Wigner function = Fourier transform + Coordinate rotation

[Follow-up post: In what sense is the Wigner function a quasiprobability distribution?]

I’ve never liked how people introduce the Wigner function (aka the Wigner quasi-probability distribution). Usually, they just write down a definition like

(1) $\begin{align*} W(x,p) = \frac{1}{\pi \hbar} \int \mathrm{d}y \rho(x+y, x-y) e^{-2 i p y/\hbar} \end{align*}$

and say that it’s the “closest phase-space representation” of a quantum state. One immediately wonders: What’s with the weird factor of $2$ , and what the heck is $y$ ? Usually, the only justification given for the probability interpretation is that integrating over one of the variables recovers the probability distribution for the other (if it were measured):

(2) $\begin{align*} \int \! \mathrm{d}p \, W(x,p) = \vert \rho(x,x) \vert^2 , \\ \int \! \mathrm{d}x \, W(x,p) = \vert \hat{\rho}(p,p) \vert^2 , \end{align*}$

where $\hat{\rho}(p,p')$ is just the density matrix in the momentum basis. But of course, that doesn’t really tell us why we should think of $W(x,p)$ , as having anything to do with the (rough) value of $x$ conditional on a (rough) value of $p$ .

Well now I have a much better idea of what the Wigner function actually is and how to interpret it. We start by writing it down in sane variables (and suppress $\hbar$ ):

(3) $\begin{align*} W(\bar{x},\bar{p}) = \frac{1}{2 \pi} \int \! \mathrm{d}\Delta x \,\rho \left(\bar{x}+\frac{\Delta x}{2}, \bar{x}-\frac{\Delta x}{2} \right) e^{-i \bar{p} \Delta x}. \end{align*}$

So the first step in the interpretation is to consider the function

(4) $\begin{align*} M(\bar{x},\Delta x) \equiv \rho \left(\bar{x}+\frac{\Delta x}{2}, \bar{x}-\frac{\Delta x}{2} \right) , \end{align*}$

which appears in the integrand. This is just the (position-space) density matrix in rotated coordinates $\bar{x} \equiv (x+x')/2$ and $\Delta x = x-x'$ . There is a strong sense in which the off-diagonal terms of the density matrix represent the quantum coherence of the state between different positions, so $\Delta x$ indexes how far this coherence extends; large values of $\Delta x$ indicate large spatial coherence. On the other hand, $\bar{x}$ indexes how far down the diagonal of the density matrix we move; it’s the average position of the two points between which the off-diagonal terms of the density matrix measures coherence. (See the figure below.)

The function $M(\bar{x},\Delta x)$ is just the position-space density matrix $\rho(x,x')$ rotated in new coordinates: $(\bar{x},\Delta x) = ((x+x')/2,x-x')$ . $\bar{x}$ moves down the diagonal, and $\Delta x$ reaches off-diagonal.

So then we see that the Wigner function $W(\bar{x},\bar{p})$ is just the Fourier transform of $M(\bar{x},\Delta x)$ with respect to the coherence variable $\Delta x$ .

(5) $\begin{align*} W(\bar{x},\bar{p}) = \int \! \mathrm{d}\Delta x \, M(\bar{x},\Delta x) e^{-i \bar{p} \Delta x}. \end{align*}$

We can express this more compactly as

(6) $\begin{align*} W = \mathcal{F}_{\Delta x \to p}[R_{x,x' \to \bar{x},\Delta x}[\rho]]. \end{align*}$

where $\mathcal{F}_{\Delta x \to p}$ denotes the Fourier transform from $\Delta x$ to $\bar{p}$ and $R_{x,x' \to \bar{x},\Delta x}$ denotes the relevant $45^\circ$ coordinate rotation from $(x,x')$ to $(\bar{x},\Delta x)$ . This interpretation has several of advantages.

First, all the numerical constants in the formula for the Wigner function can be immediately traced back to the Fourier transform. (Different Fourier transform conventions will of course have different constants.)

Second, transforming the density matrix into a Wigner function is a manifestly reversible operation, since both the Fourier transform and the rotation are reversible.^a So the Wigner function is just another lens with which we can look at the state of a quantum system. We could just as well always work with Wigner functions when analyzing the dynamics of a quantum system, although the Hamiltonian evolution turns out to be a little uglier to express.

