TheWigner quasiprobability distribution, also called theWigner function or theWigner–Ville distribution, afterEugene Wigner andJean-André Ville, is aquasiprobability distribution. It was introduced by Eugene Wigner in 1932^[1] to studyquantum corrections to classicalstatistical mechanics. The goal was to link thewavefunction that appears in theSchrödinger equation to a probability distribution inphase space.

It is agenerating function for all spatialautocorrelation functions of a given quantum-mechanical wavefunctionψ(x).Thus, it maps^[2] on the quantumdensity matrix in the map between real phase-space functions andHermitian operators introduced byHermann Weyl in 1927,^[3] in a context related torepresentation theory in mathematics (seeWeyl quantization). In effect, it is theWigner–Weyl transform of the density matrix, so the realization of that operator in phase space.

It has applications instatistical mechanics,quantum chemistry,quantum optics, classicaloptics and signal analysis in diverse fields, such aselectrical engineering,seismology,time–frequency analysis for music signals,spectrograms inbiology and speech processing, andengine design.

A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation failsfor a quantum particle, due to theuncertainty principle. Instead, the above quasiprobability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions.

For instance, the Wigner distribution can and normally does take on negative values for states which have no classical model—and is a convenient indicator of quantum-mechanical interference. (See below for a characterization of pure states whose Wigner functions are non-negative.)Smoothing the Wigner distribution through a filter of size larger thanħ (e.g., convolving with aphase-space Gaussian, aWeierstrass transform, to yield theHusimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.^[a]

Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a fewħ, and hence disappear in theclassical limit. They are shielded by theuncertainty principle, which does not allow precise location within phase-space regions smaller thanħ, and thus renders such "negative probabilities" less paradoxical.

The Wigner distributionW(x,p) of a pure state is defined as

whereψ is the wavefunction, andx andp are position and momentum, but could be any conjugate variable pair (e.g. real and imaginary parts of the electric field or frequency and time of a signal). Note that it may have support inx even in regions whereψ has no support inx ("beats").

It is symmetric inx andp:

${\displaystyle W(x,p)=\frac{1}{\pi \hslash }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\phi }^{\ast }(p+q)\phi (p-q)e^{-2ixq/\hslash }dq,}$

whereφ is the normalized momentum-space wave function, proportional to theFourier transform ofψ.

In 3D,

${\displaystyle W(\overrightarrow{r},\overrightarrow{p})=\frac{1}{(2\pi {)}^3}\int {\psi }^{\ast }(\overrightarrow{r}+\hslash \overrightarrow{s}/2)\psi (\overrightarrow{r}-\hslash \overrightarrow{s}/2)e^{i\overrightarrow{p}\cdot \overrightarrow{s}}{d}^{3}s.}$

In the general case, which includes mixed states, it is the Wigner transform of thedensity matrix:$${\displaystyle W(x,p)=\frac{1}{\pi \hslash }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}⟨x-y|\hat{\rho }|x+y⟩e^{2ipy/\hslash }dy,}$$where ⟨x|ψ⟩ =ψ(x). ThisWigner transformation (or map) is the inverse of theWeyl transform, which maps phase-space functions toHilbert-space operators, inWeyl quantization.

Thus, the Wigner function is the cornerstone ofquantum mechanics inphase space.

In 1949,José Enrique Moyal elucidated how the Wigner function provides the integration measure (analogous to aprobability density function) in phase space, to yieldexpectation values from phase-spacec-number functionsg(x, p) uniquely associated to suitably ordered operatorsĜ through Weyl's transform (seeWigner–Weyl transform and property 7 below), in a manner evocative of classicalprobability theory.

Specifically, an operator'sĜ expectation value is a "phase-space average" of the Wigner transform of that operator:$${\displaystyle ⟨\hat{G}⟩=\int dxdpW(x,p)g(x,p).}$$

1.W(x, p) is a real-valued function.

2. Thex andp probability distributions are given by themarginals:

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dpW(x,p)=⟨x|\hat{\rho }|x⟩.}$ If the system can be described by apure state, one gets${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dpW(x,p)=|\psi (x){|}^2.}$

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dxW(x,p)=⟨p|\hat{\rho }|p⟩.}$ If the system can be described by apure state, one has${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dxW(x,p)=|\phi (p){|}^2.}$

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dx{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dpW(x,p)=\mathrm{Tr}(\hat{\rho }).}$

Typically the trace of the density matrix${\displaystyle \hat{\rho }}$ is equal to 1.

