Graduate Quantum Optics Question and Wigner Functions

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The discussion clarifies the differences between the Wigner function and the Q function in quantum optics, noting that the Wigner function is a quasi-probability distribution that can take negative values, while the Q function is always positive. Despite the positivity of the Q function, it does not imply that quantum mechanics simplifies to classical statistical mechanics, as calculating expectation values requires integrating against the P-function, which can exhibit negative values and singularities. Wigner, Q, and P functions form a one-parameter family of quasi-probability distributions that can be interconverted through convolution or deconvolution with Gaussians. When representing quantum states with one of these functions, observables must be represented by the dual quasi-probability representation, with the Wigner representation being unique in its self-duality. This explanation enhances understanding of the relationships between these functions in quantum optics.
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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.
 
Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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