In what sense is the Wigner function a quasiprobability distribution?

By Jess Riedel September 22, 2014 May 22, 2020 Physics

For the upteenth time I have read a paper introducing the Wigner function essentially like this:

The Wigner-representation of a quantum state $\rho$ is a real-valued function on phase space defined^a (with $\hbar=1$ ) as
(1) $\begin{align*} W_\rho(x,p) \equiv \int \! \mathrm{d}\Delta x \, e^{i p \Delta x} \langle x+\Delta x /2 \vert \rho \vert x-\Delta x /2 \rangle. \end{align*}$
It’s sort of like a probability distribution because the marginals reproduce the probabilities for position and momentum measurements:
(2) $\begin{align*} P(x) \equiv \langle x \vert \rho \vert x \rangle = \int \! \mathrm{d}p \, W_\rho(x,p) \end{align*}$
and
(3) $\begin{align*} P(p) \equiv \langle p\vert \rho \vert p \rangle = \int \! \mathrm{d}x \, W_\rho(x,p). \end{align*}$
But the reason it’s not a real probability distribution is that it can be negative.

The fact that $W_\rho(x,p)$ can be negative is obviously a reason you can’t think about it as a true PDF, but the marginals property is a terribly weak justification for thinking about $W_\rho$ as a “quasi-PDF”. There are all sorts of functions one could write down that would have this same property but wouldn’t encode much information about actual phase space structure, e.g., the Jigner^b function

$J_\rho(x,p) \equiv P(x)P(p) = \langle x \vert \rho \vert x \rangle \langle p \vert \rho \vert p \rangle,$

which tells as nothing whatsoever about how position relates to momentum.

Here is the real reason you should think the Wigner function $W_\rho$ is almost, but not quite, a phase-space PDF for a state $\rho$ :

Consider an arbitrary length scale $\sigma_x$ , which determines a corresponding momentum scale $\sigma_p = 1/2\sigma_x$ and a corresponding set^c of coherent states $\{ \vert \alpha \rangle \}$ .
If a measurement is performed on $\rho$ with the POVM of coherent states $\{ \vert \alpha \rangle \langle \alpha \vert \}$ , then the probability of obtaining outcome $\alpha$ is given by the Husimi Q function representation of $\rho$ :
(4) $\begin{align*} Q_\rho(\alpha) = \langle \alpha \vert \rho \vert \alpha \rangle. \end{align*}$
If $\rho$ can be constructed as a mixture of the coherent states $\{ \vert \alpha \rangle \}$ , then^d it is given by
(5) $\begin{align*} \rho = \int \mathrm{d}\alpha P_\rho(\alpha) \vert \alpha \rangle \langle \alpha \vert, \end{align*}$
where $P_\rho(\alpha)$ is the Glauber P function representation of $\rho$ .
Because the coherent states obey $\vert \langle \beta \vert \alpha \rangle \vert^2 = g_{\sigma_x}(\alpha - \beta)$ where $g_{s}(\gamma) = \mathrm{exp}(-\vert \gamma \vert^2/2 s^2)/s \sqrt{2 \pi}$ is the Gaussian function of width $s$ centered on the origin, we have that the Q function is just the P function convolved with a Gaussian of width $\sigma_x$ :
(6) $\begin{align*} Q_\rho(\alpha) &= \langle \alpha \vert \rho \vert \alpha \rangle = \int \mathrm{d}\beta \vert \langle \beta \vert \alpha \rangle \vert^2 P(\beta) \\ &= (P_\rho \circ g_{\sigma_x}) (\alpha), \end{align*}$
where $\circ$ denotes the convolution. Convolution is equivalent to multiplication in Fourier space, so convolving with the Gaussian $g_{\sigma_x}$ is equivalent to smoothly suppressing high-frequency modes with a frequency cut-off scale set by $1/\sigma_x$ .
The Wigner function $W_\rho$ is exactly in between the two, in the sense that it is both the convolution of the P function $P_\rho$ with a (half-as-wide) Gaussian $g_{\sigma_x/2}$ and the inverse convolution of the Q function $Q_\rho$ with $g_{\sigma_x/2}$ :
(7) $\begin{align*} Q_\rho = W_\rho \circ g_{\sigma_x/2} \quad , \quad W_\rho = P_\rho \circ g_{\sigma_x/2} \end{align*}$
which immediately recovers $Q_\rho = P_\rho \circ g_{\sigma_x/2} \circ g_{\sigma_x/2} = P_\rho \circ g_{\sigma_x}$ .
Strikingly, the Wigner function $W_\rho$ is well-defined for all states $\rho$ and (unlike the P and Q functions) does not depend^e on our choice of $\sigma_x$ .

So here’s a catch phrase: The Wigner function is the unique, scale-independent “convolutionary mean” of the Glauber P function and the Husimi Q function^f. This is highly illuminating because the P and Q functions have immediate operational interpretations as honest phase-space PDFs, respectively representing the preparation and measurement descriptions of the state $\rho$ . It also tells us that the Wigner function cannot have negative regions larger than about $\hbar$ , since regions of that size would not become positive under convolution into the Q function. More generally, the Q, W, and (when it exists) P functions cannot differ except in high frequency phase-space modes that are ironed out by the convolutions.

Edit 2014-9-24: Minor changes, with “Gaussian-convolution” –> “convolutionary”, because that sounds like a cooler adjectives.