3.W(x,p) has the following reflection symmetries:

• Time symmetry:${\displaystyle \psi (x)\to \psi (x{)}^{\ast }\Rightarrow W(x,p)\to W(x,-p).}$
• Space symmetry:${\displaystyle \psi (x)\to \psi (-x)\Rightarrow W(x,p)\to W(-x,-p).}$

4.W(x,p) isGalilei-covariant:

${\displaystyle \psi (x)\to \psi (x+y)\Rightarrow W(x,p)\to W(x+y,p).}$

It is notLorentz-covariant.

5. The equation of motion for each point in the phase space is classical in the absence of forces:

${\displaystyle \frac{\mathrm{\partial }W(x,p)}{\mathrm{\partial }t}=\frac{-p}{m}\frac{\mathrm{\partial }W(x,p)}{\mathrm{\partial }x}.}$

In fact, it is classical even in the presence of harmonic forces.

6. State overlap is calculated as

${\displaystyle |⟨\psi |\theta ⟩{|}^2=2\pi \hslash {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dx{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dp{W}_{\psi }(x,p)W_{\theta }(x,p).}$

7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms:

${\displaystyle g(x,p)\equiv {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dy⟨x-\frac{y}{2}|\hat{G}|x+\frac{y}{2}⟩e^{ipy/\hslash },}$

${\displaystyle ⟨\psi |\hat{G}|\psi ⟩=\mathrm{Tr}(\hat{\rho }\hat{G})={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dx{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dpW(x,p)g(x,p).}$

8. ForW(x,p) to represent physical (positive) density matrices, it must satisfy

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dx{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dpW(x,p)W_{\theta }(x,p)\ge 0}$

for all pure states |θ⟩.

9. By virtue of theCauchy–Schwarz inequality, for a pure state, it is constrained to be bounded:

${\displaystyle -\frac{2}{h}\le W(x,p)\le \frac{2}{h}.}$

This bound disappears in the classical limit,ħ → 0. In this limit,W(x, p) reduces to the probability density in coordinate spacex, usually highly localized, multiplied byδ-functions in momentum: the classical limit is "spiky". Thus, this quantum-mechanical bound precludes a Wigner function which is a perfectly localized δ-function in phase space, as a reflection of the uncertainty principle.^[4]

10. The Wigner transformation is simply theFourier transform of theantidiagonals of the density matrix, when that matrix is expressed in a position basis.^[5]

Let${\displaystyle |m⟩\equiv \frac{a^{†m}}{\sqrt{m!}}|0⟩}$ be the${\displaystyle m}$-thFock state of aquantum harmonic oscillator. Groenewold (1946) discovered its associated Wigner function, in dimensionless variables:

${\displaystyle W_{|m⟩}(x,p)=\frac{(-1{)}^m}{\pi }{e}^{-(x^2+p^2)}{L}_m(2(p^2+x^2)),}$

where${\displaystyle L_m(x)}$ denotes the${\displaystyle m}$-thLaguerre polynomial.

This may follow from the expression for the static eigenstate wavefunctions,

${\displaystyle u_m(x)={\pi }^{-1/4}{H}_m(x)e^{-x^2/2},}$

where${\displaystyle H_m}$ is the${\displaystyle m}$-thHermite polynomial. From the above definition of the Wigner function, upon a change of integration variables,

${\displaystyle W_{|m⟩}(x,p)=\frac{(-1{)}^m}{{\pi }^{3/2}{2}^{m}m!}{e}^{-x^2-p^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}d\zeta e^{-{\zeta }^2}{H}_m(\zeta -ip+x)H_m(\zeta -ip-x).}$

The expression then follows from the integral relation between Hermite and Laguerre polynomials.^[6]

TheWigner transformation is a general invertible transformation of an operatorĜ on aHilbert space to a functiong(x, p) onphase space and is given by

${\displaystyle g(x,p)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}ds{e}^{ips/\hslash }⟨x-\frac{s}{2}|\hat{G}|x+\frac{s}{2}⟩.}$

Hermitian operators map to real functions. The inverse of this transformation, from phase space to Hilbert space, is called theWeyl transformation:

${\displaystyle ⟨x|\hat{G}|y⟩={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dp}{h}{e}^{ip(x-y)/\hslash }g\left(\frac{x+y}{2},p\right)}$

(not to be confused with the distinctWeyl transformation in differential geometry).

