Wigner Functions

Definition

Wigner functions are a distribution function on phase space that allow to represent the state of a quantum-mechanical system. They are in many ways similar to classical phase space probability distributions, but can, in contrast to these, be negative. A description of a quantum system in terms of Wigner functions is equivalent to the more widely used one in terms of density operators or wave functions, but has advantages in visualizing properties of a quantum state and in studying the quantum–classical transition.

1. Introduction

The Wigner function is a function that represents the state of a quantum-mechanical system via a real-valued quasiprobability distribution on phase space. It was introduced by Wigner [1] in 1932 in an article entitled “On the quantum correction for thermodynamic equilibrium”. This title already hints at the original context in which this formalism was developed, namely statistical mechanics. Consequently, although presented in this entry mainly for the single-particle case (which is relevant in many applications), the Wigner function was from the beginning also a many-body formalism. Specifically, Wigner’s aim was to be able to systematically obtain quantum corrections to classical statistical mechanics [1]. Wigner was not the only one who worked on reformulating quantum mechanics in a way more akin to classical mechanics. For instance, Husimi [2] introduced a different phase space distribution in 1940, and Madelung [3] showed in 1927 that the Schrödinger equation for a one-particle system can be re-written as a hydrodynamic equation resembling compressible Euler equations. The most important approach of this type is probably the formalism developed by David Bohm [4,5], which involves, in addition to the quantum-mechanical wavefunction, classical particle trajectories as hidden variables and constitutes the basis for a possible solution of the quantum-mechanical measurement problem [6] (see below). Major contributions to the quantum-mechanical phase space formalism used today were made by Groenewold [7] and Moyal [8].

Wigner functions have developed into a widely used tool in quantum information theory and quantum technology. A key advantage of Wigner functions is that they allow for an easier visualization of quantum states and for a study of the classical limit. A disadvantage is that the Schrödinger equation for the quantum-mechanical wavefunctions (the main alternative) is in many cases easier to solve [9], for example, due to the availability of many well-developed solution techniques and due to the fact that the wavefunction depends on fewer coordinates. (More specifically, for a particle in one spatial dimension, the wavefunction is a function of position x, while the Wigner function is a function of position x and momentum p. Higher-dimensional partial differential equations are harder to solve. This advantage is relevant only for systems that can be attributed to a wavefunction and therefore not generally for systems entangled with their environment.) There are many excellent reviews on this topic [9,10,11,12,13,14], which this entry does not intend to compete with—the author will, here, in the style of a dictionary entry, provide a brief glance at the topic for the impatient reader. The presentation of the formalism here mostly follows that in Refs. [15,16]. For historical aspects, the author mostly follows Ref. [11]. A focus will be the use of Wigner functions in studying the relation between classical and quantum mechanics.

2. Classical vs. Quantum Mechanics

In classical mechanics, the state of a system is specified by its position $\overrightarrow{\mathrm{\Gamma }}$ in phase space—a space (in general, high-dimensional) spanned by the positions and momenta of all particles. Especially for many-particle systems, the description is usually cast in terms of a distribution function $\rho \left(\overrightarrow{\mathrm{\Gamma }}\right)$, which gives the probability finding the system in the state $\overrightarrow{\mathrm{\Gamma }}$. Being a probability distribution, this distribution satisfies

$$\int \mathrm{d}\mathrm{\Gamma }\rho \left(\overrightarrow{\mathrm{\Gamma }}\right)=1.$$

(1)

Moreover, if $A\left(\overrightarrow{\mathrm{\Gamma }}\right)$ is an observable (which in classical mechanics is a function of the phase space variables and occasionally also of time), the expectation value of this observable is given by

$$\langle A\rangle =\int \mathrm{d}\mathrm{\Gamma }\rho \left(\overrightarrow{\mathrm{\Gamma }}\right)A\left(\overrightarrow{\mathrm{\Gamma }}\right).$$

(2)

In the Schrödinger picture of classical mechanics [17], the distribution function evolves according to the Liouville equation

$${\partial }_t\rho \left(\overrightarrow{\mathrm{\Gamma }}\right)=\{H\left(\overrightarrow{\mathrm{\Gamma }}\right),\rho \left(\overrightarrow{\mathrm{\Gamma }}\right)\},$$

(3)

where H is the system’s Hamiltonian and $\{·,·\}$ is the Poisson bracket. (We never write the dependence on time explicitly to keep the notation short.) If the system is a point particle in one spatial dimension with position x and momentum p—which we shall assume for the rest of this entry to simplify the notation—the Poisson bracket is defined as

$$\{f,g\}={\displaystyle \frac{\partial f}{\partial x}}{\displaystyle \frac{\partial g}{\partial p}}-{\displaystyle \frac{\partial f}{\partial p}}{\displaystyle \frac{\partial g}{\partial x}}$$

(4)

for two arbitrary functions $f(x,p)$ and $g(x,p)$.

