Quantum Optics Question and Wigner Functions

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SUMMARY

The discussion clarifies the distinctions between the Wigner function and the Q function in quantum optics. The Wigner function is a quasi-probability distribution that can take negative values, while the Q function is always positive. To compute the expectation value of an observable using the Q function, one must integrate it against the P-function of the observable, which can exhibit negative values and singularities. The Wigner, Q, and P functions form a one-parameter family of quasi-probability distributions, allowing transitions between them through convolution or deconvolution with Gaussians.

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  • Understanding of quasi-probability distributions
  • Familiarity with quantum optics concepts
  • Knowledge of Wigner, Q, and P functions
  • Basic calculus for integration of functions
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Raptor112
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I understand that Wigner function is a quasi-probability distibution as it can take negative values, but in quantum optics I see that the Q function is mentioned as often. So what is the difference between the Q function and the Wigner Function?
 
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Q-functions are always positive. This does not mean however that quantum mechanics reduces to classical statistical mechanics, because to calculate the expectation value of an observable from it, you have to integrate it against the P-function of the observable which not only can take negative values but can be "worse" than that (e.g. being more singular than a delta distribution).

Wigner, Q- and P-functions from a one-parameter family of quasi-probability distributions and you can move between them by a convolution or deconvolution of the functions with Gaussians. Generally if you represent quantum states by one of them, you have to represent observables by the quasi-probability representation dual to it. Only the Wigner representation is self-dual.

I hope this helps.
 

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