TheWigner functionW(x,p) discussed here is thus seen to be the Wigner transform of thedensity matrix operatorρ̂. Thus the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap ofg(x, p) with the Wigner function.

The Wigner transform of thevon Neumann evolution equation of the density matrix in theSchrödinger picture isMoyal's evolution equation for the Wigner function:

whereH(x,p) is the Hamiltonian, and {{⋅, ⋅}} is theMoyal bracket. In the classical limit,ħ → 0, the Moyal bracket reduces to thePoisson bracket, while this evolution equation reduces to theLiouville equation of classical statistical mechanics.

Formally, the classical Liouville equation can be solved in terms of the phase-space particle trajectories which are solutions of the classical Hamilton equations. This technique of solving partial differential equations is known as themethod of characteristics. This method transfers to quantum systems, where the characteristics' "trajectories" now determine the evolution of Wigner functions. The solution of the Moyal evolution equation for the Wigner function is represented formally as

${\displaystyle W(x,p,t)=W(\star (x_{-t}(x,p),p_{-t}(x,p)),0),}$

where${\displaystyle x_t(x,p)}$ and${\displaystyle p_t(x,p)}$ are the characteristic trajectories subject to thequantum Hamilton equations with initial conditions${\displaystyle x_{t=0}(x,p)=x}$ and${\displaystyle p_{t=0}(x,p)=p}$, and where${\displaystyle \star }$-product composition is understood for all argument functions.

Since${\displaystyle \star }$-composition of functions isthoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal), vestiges of local trajectories in quantum systems are barely discernible in the evolution of the Wigner distribution function.^[b] In the integral representation of${\displaystyle \star }$-products, successive operations by them have been adapted to a phase-space path integral, to solve the evolution equation for the Wigner function^[7] (see also^[8][9][10]). This non-local feature of Moyal time evolution^[11] is illustrated in the gallery below, for Hamiltonians more complex than the harmonic oscillator. In the classical limit, the trajectory nature of the time evolution of Wigner functions becomes more and more distinct. Atħ = 0, the characteristics' trajectories reduce to the classical trajectories of particles in phase space.

In the special case of thequantum harmonic oscillator, however, the evolution is simple and appears identical to the classical motion: a rigid rotation in phase space with a frequency given by the oscillator frequency. This is illustrated in the gallery below. This same time evolution occurs withquantum states of light modes, which are harmonic oscillators.

The Wigner function allows one to study theclassical limit, offering a comparison of the classical and quantum dynamics in phase space.^[12][13]

It has been suggested that the Wigner function approach can be viewed as a quantum analogy to the operatorial formulation of classical mechanics introduced in 1932 byBernard Koopman andJohn von Neumann: the time evolution of the Wigner function approaches, in the limitħ → 0, the time evolution of theKoopman–von Neumann wavefunction of a classical particle.^[14]

Moments of the Wigner function generate symmetrized operator averages, in contrast to the normal order and antinormal order generated by theGlauber–Sudarshan P representation andHusimi Q representation respectively. The Wigner representation is thus very well suited for making semi-classical approximations in quantum optics^[15] and field theory of Bose-Einstein condensates where high mode occupation approaches a semiclassical limit.^[16]

As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if${\displaystyle W(x,p)\ge 0}$ for all${\displaystyle x}$ and${\displaystyle p}$, then the wave function must have the form

${\displaystyle \psi (x)=e^{-a{x}^2+bx+c}}$

for some complex numbers${\displaystyle a,b,c}$ with${\displaystyle \mathrm{Re}(a)>0}$ (Hudson's theorem^[17]). Note that${\displaystyle a}$ is allowed to be complex. In other words, it is a one-dimensionalgaussian wave packet. Thus, pure states with non-negative Wigner functions are not necessarily minimum-uncertainty states in the sense of theHeisenberg uncertainty formula; rather, they give equality in theSchrödinger uncertainty formula, which includes an anticommutator term in addition to the commutator term. (With careful definition of the respective variances, all pure-state Wigner functions lead to Heisenberg's inequality all the same.)

