View source for Wigner quasiprobability distribution

{{other uses|Wigner distribution (disambiguation)}}
{{Short description|Wigner distribution function in physics as opposed to in signal processing}}
[[Image:Wignerfunction catstate even 2.0.png|thumb|Wigner function of a so-called [[cat state]]]]
The '''Wigner quasiprobability distribution''' (also called the '''Wigner function''' or the '''Wigner–Ville distribution''', after [[Eugene Wigner]] and [[Jean Ville|Jean-André Ville]]) is a [[quasiprobability distribution]]. It was introduced by Eugene Wigner in 1932<ref name=":0">
{{cite journal
 |author=E. P. Wigner
 |year=1932
 |title=On the quantum correction for thermodynamic equilibrium
 |journal=Physical Review
 |volume=40 |issue=5 |pages=749–759
 |doi=10.1103/PhysRev.40.749 |bibcode=1932PhRv...40..749W |hdl=10338.dmlcz/141466
 |hdl-access=free
 }}
</ref> to study [[quantum]] corrections to classical [[statistical mechanics]]. The goal was to link the [[wavefunction]] that appears in the [[Schrödinger equation]] to a probability distribution in [[phase space]].

It is a [[generating function]] for all spatial [[autocorrelation]] functions of a given quantum-mechanical wavefunction {{math|''ψ''(''x'')}}.
Thus, it maps<ref>
{{cite journal
 |author=H. J. Groenewold
 |year=1946
 |title=On the principles of elementary quantum mechanics
 |journal=Physica
 |volume=12 |issue=7 |pages=405–460
 |doi=10.1016/S0031-8914(46)80059-4|bibcode=1946Phy....12..405G}}
</ref> on the quantum [[density matrix]] in the map between real phase-space functions and [[Hermitian]] operators introduced by [[Hermann Weyl]] in 1927,<ref>
{{cite journal
 |author=H. Weyl
 |title=Quantenmechanik und gruppentheorie
 |year=1927
 |journal=Zeitschrift für Physik
 |volume=46 |issue=1–2
 |page=1
 |doi=10.1007/BF02055756|bibcode=1927ZPhy...46....1W|s2cid=121036548
 }};  H. Weyl, ''[[Gruppentheorie und Quantenmechanik]]'' (Leipzig: Hirzel) (1928); H. Weyl, ''The Theory of Groups and Quantum Mechanics'' (Dover, New York, 1931).</ref> in a context related to [[Group representation|representation theory]] in mathematics (see [[Weyl quantization]]). In effect, it is the [[Wigner–Weyl transform]] of the density matrix, so the realization of that operator in phase space.

It has applications in [[statistical mechanics]], [[quantum chemistry]], [[quantum optics]], classical [[optics]] and signal analysis in diverse fields, such as [[electrical engineering]], [[seismology]], [[time–frequency analysis for music signals]], [[spectrogram]]s in [[biology]] and speech processing, and [[Waterfall plot|engine design]].

== Relation to classical mechanics ==

A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection ([[Statistical ensemble (mathematical physics)|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 fails
for a quantum particle, due to the [[uncertainty 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 than {{mvar|ħ}} (e.g., convolving with a
phase-space Gaussian, a [[Weierstrass transform]], to yield the [[#Other related quasiprobability distributions|Husimi representation]], below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.{{efn|Specifically, since this convolution is invertible, in fact, no information has been sacrificed, and the full quantum entropy has not increased yet. However, if this resulting Husimi distribution is then used as a plain measure in a phase-space integral evaluation of expectation values ''without the requisite star product of the Husimi representation'', then, at that stage, quantum information ''has been forfeited'' and the distribution ''is a semi-classical one'', effectively. That is, depending on its usage in evaluating expectation values, the very ''same distribution may serve as a quantum or a classical distribution function''.}}

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 few {{mvar|ħ}}, and hence disappear in the [[classical limit]]. They are shielded by the [[uncertainty principle]], which does not allow precise location within phase-space regions smaller than {{mvar|ħ}}, and thus renders such "[[negative probability|negative probabilities]]" less paradoxical.

