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?


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. (There are further well-known deficiencies of the Husimi distribution, such as the substantial deformation of the square of the angular momentum, in some contrast to the plain Wigner map, which simply shifts that from its classical value by a mere constant.)

Or a discrete analog thereof.

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


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