Footnotes

(↵ returns to text)

Actually, they usually use a more confusing definition. See my post on the intuitive definition of the Wigner function.↵
“Jess” + “Wigner” = “Jigner”. Ha!↵
Not just a set of states, actually, but a Parseval tight frame. They have a characteristic spatial and momentum width $\sigma_x$ and $\sigma_p$ , and are indexed by $\alpha = (x,p)$ as it ranges over phase space.↵
Of course, the P function cannot always be defined, and sometimes it can be defined but only if it takes negative values. (Inverse convolutions do not always exist.) But the Wigner function is always well-defined, forming (up to mathematical technicalities) a one-to-one representation of density matrices $\rho$ .↵
In fact I think you can use, not just the set of coherent states, but a set of states generated by the phase-space displacement operator from any normalizable pure state. (This “base” state, which is just the vacuum for coherent states, is used to construct the kernel of the corresponding convolutions.) See [arXiv:quant-ph/0612037]. Neat stuff.↵
Here’s I’m defining the “convolutionary mean” analogously to the “geometric mean”. Is this a thing?↵

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4 Comments

Francesco Albarelli
September 9, 2015 at 6:44 am

Hi, sorry if I comment this oldish post, but I have been reading many posts in your blog and this one seems the most related to the thoughts I want to express.
Like many others I have been introduced to quasi-probability distributions in the context of quantum optics and they are used to define the concept of “nonclassicality” of a state.
In his seminal works, Glauber argued that the P function is the one that truly defines a nonclassical state (this has view also be reinforced by this for example http://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.062114 ), and because of the Gaussian convolution you talk about, negativity of the Wigner function is just a sufficient condition.
However negativity of the Wigner function seems to have an operative meaning related to the efficient simulation of the system ( http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.230503 and https://iopscience.iop.org/article/10.1088/1367-2630/15/1/013037 ).
In quantum optics “nonclassical light” mostly refers to squeezed states, that actually have a practical use to improve tasks in metrology.
From another point of view though they are not that “nonclassical” since they are generated by quadratic operations in the phase space representation that can be represented as symplectic transformations on the modes, so in a sense there exists a “classical” picture to visualize them…
I am writing this because I have been reading a lot about these things, but I still have a big mess in my head, so I am sorry if my point it not very clear.
I want to hear your opinion on these matters and maybe let me know if you have an answer to the question: have optical systems have anything special? I mean, is it different to recover the classical equations of motion of a particle in some suitable limit, than recover the Maxwell equations and their phenomenology? In this sense does it make sense to define a nonclassical state differently for optical systems than for other (mechanical for example) ones?
I decided to write because you seem to be knowledgeable enough to shed a little light on this, or at least point me to some relevant literature, or maybe just tell me that I have not understood anything at all (a possibility I am completely open to).
Keep up your nice work with this blog :)
Reply
- Jess Riedel
  September 10, 2015 at 11:04 am
  
  Hi Francesco, thanks for the kind words.
  The quantum-classical limit is a mapping from a very big space (Hilbert space) to a much small space (classical phase space) that only makes sense in a full ℏ→0 limit. I don’t think it is useful to say that individual quantum states suddenly qualify as “classical” when they cross some exact threshold. (In particular, the fact that a pseudo-probability distribution like W can be interpreted as an honest PDF when it is positive is unimportant because, for finite ℏ, it’s not actually the PDF of any measurement!) Rather, some aspects of classicality will be recovered (approximately or exactly) at different thresholds.
  Likewise, a material made of microscopic atoms doesn’t suddenly qualify as “continuous” when the atoms are smaller than some size; it just becomes a better and better approximation as the atoms get smaller. There are some configurations of atoms that do not have a nice continuous limit: if the atomic spacing goes to zero but the gradient of some quantity diverges (say, the change-per-atom stays fixed except flipping back-and-forth in sign), then this just has no continuum interpretation. I’ve written more along these lines here: https://blog.jessriedel.com/2015/03/24/how-to-think-about-quantum-mechanics-part-4-quantum-indeterminism-as-an-anomaly/
  Two more comments:
  (1) Any threshold involving the P function is suspect because the P function isn’t a function of just the quantum state. As I mention in point #6 above, the P function is only defined with respect to a given quadratic Hamiltonian. (You can check that the spatial width of the ground state, or equivalently the frequency, are necessary to convert a density matrix into a P function; this is not true for the Wigner function.) Changing the Hamiltonian can suddenly flip the the P function from positive to non-positive even before the system has had a chance to evolve according to the Hamiltonian. The paper you link claims just that “only the Glauber-Sudarshan P function behaves under the action of an attenuator [beamsplitter] like a classical phase-space distribution”. Well, OK. I wager if I spent an afternoon I could find a similarly compelling statement about the Q function satisfying some vaguely nice property in certain optics setups. So long as ℏ is finite, it’s really just a matter of preference.
  [By the way, two of your links are broken because the closing “)” has been pulled into the URL.]
  (2) With regard to squeezed light: the ℏ→0 limit sends phase-space localized wavepackets into phase-space delta functions. Whether squeezed are sent to delta function (and what to do about the necessarily many-to-one nature of that mapping) depends on the details of the limiting procedure. In particular, if you hold the squeezing parameter fixed it probably works fine, but if you hold the momentum uncertainty fixed (so that the position uncertainty becomes compensatingly small as ℏ→0) then it does not.
  Much more important is to understand the full ℏ→0 limit. Can we recover classical mechanics, in all its glory?
  Reply
Andrey
May 22, 2020 at 10:19 am

Hi Jess, thanks for a great post. I think, there should be a minus sign in the exponential in the definition of g_s in point 4.
Reply
- Jess Riedel
  May 22, 2020 at 11:37 am
  
  Good eye. Fixed. Thanks!
  Reply