== Definition and meaning ==

The Wigner distribution {{math|''W''(''x'',''p'')}} of a pure state is defined as

{{Equation box 1
|indent =:
|equation = <math>W(x, p)~\stackrel{\text{def}}{=}~\frac{1}{\pi\hbar} \int_{-\infty}^\infty \psi^*(x + y) \psi(x - y) e^{2ipy/\hbar}\,dy,</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
<!-- PLEASE DESIST FROM WANTONLY FLIPPING SIGNS BEFORE DISCUSSING IN TALKPAGE AND ASCERTAINING CONSISTENCY OF DEFINITIONS: YOU HAVE PROBABLY IGNORED SOMETHING IF THE SIGNS DISAGREE WITH YOU -->
where {{mvar|ψ}} is the wavefunction, and {{mvar|x}} and {{mvar|p}} 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 in {{mvar|x}} even in regions where {{mvar|ψ}} has no support in {{mvar|x}} ("beats").

It is symmetric in {{mvar|x}} and {{mvar|p}}:
: <math>W(x, p) = \frac{1}{\pi\hbar} \int_{-\infty}^\infty \varphi^*(p + q) \varphi(p - q) e^{-2ixq/\hbar} \,dq,</math>
where {{mvar|φ}} is the normalized momentum-space wave function, proportional to the [[Fourier transform]] of {{mvar|ψ}}.

In 3D,
: <math>W(\vec{r}, \vec{p}) = \frac{1}{(2\pi)^3} \int \psi^*(\vec{r} + \hbar\vec{s}/2) \psi(\vec{r} - \hbar\vec{s}/2) e^{i\vec{p} \cdot \vec{s}} \,d^3 s.</math>

In the general case, which includes mixed states, it is the Wigner transform of the [[density matrix]]:
<math display="block">W(x, p) = \frac{1}{\pi\hbar} \int_{-\infty}^\infty \langle x - y| \hat{\rho} |x + y \rangle e^{2ipy/\hbar} \,dy,</math>
where ⟨''x''|''ψ''⟩ = {{math|''ψ''(''x'')}}. This [[#The Wigner–Weyl transformation|Wigner transformation]] (or map) is the inverse of the [[Weyl quantization|Weyl transform]], which maps phase-space functions to [[Hilbert space|Hilbert-space]] operators, in [[Weyl quantization]].

Thus, the Wigner function is the cornerstone of [[quantum mechanics]] in [[phase space]].

In 1949, [[José Enrique Moyal]] elucidated how the Wigner function provides the integration measure (analogous to a [[probability density function]]) in phase space, to yield [[expected value|expectation values]] from phase-space [[c-number]] functions {{math|''g''(''x'',&nbsp;''p'')}} uniquely associated to suitably ordered operators {{math|''Ĝ''}} through Weyl's transform (see [[Wigner–Weyl transform]] and property&nbsp;7 below), in a manner evocative of classical [[probability theory]].

Specifically, an operator's {{math|''Ĝ''}} expectation value is a "phase-space average" of the Wigner transform of that operator:
<math display="block">\langle \hat{G} \rangle = \int dx\,dp\, W(x, p) g(x, p).</math>

== Mathematical properties ==

[[File:Number state Wigner function.png|thumb|280px|Wigner function for number states a) ''n''&nbsp;=&nbsp;0, b) ''n''&nbsp;=&nbsp;1, and c) ''n''&nbsp;=&nbsp;19. Marginal distributions for ''x'' and ''p'' are recovered by integrating over ''p'' and ''x'' respectively.]]

1. ''W''(''x'',&nbsp;''p'') is a real-valued function.

2. The ''x'' and ''p'' probability distributions are given by the [[Marginal distribution|marginals]]:
:: <math>\int_{-\infty}^\infty dp\, W(x, p) = \langle x|\hat{\rho}|x \rangle.</math> If the system can be described by a [[pure state]], one gets <math>\int_{-\infty}^\infty dp\, W(x, p) = |\psi(x)|^2.</math>
:: <math>\int_{-\infty}^\infty dx\, W(x, p) = \langle p|\hat{\rho}|p \rangle.</math> If the system can be described by a [[pure state]], one has <math>\int_{-\infty}^{\infty} dx\, W(x, p) = |\varphi(p)|^2.</math>
:: <math>\int_{-\infty}^\infty dx \int_{-\infty}^\infty dp\, W(x, p) = \operatorname{Tr}(\hat{\rho}).</math>
: Typically the trace of the density matrix ''<math>\hat{\rho}</math>'' is equal to 1.

3. ''W''(''x'', ''p'') has the following reflection symmetries:
:* Time symmetry: <math>\psi(x) \to \psi(x)^* \Rightarrow W(x, p) \to W(x, -p).</math>
:* Space symmetry: <math>\psi(x) \to \psi(-x) \Rightarrow W(x, p) \to W(-x, -p).</math>

4. ''W''(''x'', ''p'') is [[Galilean transformation|Galilei-covariant]]:
:: <math>\psi(x) \to \psi(x + y) \Rightarrow W(x, p) \to W(x + y, p).</math>
: It is not [[Lorentz-covariant]].