Quantum mechanics, in the way it is usually presented in undergraduate lectures, obeys a rather different formalism. The state of a system is represented herein by a density operator $\widehat{\rho }$ (the hat ^ denotes a quantum-mechanical operator). It has the property

$$Tr\left(\widehat{\rho }\right)=1,$$

(5)

where Tr is the quantum-mechanical trace. The expectation value of a quantum-mechanical observable $\widehat{A}$—which corresponds to an operator acting on the Hilbert space of quantum states—reads as

$$\langle \widehat{A}\rangle =Tr\left(\widehat{A}\widehat{\rho }\right).$$

(6)

The time evolution of $\widehat{\rho }$ follows the Liouville–von Neumann equation

$${\partial }_t\widehat{\rho }={\displaystyle \frac{\mathrm{i}}{\hslash }}[\widehat{H},\widehat{\rho }]$$

(7)

with the reduced Planck constant ℏ, the Hamiltonian $\widehat{H}$, the imaginary unit $\mathrm{i}$, and the commutator $[·,·]$.

3. Theory of Wigner Functions

While there is an obvious similarity between Equations (1) and (5), between Equations (2) and (6), and between Equations (3) and (7), there is also an obvious difference—the classical formalism operates on phase space and uses phase space distributions, while the quantum formalism uses operators. Intuitively, one might attribute the fact that quantum mechanics does not operate in phase space to the Heisenberg uncertainty relation, which implies that a system cannot simultaneously have a well-defined position and momentum, and that the state of a quantum system can therefore not be represented by a point in phase space.

However, this does not exclude the possibility of representing a quantum state on phase space by something other than a point [18]. This other thing is the Wigner function. A Wigner function is a distribution function on phase space that describes the state of a quantum-mechanical system. The definition of the Wigner function for a particle in one spatial dimension whose only degrees of freedom are position and momentum and whose quantum state is given by $\widehat{\rho }$ is [19]

$$W(x,p)={\displaystyle \frac{1}{2\pi \hslash }}\int \mathrm{d}{x}^{\prime }\langle x-{\displaystyle \frac{1}{2}}{x}^{\prime }\left|\widehat{\rho }\right|x+{\displaystyle \frac{1}{2}}{x}^{\prime }\rangle e^{\mathrm{i}{\displaystyle \frac{x^{\prime }p}{\hslash }}}.$$

(8)

(The author uses here the bra–ket notation, which is explained in any textbook on quantum mechanics [20].)

More generally, the Wigner function is a function of the phase space coordinates $\overrightarrow{\mathrm{\Gamma }}$. The representation of the quantum state in terms of W contains the same information to that in terms of $\widehat{\rho }$, and thus, the operator-based formalism and the Wigner description are equivalent. However, the Wigner description has much stronger similarities to the classical case.

As an example, we can consider the ground state of the quantum-mechanical harmonic oscillator (discussed in Ref. [9], which the author follows here). The wavefunction is given by

$$\psi \left(x\right)={\displaystyle \frac{1}{\sqrt[4]{a^2\pi }}}{e}^{-{\displaystyle \frac{x^2}{2a^2}}}.$$

(9)

Here, $a^2=\hslash /\left(m\omega \right)$, with mass m and frequency $\omega$. The density operator in the position representation is

$$\rho (x,x^{\prime })=\langle x|\widehat{\rho }|x^{\prime }\rangle =\psi \left(x\right)\overline{\psi }\left(x^{\prime }\right)={\displaystyle \frac{1}{\sqrt{a^2\pi }}}{e}^{-{\displaystyle \frac{x^2+x^{\prime 2}}{2a^2}}},$$

(10)

where the bar designates a complex conjugate. Equations (8) and (10) give

$$W(x,p)={\displaystyle \frac{1}{\pi \hslash }}{e}^{-{\displaystyle \frac{a^2{p}^2}{{\hslash }^2}}-{\displaystyle \frac{x^2}{a^2}}}.$$

(11)