Third, we can also see how the Wigner function is related to the momentum-space representation of the density matrix by “filling in” all the possible applications of Fourier transforms and coordinate rotations. (See the next figure.) This lets us immediately and transparently derive that

(7) $\begin{align*} W(\bar{x},\bar{p}) &= \int \! \mathrm{d}\Delta p \, \hat{M}(\bar{p},\Delta p) e^{+i \bar{x} \Delta p} \\ &= \int \! \mathrm{d}\Delta p \, \hat{\rho} \left(\bar{p}+\frac{\Delta p}{2},\bar{p}-\frac{\Delta p}{2} \right) e^{+i \bar{x} \Delta p}. \end{align*}$

where $\hat{M}(\bar{p},\Delta p)$ is the relevant coordinate rotation of $\hat{\rho},(p,p')$ .

Diagram of how the position- and momentum-space representations of the density matrix are related to the Wigner function by Fourier transforms and coordinate rotations. Auxillary functions $\sigma$ , $\hat{\sigma}$ , $M$ , $\hat{M}$ , and $\hat{W}$ have been introduced to fill in possibilities. Each function can be transformed into any other.

The (doubly) Fourier transformed Wigner function^b

(8) $\begin{align*} \hat{W}(\Delta p,\Delta x) \propto \int \int \! \mathrm{d}\bar{x} \, \mathrm{d}\bar{p} \, W(\bar{x},\bar{p}) e^{+i \bar{x} \Delta p} e^{-i \bar{p} \Delta x} \\ \end{align*}$

(aka the Moyal characteristic function, aka the Wigner characteristic function, especially when the arguments are swapped) also has a nice relationship to the Husimi Q function $Q(\vec{\alpha}) = \langle \vec{\alpha} \vert \rho \vert \vec{\alpha} \rangle$ . Here, $\vec{\alpha} = (\bar{x},\bar{p})$ is just a condensed notation for phase space coordinates and $\vert \vec{\alpha} \rangle$ is the (unsqueezed) coherent state centered at $(\bar{x},\bar{p})$ . It’s well known that the Husimi Q function is just the Wigner function convolved with a Gaussian:

(9) $\begin{align*} Q(\vec{\alpha}) \propto \int \! \mathrm{d}\vec{\beta} \, W(\beta) e^{-\vert \alpha - \beta \vert^2/2} \\ \end{align*}$

where $\mathrm{d}\vec{\beta} = \mathrm{d}\bar{x} \, \mathrm{d}\bar{p}$ and $\vert \vec{\alpha} - \vec{\beta} \vert^2 = \bar{x}^2+\bar{p}^2$ . (Up to normalization, the Wigner representation of the coherent state at $\vec{\alpha}_0$ is a Gaussian: $e^{-\vert \vec{\alpha}_0 - (\bar{x},\bar{p}) \vert^2/2}$ .)

But of course, by the convolution theorem, a convolution in real space is just multiplication in Fourier space:

(10) $\begin{align*} \hat{Q}(\vec{\nu}) \propto e^{-\vert \vec{\nu} \vert^2/2} \hat{W}(\vec{\nu}) \\ \end{align*}$

where

(11) $\begin{align*} \hat{Q}(\vec{\mu}) &\propto \int \! \mathrm{d}\vec{\alpha} Q(\vec{\alpha}) e^{-i \vec{\mu} \cdot \vec{\alpha}}, \\ \hat{W}(\vec{\mu}) &\propto \int \! \mathrm{d}\vec{\alpha} W(\vec{\alpha}) e^{-i \vec{\mu} \cdot \vec{\alpha}}. \end{align*}$

In fact, the Fourier transformed Wigner function $\hat{W}$ has more meaning than you might think. Structure of the Wigner function inside areas smaller than $\hbar$ are known to be associated with the type of long-scale coherence that is typically destroyed by environmental decoherence.^c But the Fourier transformed Wigner function inverts this; the sub-Planckian structure now is on large scales, while the supra-Planckian structures (which gives the classical phase space probability distribution) is compressed to small scales. When calculating the Husimi Q function, we see that the Gaussian just suppresses the large-scale structure (i.e. the coherence) while leaving the small-scale structure untouched. (Of course, we could probably have read this off directly from the convolution.)