5. The equation of motion for each point in the phase space is classical in the absence of forces:
:: <math>\frac{\partial W(x, p)}{\partial t} = \frac{-p}{m} \frac{\partial W(x, p)}{\partial x}.</math>
: In fact, it is classical even in the presence of harmonic forces.

6. State overlap is calculated as
:: <math>|\langle \psi|\theta \rangle|^2 = 2\pi\hbar \int_{-\infty}^\infty dx \int_{-\infty}^\infty dp\, W_\psi(x, p) W_\theta(x, p).</math>

7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms:
:: <math>g(x, p) \equiv \int_{-\infty}^\infty dy\, \left\langle x - \frac{y}{2} \right| \hat{G} \left| x + \frac{y}{2} \right\rangle e^{ipy/\hbar},</math>
:: <math>\langle \psi|\hat{G}|\psi\rangle = \operatorname{Tr}(\hat{\rho} \hat{G}) = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dp\, W(x, p) g(x, p).</math>

8. For ''W''(''x'', ''p'') to represent physical (positive) density matrices, it must satisfy
:: <math>\int_{-\infty}^\infty dx\, \int_{-\infty}^\infty dp\, W(x, p) W_\theta(x, p) \ge 0</math>
: for all pure states |θ⟩.

9. By virtue of the [[Cauchy–Schwarz inequality]], for a pure state, it is constrained to be bounded:
:: <math>-\frac 2 h \leq W(x, p) \leq \frac 2 h.</math>
: This bound disappears in the classical limit, ''ħ'' → 0.  In this limit, ''W''(''x'',&nbsp;''p'') reduces to the probability density in coordinate space ''x'', usually highly localized, multiplied by [[Dirac delta function|δ-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.<ref>{{cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1142/S2251158X12000069 | title = Quantum Mechanics in Phase Space | journal = Asia Pacific Physics Newsletter | volume = 1 | page = 37 | year = 2012 | arxiv = 1104.5269 | s2cid = 119230734 }};  [[Cosmas Zachos|C. Zachos]], D. Fairlie, and [[Thomas Curtright|T. Curtright]], ''Quantum  Mechanics in Phase Space'' (World Scientific, Singapore, 2005). {{isbn|978-981-238-384-6}}.</ref>

10. The Wigner transformation is simply the [[Fourier transform]] of the [[antidiagonal]]s of the density matrix, when that matrix is expressed in a position basis.<ref>{{cite book |last1=Hawkes |first1=Peter W. |title=Advances in Imaging and Electron Physics |date=2018 |publisher=Academic Press |isbn=9780128155424 |pages=47}}</ref>

== Examples ==
{{See also|Phase-space formulation#Simple harmonic oscillator}}
Let <math>|m\rangle \equiv \frac{a^{\dagger m}}{\sqrt{m!}} |0\rangle</math> be the <math>m</math>-th [[Fock state]] of a [[quantum harmonic oscillator]]. Groenewold (1946) discovered its associated Wigner function, in dimensionless variables:
: <math>W_{|m\rangle}(x, p) = \frac{(-1)^m}{\pi} e^{-(x^2 + p^2)} L_m\big(2(p^2 + x^2)\big),</math>
where <math>L_m(x)</math> denotes the <math>m</math>-th [[Laguerre polynomials|Laguerre polynomial]].   
 
This may follow from the expression for the static eigenstate wavefunctions,
: <math>u_m(x) = \pi^{-1/4} H_m(x) e^{-x^2/2},</math>
where <math>H_m</math> is the <math>m</math>-th [[Hermite polynomials|Hermite polynomial]]. From the above definition of the Wigner function, upon a change of integration variables,  
: <math>W_{|m\rangle}(x, p) = \frac{(-1)^m}{\pi^{3/2} 2^m m!} e^{-x^2 - p^2} \int_{-\infty}^\infty d\zeta\, e^{-\zeta^2} H_m(\zeta - ip + x) H_m(\zeta - ip - x).</math>
The expression then follows from the integral relation between Hermite and Laguerre polynomials.<ref>{{Cite book |last=Schleich |first=Wolfgang P. |url=https://onlinelibrary.wiley.com/doi/book/10.1002/3527602976 |title=Quantum Optics in Phase Space |date=2001-02-09 |publisher=Wiley |isbn=978-3-527-29435-0 |edition=1st |language=en |doi=10.1002/3527602976 |page=105}}</ref>

