# Measuring the Wigner Function

• A
• neils
neils
TL;DR Summary
How do you measure the Wigner Function of a quantum optical state?
Suppose i measure the phase and amplitude of some radiation, this might be a coherent state of an entangled state, how would i construct the Wigner function from these measurements?

• vanhees71
Gold Member
• vanhees71 and atyy
neils
Many thanks for the good links and works in this area. From looking at these i can now see that what I'm measuring is the Q-function. I can now appreciate both the Q and the Wigner function are both convolutions with the P-function.

However, i looking at the texts of the above i didnt see any method of generating the Wigner function from the Q-function. I now see this would require some kind of deconvolution, but it there a tried and tested method for this?

A further question arising is what might be the best function to analyse the quantum state, Q, P, or Wigner function, given that I'd be interested confirming measured entangled photon pair states?

thank you for any help.
Neil

However, i looking at the texts of the above i didnt see any method of generating the Wigner function from the Q-function. I now see this would require some kind of deconvolution, but it there a tried and tested method for this?

You can do so, but as any deconvolution process doing so may introduce errors and may involve ambiguities. For example, having negative values in the Wigner function is a flag for nonclassicality, while the Husimi Q function is necessarily non-negative. So reconstructing the negative parts via deconvolution does not really work well. The information content of all functions is the same, anyway. The Husimi Q function, the density matrix, the Wigner function and the Glauber-Sudarshan P distribution contain the same amount of information about the light field. However, in some instances it is easier to identify non-classicality using the Wigner function.

If you want to measure the Wigner function, the way to do it is balanced homodyne detection. You do not measure amplitude or phase, but a quadrature distribution which is the projection of the Wigner function along one axis. You may now repeat this for several axes by changing the relaitve phase between your signal and your local oscillator and can then perform quantum state tomography, where you reconstruct the Wigner function (or more typically the density matrix as it is easier) from this set of projections. A standard overview can be found here:
Reviews of Modern Physics paper by Lvovsky

A further question arising is what might be the best function to analyse the quantum state, Q, P, or Wigner function, given that I'd be interested confirming measured entangled photon pair states?

Entangled in what? The Wigner function is a phase space description, so it is well suited for looking at continuous degrees of freedom. If you consider an optical Schrödinger cat state to be entangled, the Wigner function would be a good way to look at it. However, if you really want to have a look at pair states, you need correlations between different modes, so you need to measure a two-mode Wigner function. This works well for twin beams in well defined modes, where you can measure pairs of quadratures (e.g. a joint measurement of the same quadrature for both beams followed by a measurement of the joint orthogonal quadratures in both beams), but if the entanglement includes many modes it is tedious and not really helpful to do homodyne detection due to its inherent mode sensitivity.

neils