[Edited 2014-6-10]

Footnotes

(↵ returns to text)

Remember that, aside from being a complex-valued function on $R^2$ , the density matrix $\rho$ must also be positive. I haven’t bothered to figure out how that requirement translate into restrictions on the Wigner function beyond the simple fact that it’s a real-valued function on $R^2$ . The unit-trace requirement on $\rho$ is equivalent to the requirement that integrating $W(\bar{x},\bar{p})$ over its domain equals unity.↵
I’m going to stop trying to bother with normalization.↵
Furthermore, this structure is responsible for superpositions being useful in quantum enhanced measurements.↵

6 Comments

Vineeth Bhaskara
January 26, 2017 at 6:30 pm

Oh.. man! That is one of the *BEST* explanations I have seen so passionately explained on the internet on that too a serious subject! Thanks a bunch! Keep writing. :)
Lucy Keer
August 26, 2018 at 7:40 am

Thanks very much for writing this up! I had very similar frustrations recently in trying to understand the Wigner function and am currently writing my own notes – I’ve ended up taking a fairly different route but have referred back to yours many times.
(I tried to attach my notes but the spam filter didn’t like them – anyway you can get to them from the website link I included.)
The most elegant characterisation I found is described in e.g. this paper by Wootters: https://www.sciencedirect.com/science/article/pii/000349168790176X – as well as reproducing the position and momentum marginals, you can also integrate along any line in phase space and get the marginal for an observable that’s the appropriate linear combination of position and momentum. (Well, the actual statement in the paper is a bit more involved and involves a strip of phase space instead of a line, but still pretty close.)
- Jess Riedel
  August 26, 2018 at 8:39 am
  
  Thanks Lucy. Sorry about the spam filter; the options provided by WordPress are crummy and I haven’t found a good balance between ease-of-use and not being overwhelmed by spam. For others, your notes can be found here.
  The Wooters paper is a classic, but I actually think the property of recovering the marginal distribution along any straight line in phase space isn’t as important for establishing intuition as the fact that the Wigner function is the “convolutionary mean” of the Husimi Q function and the Shudarshan P function.
Lucy Keer
August 26, 2018 at 7:45 am

Oops, that was a bit unclear – meant to say that I tried to add a link to my notes, but the spam filter didn’t like it, but they can be reached from the website link attached to my name.
Mahmoud
September 13, 2018 at 1:18 pm

Now I know that the negative values of Wigner function represents of the non-classical state for the density operator, or ‘quantumness’ of states. While, if the Glauber-P is non-negative, then the quantum system has a classical analog, and if the Glauber-P is negative, the quantum system has no classical analog. So, I need to know what is the different between them? Why studied together if one of them is enough? In addition to what is the physical meaning of Husimi-Q function?
Please!!
- Ninnat Dangniam
  December 28, 2018 at 1:54 am
  
  Mahmoud, one has to be careful how to define “quantumness” based on the negavitity of a quasi-probability distribution. But let’s say that a state is “classical” if any measurement on it can be modeled as the phase space average $\int d^2 \vec{\alpha}\, \mu(\vec{\alpha})\xi(\vec{\alpha})$ of a positive quasi-probability $\mu(\vec{\alpha})$ of the state and a positive quasi-probability $\xi(\vec{\alpha})$ of the measurement, then states with positive P-functions are classical because the corresponding quasi-probability representations for measurements in the integral must be Q-functions, which are always positive for positive operators. (I believe that one has to put in some assumption such as Heisenberg-group covariance to prove the necessity.) This is the duality of the P and the Q representations, whereas the Wigner representation is self-dual.
  Generally there are infinitely many quasi-probability representations that you can define for your system, and states that have positive/negative quasi-probabilities in one aren’t necessarily positive/negative in the others, so negativity alone can’t tell you about quantumness. This review paper by Chris Ferrie has a good exposition along this line.