== Evolution equation ==
{{Main|Wigner–Weyl transform|Phase space formulation}}

The '''Wigner transformation''' is a general invertible transformation of an operator {{math|''Ĝ''}} on a [[Hilbert space]] to a function ''g''(''x'',&nbsp;''p'') on [[phase space]] and is given by
: <math>g(x, p) = \int_{-\infty}^\infty ds\, e^{ips/\hbar} \left\langle x - \frac s2\right| \hat G \left|x + \frac s2\right\rangle.</math>

Hermitian operators map to real functions. The inverse of this transformation, from phase space to Hilbert space, is called the '''[[Wigner–Weyl transform|Weyl transformation]]''':
: <math>\langle x | \hat G | y \rangle = \int_{-\infty}^\infty \frac{dp}{h} e^{ip(x - y)/\hbar} g\left(\frac{x + y}{2}, p\right)</math>
(not to be confused with the distinct [[Weyl transformation|Weyl transformation in differential geometry]]).

The '''Wigner function''' {{math|''W''(''x'', ''p'')}} discussed here is thus seen to be the Wigner transform of the [[density matrix]] operator ''ρ̂''. Thus the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of {{math|''g''(''x'',&nbsp;''p'')}} with the Wigner function.

The Wigner transform of the [[Density matrix|von Neumann evolution equation]] of the density matrix in the [[Schrödinger picture]] is '''Moyal's evolution equation''' for the Wigner function:
{{Equation box 1
|indent =:
|equation =  <math>\frac{\partial W(x, p, t)}{\partial t} = - \{\{W(x, p, t), H(x, p)\}\},</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
where {{math|''H''(''x'', ''p'')}} is the Hamiltonian, and <nowiki>{{⋅,&nbsp;⋅}}</nowiki> is the [[Moyal bracket]]. In the classical limit, {{math|''ħ'' → 0}}, the Moyal bracket reduces to the [[Poisson bracket]], while this evolution equation reduces to the [[Liouville's theorem (Hamiltonian)#Poisson bracket|Liouville 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 the [[method 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
: <math>W(x, p, t) = W\big(\star\big(x_{-t}(x, p), p_{-t}(x, p)\big), 0\big),</math>
where <math>x_t(x, p)</math> and <math>p_t(x, p)</math> are the characteristic trajectories subject to the [[Method of quantum characteristics|quantum Hamilton equations]] with initial conditions <math>x_{t=0}(x, p) = x</math> and <math>p_{t=0}(x, p) = p</math>, and where [[Moyal product|<math>\star</math>-product]] composition is understood for all argument functions.

