matrices which are positive semidefinite operators, the Wigner function in phase space is not. (Conversely, a classical positive definite phase space Liouville density Wigner transforms to a non-positive definite "Groenewold operator", Bracken and Wood.)

You might consider, if you already haven't, the Husimi distribution, which, being a Weierstrass transform of the Wigner function, corresponds to low-pass filtering thereof, and is, in fact, positive semidefinite---the price paid for loss of information, aggressively non-unitary to be sure, due to this Gaussian blurring.

But it might help if your question were narrower and more specific.

This post imported from StackExchange Physics at 2016-03-28 19:42 (UTC), posted by SE-user Cosmas Zachos