# Wigner characteristic function

I came across the "representation of a Gaussian state by its Wigner characteristic function". I don't know what Wigner characteristic function is and google results are not precise enough. Neither have I been able to find this in some quantum mechanics book.

It will be great if you could explain what it means, but even better would be citing a source from where I could read it up. I came across this term in this paper:

"Inseparability criterion for continuous variable systems". Duan, Giedke, Chirac and Zoller. Physical Review Letters (year 2000)

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A preprint version is available here: arxiv.org/abs/quant-ph/9908056 – Qmechanic May 24 '13 at 15:28
Some background material in this question physics.stackexchange.com/q/14906 – twistor59 May 24 '13 at 16:34
It's Ignacio Cirac, not Chirac! ;) – Vibert May 24 '13 at 17:37
@ Vibert: Yeah! Sorry :) – Man May 25 '13 at 13:15
A great reference is Wigner functions and Weyl transforms for pedestrians – bianchira Jul 25 '14 at 18:34

There are many ways to define the Wigner characteristic function corresponding to a density operator $\rho$. In the case of a single degree of freedom (i.e. one optical mode, or one point particle moving in one dimensions), one way is

$$\chi (\xi) = \mathrm{Tr}[\rho \hat{D}_{\xi}]$$

where

$$D_{\xi} =\mathrm{exp}(i \hat{X}^{\mathrm{T}} \! \omega \, \xi)$$

is the phase-space displacement operator,

$$\omega = \left( \begin{array}{cc} 0& 1\\-1&0 \end{array}\right)$$

is the symplectic form on phase space, and $\hat{X} = (\hat{q},\hat{p})$ is the phase-space coordinate operator giving the position-momentum coordinate pair. The coordinate $\xi=(\xi_{q},\xi_{p})$ ranges over conjugate phase space, i.e. the space of two-real coordinates over which symplectic Fourier transforms of functions are defined.

The symplectic Fourier transform $\hat{G}(\xi)$ of a function on phase space $G(X)$ is just like the normal Fourier transform of a function of two real variables except that the symplectic form is used in place of the regular dot product

$$\hat{G}(\xi) = \int \mathrm{d}X \, G(X) e^{i X^T \omega \, \xi} = \int \mathrm{d}X \, G(X) e^{ i X \wedge \xi}$$

Here, we have used the simplified notation $a \wedge b = a^{\mathrm{T}} \cdot \omega \cdot b$. Since $\omega$ just rotates in phase space by 90 degrees, the symplectic Fourier transform differs from the normal one just by this change of coordinates, $\xi \to \omega \,\xi$. (This isn't that important in the case of a single-degree of freedom, but it helps when things get more complicated.)

A very important property of the Wigner characteristic function is that it is the symplectic Fourier transform of the the Wigner quasiprobability distribution:

$$\chi(\xi) = \hat{W}(\chi) = \int \mathrm{d}X \, W(X) e^{+ i X \wedge \xi}$$

Confusingly, the Wigner quasiprobability distribution is usually just called "the Wigner function". (Jorge made this mistake in the other answer.) Wikipedia does an OK job describing the relation between the two in the article on quasiprobability distributions. In that article, they use complex coordinates $\alpha = x + i p$, and the complex conjugate operation, to encode the symplectic structure. This is equivalent to using $\omega$ or $\wedge$.

Using the most popular definition of $W(X)$, we can then write down an alternate form of the Wigner characteristic function for a pure state with wavefunction $\Psi(x)$:

$$\chi(\xi)=\frac{1}{2\pi\hbar}\int \Psi\left(x+\frac{\xi_x}{2} \right) \Psi^*\left(x-\frac{\xi_x}{2} \right)e^{+i x \xi_p } \mathrm{d} x$$

An excellent introduction to Wigner characteristic functions can be found in section II.A of

Weedbrook et al. "Gaussian Quantum Information", Rev. Mod. Phys. 84, 621 (2012) [arXiv:1110.3234].

Weedbrook et al. discuss this in the context of Gaussian states and information processing, but the introduction does not depend on that context. As they show, all of this can be generalized straightforwardly to multiple modes.

You also might like

J. Eisert, M.M. Wolf. "Gaussian quantum channels", Quantum Information with Continous Variables of Atoms and Light, p. 23-42 [arXiv:quant-ph/0505151].

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I gave an intuitive discussion of the Wigner quasiprobability distribution on my blog: blog.jessriedel.com/2014/04/01/… – Jess Riedel Jul 25 '14 at 18:29

From a problem sheet I have, the definition of the Winger distribution function is (for $\Psi$ normalized)

$$W(\vec{r},\vec{p})=\frac{1}{(2\pi\hbar)^3}\int\Psi^*\left(\vec{r}-\frac{\vec{r}^{'}}{2} \right) \Psi\left(\vec{r}+\frac{\vec{r}^{'}}{2} \right)e^{-i\vec{p}\cdot\vec{r}'/\hbar}d^3r'$$

Sadly it is quoted without reference.

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You've written down the Wigner function (sometimes called the Wigner quasiprobability distribution), not the Wigner characteristic function. They are symplectic Fourier transforms of each other. See Wikipedia. – Jess Riedel Jul 25 '14 at 16:34