Since <math>\star</math>-composition of functions is ''thoroughly 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.{{efn|Quantum characteristics should not be confused with trajectories of the [[Path integral formulation|Feynman path integral]], or trajectories of the [[de Broglie–Bohm theory]]. This three-fold ambiguity allows better understanding of the position of [[Niels Bohr]], who vigorously but counterproductively opposed the notion of trajectory in the atomic physics. At the 1948 Pocono Conference, e.g., he said to [[Richard Feynman]]: "... one could not talk about the trajectory of an electron in the atom, because it was something not observable". ("The Beat of a Different Drum: The Life and Science of Richard Feynman", by Jagdish Mehra (Oxford, 1994, pp.&nbsp;245–248)). Arguments of this kind were widely used in the past by [[Ernst Mach]] in his criticism of an atomic theory of physics and later, in the 1960s, by [[Geoffrey Chew]], [[Tullio Regge]] and others to motivate replacing the local quantum field theory by the [[S-matrix]] theory. Today, statistical physics entirely based on atomistic concepts is included in standard courses, the S-matrix theory went out of fashion, while the Feynman path-integral method has been recognized as the most efficient method in [[Gauge theory|gauge theories]].}} In the integral representation of <math>\star</math>-products, successive operations by them have been adapted to a phase-space path integral, to solve the evolution equation for the Wigner function<ref>
{{cite journal
 |author=B. Leaf
 |year=1968
 |title=Weyl transform in nonrelativistic quantum dynamics
 |journal=Journal of Mathematical Physics
 |volume=9 |issue=5 |pages=769–781
 |doi=10.1063/1.1664640 |bibcode=1968JMP.....9..769L}}
</ref> (see also <ref>
{{cite journal
 |author=P. Sharan
 |year=1979
 |title=Star-product representation of path integrals
 |journal=Physical Review D
 |volume=20 |issue=2 |pages=414–418
 |bibcode=1979PhRvD..20..414S
 |doi=10.1103/PhysRevD.20.414}}
</ref><ref>
{{cite journal
 |author=M. S. Marinov
 |year=1991
 |title=A new type of phase-space path integral
 |journal=Physics Letters A
 |volume=153 |issue=1 |pages=5–11
 |doi=10.1016/0375-9601(91)90352-9 |bibcode=1991PhLA..153....5M}}
</ref><ref name="OlshanetskyVainshtein2002">B. Segev: ''Evolution kernels for phase space distributions''. In: {{cite book |author1=M. A. Olshanetsky |author2=Arkady Vainshtein |title=Multiple Facets of Quantization and Supersymmetry: Michael Marinov Memorial Volume |url=https://books.google.com/books?id=_pwmuPF1DY4C&pg=PA68 |access-date=26 October 2012 |year=2002 |publisher=World Scientific |isbn=978-981-238-072-2 |pages=68–90}} See especially section&nbsp;5. "Path integral for the propagator" on [https://books.google.com/books?id=_pwmuPF1DY4C&pg=PA86 pages 86–89].
Also [http://homepages.spa.umn.edu/~vainshte/Marinov/Segev/segev.pdf online].</ref>). This non-local feature of Moyal time evolution<ref>{{cite journal |author=M. Oliva, D. Kakofengitis, and O. Steuernagel |year=2018 |title=Anharmonic quantum mechanical systems do not feature phase space trajectories |journal=Physica A | volume=502 |pages=201–210 |doi=10.1016/j.physa.2017.10.047  |arxiv=1611.03303 |bibcode=2018PhyA..502..201O |s2cid=53691877 }}</ref> 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. At ''ħ''&nbsp;=&nbsp;0, the characteristics' trajectories reduce to the classical trajectories of particles in phase space.

{{Gallery
|title=Examples of time evolution
|align=center
|File:Wigner function propagation for morse potential.ogv|Pure state in a [[Morse potential]]. The green dashed lines represent [[level set]] of the [[Hamiltonian mechanics|Hamiltonian]].
|File:Wigner function for quartic potential.ogv|Pure state in a quartic potential. The solid lines represent the level set of the Hamiltonian.
|File:Wigner function for tunnelling.ogv|[[Quantum tunnelling|Tunnelling]] of a [[wave packet]] through a potential barrier. The solid lines represent the level set of the Hamiltonian.
|File:Hamiltonian flow quantum.webm|Long-time evolution of a mixed state ''ρ'' in an anharmonic potential well. Marginals are plotted on the right (''p'') and top (''x'').
|File:Hamiltonian flow quantum - equilibrated.png|An equilibrium mixed state ''ρ'' (evolves to itself), in the same anharmonic potential.
}}

===Harmonic-oscillator time evolution===

In the special case of the [[quantum 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 with [[quantum optics|quantum states of light modes]], which are harmonic oscillators.

{{Gallery
|title=Examples of Wigner-function time evolutions in a quantum harmonic oscillator
|align=center
|File:Wigner quasiprobability distribution of a coherent state.webm|Wigner quasiprobability distribution of a coherent state.
|File:Wigner quasiprobability distribution of cat state.webm|A [[cat state]]; the [[Marginal distribution|marginals]] are plotted on the right (''p'') and underneath (''x'').
|File:Wigner quasiprobability distribution of cat states, grid.webm|Cat states with 2, 3, 4 cats. The separation between cats range from 0.5, 1, 2, 4, showing increasingly sharp inference.
|7=File:Wigner quasiprobability distribution of cat state, n = 10, a = 10.webm|8=A very big cat state, with 10 cats separated at <math>\alpha = 10</math>.
}}

{{Gallery
|title=Fock states
|align=center
|File:Wigner quasiprobability distribution of superpositioned Fock states (1, 2).webm|Superpositioned Fock states, for <math> (n, m) = (1, 2)</math>.
|File:Wigner quasiprobability distribution of superpositioned Fock states.webm|<math> 0 \leq n \leq m \leq 2</math>.
|File:Wigner quasiprobability distribution of superpositioned Fock states, grid.png|<math> 0 \leq n \leq m \leq 4</math>.
}}