In higher dimensions, the characterization of pure states with non-negative Wigner functions is similar; the wave function must have the form

${\displaystyle \psi (x)=e^{-(x,Ax)+b\cdot x+c},}$

where${\displaystyle A}$ is a symmetric complex matrix whose real part is positive-definite,${\displaystyle b}$ is a complex vector, andc is a complex number.^[18] The Wigner function of any such state is a Gaussian distribution on phase space.

Francisco Soto and Pierre Claverie^[18] give an elegant proof of this characterization, using theSegal–Bargmann transform. The reasoning is as follows. TheHusimi Q function of${\displaystyle \psi }$ may be computed as the squared magnitude of the Segal–Bargmann transform of${\displaystyle \psi }$, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of${\displaystyle \psi }$ is non-negative everywhere on phase space, then the Husimi Q function will be strictly positive everywhere on phase space. Thus, the Segal–Bargmann transform${\displaystyle F(x+ip)}$ of${\displaystyle \psi }$ will be nowhere zero. Thus, by a standard result from complex analysis, we have

${\displaystyle F(x+ip)=e^{g(x+ip)}}$

for some holomorphic function${\displaystyle g}$. But in order for${\displaystyle F}$ to belong to theSegal–Bargmann space—that is, for${\displaystyle F}$ to be square-integrable with respect to a Gaussian measure—${\displaystyle g}$ must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that${\displaystyle g}$ must actually be a quadratic polynomial. Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative. We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function.

There does not appear to be any simple characterization ofmixed states with non-negative Wigner functions.

It has been shown that the Wigner quasiprobability distribution function can be regarded as anħ-deformation of another phase-space distribution function that describes an ensemble ofde Broglie–Bohm causal trajectories.^[19]Basil Hiley has shown that the quasi-probability distribution may be understood as thedensity matrix re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells".^[20][21]

There is a close connection between the description of quantum states in terms of the Wigner function and a method of quantum states reconstruction in terms ofmutually unbiased bases.^[22]

• In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simpleray tracing and the full wave analysis of the system. Herep/ħ is replaced withk = |k| sin θ ≈ |k|θ in the small-angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at positionx and angleθ while still including the effects of interference.^[23] If it becomes negative at any point, then simple ray tracing will not suffice to model the system. That is to say, negative values of this function are a symptom of theGabor limit of the classical light signal andnot of quantum features of light associated withħ.
• Insignal analysis, a time-varying electrical signal, mechanical vibration, or sound wave are represented by aWigner function. Here,x is replaced with the time, andp/ħ is replaced with the angular frequencyω = 2πf, wheref is the regular frequency.
• Inultrafast optics, short laser pulses are characterized with the Wigner function using the samef andt substitutions as above. Pulse defects such aschirp (the change in frequency with time) can be visualized with the Wigner function. See adjacent figure.
• In quantum optics,x andp/ħ are replaced with theX andP quadratures, the real and imaginary components of the electric field (seecoherent state).

• Quantum tomography
• Frequency-resolved optical gating

The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (seeTransformation between distributions in time–frequency analysis). As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications:

• Glauber P representation
• Husimi Q representation

Nevertheless, in some sense, the Wigner distribution holds a privileged position among all these distributions, since it is theonly one whose requisite star-product drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and socan be visualized as a quasiprobability measure analogous to the classical ones.

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The Wigner function was independently derived several times in different contexts. It was introduced by Eugene Wigner in 1932.^[1] Eugene Wigner was unaware that even within the context of quantum theory, it had been introduced a couple of years before byWerner Heisenberg andPaul Dirac,^[24][25] albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom.

It was later rederived by Jean Ville in 1948 as a quadratic (in signal)representation of the local time-frequency energy of a signal,^[26] effectively aspectrogram.

In 1949,José Enrique Moyal, who had derived it independently, recognized it as thequantum moment-generating functional,^[27] and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (seePhase-space formulation).

In most of his correspondence with Moyal in the mid-1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.^[28]

• M. Levanda and V. Fleurov, "Wigner quasi-distribution function for charged particles in classical electromagnetic fields",Annals of Physics,292, 199–231 (2001).arXiv:cond-mat/0105137.

• wignerArchived 2017-08-21 at theWayback Machine Wigner function implementation in QuTiP.
• Quantum Optics Gallery.
• Sonogram Visible Speech GPL-licensed freeware for the Wigner quasiprobability distribution of signal files.

• Continuous distributions
• Concepts in physics
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• Exotic probabilities
• Quantum optics

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