{{Gallery
|title=Squeezed states
|align=center
|File:Wigner quasiprobability distribution of squeezed states.webm|Wigner quasiprobability distribution of squeezed states, for varying amount of phase shift and displacement. When the phase shift is zero (leftmost column), the peak-to-peak amplitude of <math> x</math> oscillation is less uncertain, but the phase (the time at which <math> x</math> crosses the midpoint) is more uncertain. The opposite is true when the phase shift is half a cycle (rightmost column).
|File:Wigner quasiprobability distribution of superpositioned squeezed states, varying purity.webm|Varying purity, from totally pure to totally mixed.
}}

== Classical limit ==
The Wigner function allows one to study the [[classical limit]], offering a comparison of the classical and quantum dynamics in phase space.<ref>See, for example: [[Wojciech H. Zurek]], ''Decoherence and the transition from quantum to classical – revisited'', Los Alamos Science, 27, 2002, [https://arxiv.org/abs/quant-ph/0306072 arXiv:quant-ph/0306072], pp.&nbsp;15&nbsp;ff.</ref><ref>See, for example: C. Zachos, D. Fairlie, T. Curtright, ''Quantum mechanics in phase space: an overview with selected papers'', World Scientific, 2005. {{isbn|978-981-4520-43-0}}.</ref>

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 by [[Bernard Koopman]] and [[John von Neumann]]: the time evolution of the Wigner function approaches, in the limit ''ħ''&nbsp;→&nbsp;0, the time evolution of the [[Koopman–von Neumann wavefunction]] of a classical particle.<ref name="BondarCabrera2013">{{cite journal |last1=Bondar |first1=Denys I. |last2=Cabrera |first2=Renan |last3=Zhdanov |first3=Dmitry V. |last4=Rabitz |first4=Herschel A. |title=Wigner phase-space distribution as a wave function |journal=Physical Review A |volume=88 |issue=5 |year=2013 |page=052108 |issn=1050-2947 |doi=10.1103/PhysRevA.88.052108 |arxiv=1202.3628 |s2cid=119155284}}</ref>

Moments of the Wigner function generate symmetrized operator averages, in contrast to the normal order and antinormal order generated by the [[Glauber–Sudarshan P representation]] and [[Husimi Q representation]] respectively. The Wigner representation is thus very well suited for making semi-classical approximations in quantum optics<ref name="Graham1973">{{Cite book| publisher = Springer| pages = 1| last = Graham| first = R| title = Springer Tracts in Modern Physics| chapter = Statistical Theory of Instabilities in Stationary Nonequilibrium Systems with Applications to Lasers and Nonlinear Optics| date = 1973}}</ref> and field theory of Bose-Einstein condensates where high mode occupation approaches a semiclassical limit.<ref name="Blakie2008">{{Cite journal| doi = 10.1080/00018730802564254| issn = 0001-8732| volume = 57| issue = 5| pages = 363–455| last1 = Blakie†| first1 = P.B.| last2 = Bradley†| first2 = A.S.| last3 = Davis| first3 = M.J.| last4 = Ballagh| first4 = R.J.| last5 = Gardiner| first5 = C.W.| title = Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques| journal = Advances in Physics| access-date = 2023-03-01| date = 2008-09-01| url = https://doi.org/10.1080/00018730802564254| arxiv = 0809.1487}}</ref>

==Positivity of the function==
As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if <math>W(x, p) \ge 0</math> for all <math>x</math> and <math>p</math>, then the wave function must have the form
: <math>\psi(x) = e^{-ax^2+bx+c}</math>
for some complex numbers <math>a, b, c</math> with <math>\operatorname{Re}(a) > 0</math> (Hudson's theorem<ref name="Hudson1974">{{cite journal
 | author = Hudson, Robin L.
 | title = When is the Wigner quasi-probability density non-negative?
 | journal = Reports on Mathematical Physics
 | volume = 6
 | issue = 2
 | pages = 249–252
 | year = 1974
 | doi = 10.1016/0034-4877(74)90007-X
 | bibcode = 1974RpMP....6..249H
}}</ref>). Note that <math>a</math> is allowed to be complex. In other words, it is a one-dimensional [[Wave packet#quantum mechanics|gaussian wave packet]]. Thus, pure states with non-negative Wigner functions are not necessarily minimum-uncertainty states in the sense of the [[Uncertainty principle|Heisenberg uncertainty formula]]; rather, they give equality in the [[Robertson–Schrodinger relation|Schrödinger 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
: <math>\psi(x) = e^{-(x,Ax)+b\cdot x+c},</math>
where <math>A</math> is a symmetric complex matrix whose real part is positive-definite, <math>b</math> is a complex vector, and {{mvar|c}} is a complex number.<ref name="Soto,Claverie">F. Soto and P. Claverie, "When is the Wigner function of multidimensional systems nonnegative?", ''Journal of Mathematical Physics'' '''24''' (1983) 97–100.</ref> The Wigner function of any such state is a Gaussian distribution on phase space.

Francisco Soto and Pierre Claverie<ref name="Soto,Claverie"/> give an elegant proof of this characterization, using the [[Segal–Bargmann space#The Segal–Bargmann transform|Segal–Bargmann transform]]. The reasoning is as follows. The [[Husimi Q representation|Husimi Q function]] of <math>\psi</math> may be computed as the squared magnitude of the Segal–Bargmann transform of <math>\psi</math>, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of <math>\psi</math> 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 <math>F(x + ip)</math> of <math>\psi</math> will be nowhere zero. Thus, by a standard result from complex analysis, we have
: <math>F(x + ip) = e^{g(x+ip)}</math>
for some holomorphic function <math>g</math>. But in order for <math>F</math> to belong to the [[Segal–Bargmann space]]—that is, for <math>F</math> to be square-integrable with respect to a Gaussian measure—<math>g</math> must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that <math>g</math> 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 of [[density matrix|mixed states]] with non-negative Wigner functions.

== Interpretations of quantum mechanics ==
It has been shown that the Wigner quasiprobability distribution function can be regarded as an {{mvar|ħ}}-[[Deformation theory|deformation]] of another phase-space distribution function that describes an ensemble of [[De Broglie–Bohm theory|de&nbsp;Broglie–Bohm]] causal trajectories.<ref>{{cite journal | last1=Dias | first1=Nuno Costa | last2=Prata | first2=João Nuno | title=Bohmian trajectories and quantum phase space distributions | journal=Physics Letters A | volume=302 | issue=5–6 | year=2002 | issn=0375-9601 | doi=10.1016/s0375-9601(02)01175-1 | pages=261–272|arxiv=quant-ph/0208156v1| bibcode=2002PhLA..302..261D | s2cid=39936409 }}</ref> [[Basil Hiley]] has shown that the quasi-probability distribution may be understood as the [[density matrix]] re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de&nbsp;Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells".<ref>B. J. Hiley: ''Phase space descriptions of quantum phenomena'', in: A. Khrennikov (ed.): ''Quantum Theory: Re-consideration of Foundations–2'', pp.&nbsp;267–286, Växjö University Press, Sweden, 2003 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/ShadowPhaseVajxo03.pdf PDF]).</ref><ref>B. Hiley: ''Moyal's characteristic function, the density matrix and von Neumann's idempotent'' ([http://www.bbk.ac.uk/tpru/BasilHiley/Moyal&DensityMatrix.pdf preprint]).</ref>

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 of [[mutually unbiased bases]].<ref>F. C. Khanna, P. A. Mello, M. Revzen, Classical and Quantum Mechanical State Reconstruction, [https://arxiv.org/abs/1112.3164v1 arXiv:1112.3164v1] [quant-ph] (submitted December 14, 2011).</ref>

== Use outside quantum mechanics ==
[[File:Chirpedpulse.jpg|thumb|A contour plot of the Wigner–Ville distribution for a [[chirp]]ed pulse of light. The plot makes it obvious that the frequency is a linear function of time.]]
* In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple [[Ray tracing (physics)|ray tracing]] and the full wave analysis of the system. Here {{math|''p''/''ħ''}} is replaced with {{math|''k'' {{=}} {{!}}''k''{{!}}&nbsp;sin&nbsp;''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 position {{mvar|x}} and angle {{mvar|θ}} while still including the effects of interference.<ref>{{cite journal | last=Bazarov | first=Ivan V. | title=Synchrotron radiation representation in phase space | journal=Physical Review Special Topics - Accelerators and Beams | publisher=American Physical Society (APS) | volume=15 | issue=5 | date=2012-05-03 | issn=1098-4402 | doi=10.1103/physrevstab.15.050703 | page=050703| s2cid=53489256 | doi-access=free | arxiv=1112.4047 | bibcode=2012PhRvS..15e0703B }}</ref> 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 the [[Gabor limit]] of the classical light signal and ''not'' of quantum features of light associated with {{mvar|ħ}}.
* In [[signal analysis]], a time-varying electrical signal, mechanical vibration, or sound wave are represented by a [[Wigner distribution function|Wigner function]]. Here, {{mvar|x}} is replaced with the time, and {{math|''p''/''ħ''}} is replaced with the angular frequency {{math|''ω'' {{=}} 2π''f''}}, where {{mvar|f}} is the regular frequency.
* In [[ultrafast optics]], short laser pulses are characterized with the Wigner function using the same {{mvar|f}} and {{mvar|t}} substitutions as above. Pulse defects such as [[chirp]] (the change in frequency with time) can be visualized with the Wigner function. See adjacent figure.
* In quantum optics, {{mvar|x}} and {{math|''p''/''ħ''}} are replaced with the {{mvar|X}} and {{mvar|P}} quadratures, the real and imaginary components of the electric field (see [[coherent state]]).

== Characterization methods ==
* [[Quantum tomography]]
* [[Frequency-resolved optical gating]]

== Other related quasiprobability distributions ==
{{Main|Quasiprobability distribution}}

The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (see [[Transformation 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 the ''only one'' whose requisite star-product drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and so ''can'' be visualized as a quasiprobability measure analogous to the classical ones.

== History ==
{{Primary sources|section|date=May 2025}}
The Wigner function was independently derived several times in different contexts. It was introduced by Eugene Wigner in 1932.<ref name=":0" /> Eugene Wigner was unaware that even within the context of quantum theory, it had been introduced a couple of years before by [[Werner Heisenberg]] and [[Paul Dirac]],<ref>{{cite journal|author=[[Werner Heisenberg|W. Heisenberg]]|title=Über die inkohärente Streuung von Röntgenstrahlen|journal=Physikalische Zeitschrift|volume=32|pages=737–740|year=1931}}</ref><ref>{{cite journal | last=Dirac | first=P. A. M. |author-link=Paul Dirac| title=Note on Exchange Phenomena in the Thomas Atom | journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] | publisher=Cambridge University Press (CUP) | volume=26 | issue=3 | year=1930 | issn=0305-0041 | doi=10.1017/s0305004100016108 | pages=376–385| s2cid=97185766 | doi-access=free | bibcode=1930PCPS...26..376D }}</ref> 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) [[Wigner distribution function|representation of the local time-frequency energy of a signal]],<ref>J. Ville, "Théorie et Applications de la Notion de Signal Analytique", ''Câbles et Transmission'', '''2''', 61–74 (1948).</ref> effectively a [[spectrogram]].

In 1949, [[José Enrique Moyal]], who had derived it independently, recognized it as the [[quantum]] moment-generating functional,<ref>{{cite journal |last=Moyal |first=J. E. |author-link=José Enrique Moyal |year=1949 |title=Quantum mechanics as a statistical theory |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |publisher=Cambridge University Press (CUP) |volume=45 |issue=1 |pages=99–124 |bibcode=1949PCPS...45...99M |doi=10.1017/s0305004100000487 |issn=0305-0041 |s2cid=124183640}}</ref> and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (see [[Phase-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.<ref>Ann Moyal, (2006), [http://epress.anu.edu.au/maverick_citation.html "Maverick Mathematician: The Life and Science of J. E. Moyal"], ANU E-press, 2006, {{isbn|1-920942-59-9}}.</ref>

== See also ==
{{div col|colwidth=28em}}
* [[Heisenberg group]]
* [[Wigner–Weyl transform]]
* [[Phase space formulation]]
* [[Moyal bracket]]
* [[Negative probability]]
* [[Optical equivalence theorem]]
* [[Modified Wigner distribution function]]
* [[Cohen's class distribution function]]
* [[Wigner distribution function]]
* [[Transformation between distributions in time–frequency analysis]]
* [[Squeezed coherent state]]
* [[Bilinear time–frequency distribution]]
* [[Continuous-variable quantum information]]
{{div col end}}

==Footnotes==
{{notelist}}

== References ==
{{reflist}}

== Further reading ==
* 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}}.

== External links ==
* [http://qutip.org/docs/latest/guide/guide-visualization.html#wigner-function wigner] Wigner function implementation in QuTiP.
* [http://gerdbreitenbach.de/gallery/ Quantum Optics Gallery].
* [https://github.com/Christoph-Lauer/Sonogram Sonogram Visible Speech] GPL-licensed freeware for the Wigner quasiprobability distribution of signal files.

{{DEFAULTSORT:Wigner Quasi-Probability Distribution}}
[[Category:Continuous distributions]]
[[Category:Concepts in physics]]
[[Category:Mathematical physics]]
[[Category:Exotic probabilities]]
[[Category:Articles containing video clips]]
[[Category:Quantum optics]]