We continue with the general theory, first considering the stationary case. Just as we can represent the density operator $\widehat{\rho }$ as a function on phase space, we can also represent a general operator $\widehat{A}$ in this way. Such operators do (in the “standard” formalism) correspond to quantum observables. The Weyl symbol of an operator $\widehat{A}$ is given by

$$A(x,p)={\displaystyle \frac{1}{\hslash }}\int \mathrm{d}{x}^{\prime }\langle x-{\displaystyle \frac{1}{2}}{x}^{\prime }\left|\widehat{A}\right|x+{\displaystyle \frac{1}{2}}{x}^{\prime }\rangle e^{\mathrm{i}{\displaystyle \frac{x^{\prime }p}{\hslash }}}.$$

(12)

For the expectation value $\langle \widehat{A}\rangle$, which one would usually determine using Equation (6), we then have the expression [19]

$$\langle \widehat{A}\rangle =\int \mathrm{d}x\int \mathrm{d}pA(x,p)W(x,p),$$

(13)

which is fully analogous to the classical case (2). Thus, W appears to work exactly like the classical distribution function $\rho$, and indeed it is the quantum-mechanical generalization of it. However, it does not constitute a probability distribution. The reason for this is that (due to the aforementioned Heisenberg uncertainty principle), there is no point in phase space that corresponds to the system’s actual state, and thus, W cannot be interpreted as a probability distribution over such states [18]. More technically, the function W can become negative, which is not possible for a probability distribution function.

Let us now turn to dynamics. If we denote the Weyl symbol of the Hamiltonian $\widehat{H}$ by $H(x,p)$, the time evolution of W is (as can be shown in Equations (7) and (8)) given by

$${\partial }_{t}W(x,p)={\{H(x,p),W(x,p)\}}_{★}.$$

(14)

Equation (14) looks, compared to to Equation (7) (which it is equivalent to), much more similar to the classical version (3). The difference is that, instead of the Poisson bracket $\{·,·\}$, Equation (14) uses the Moyal bracket [7,8]:

$$\begin{array}{cc}\hfill & {\{A(x,p),B(x,p)\}}_{★}\hfill \\ \hfill & =-{\displaystyle \frac{\mathrm{i}}{\hslash }}(A(x,p)★B(x,p)-B(x,p)★A(x,p))\hfill \\ \hfill & ={\displaystyle \frac{2}{\hslash }}A(x,p)sin\left({\displaystyle \frac{\hslash }{2}}(\overleftarrow{{\partial }_x}\overrightarrow{{\partial }_p}-\overleftarrow{{\partial }_p}\overrightarrow{{\partial }_x})\right)B(x,p),\hfill \end{array}$$

(15)

with the star product [21]

$$A(x,p)★B(x,p)=A(x,p)exp\left({\displaystyle \frac{\mathrm{i}\hslash }{2}}(\overleftarrow{{\partial }_x}\overrightarrow{{\partial }_p}-\overleftarrow{{\partial }_p}\overrightarrow{{\partial }_x})\right)B(x,p),$$

(16)

where derivatives act in the direction indicated by the arrows. The Moyal bracket (15) is what one obtains when one applies the Wigner transformation (8) to the quantum-mechanical commutator.

Up to now, the author has discussed Wigner functions for the case of a single particle in one spatial dimension whose degrees of freedom are position and momentum. This is not the most general case. A straightforward generalization is to consider N particles in three spatial dimensions, in which case W is a function of N three-dimensional position and momentum vectors. One can also consider systems with degrees of freedom other than position and momentum, for which particles with spin or angular momentum would be typical examples. In this case, the phase space would include angular coordinates [16,22]. An application of this case is the derivation of quantum-mechanical orientational-order parameters [16].

Formally, the connection between the phase space-based and the operator-based formalism is made via the Stratonovich–Weyl kernel $\widehat{\mathrm{\Delta }}\left(\overrightarrow{\mathrm{\Gamma }}\right)$. It translates a phase space function $A\left(\overrightarrow{\mathrm{\Gamma }}\right)$ to the operator [23]

$$\widehat{A}=\int \mathrm{d}\mathrm{\Gamma }A\left(\overrightarrow{\mathrm{\Gamma }}\right)\widehat{\mathrm{\Delta }}\left(\overrightarrow{\mathrm{\Gamma }}\right).$$

(17)

For example, in the case of the one-dimensional point particle, this kernel has the form [24,25]

$$\widehat{\mathrm{\Delta }}(x,p)={\displaystyle \frac{1}{2\pi \hslash }}\int \mathrm{d}\xi \int \mathrm{d}\zeta e^{-\mathrm{i}(\xi (x-\widehat{x})+\zeta (p-\widehat{p}))},$$

(18)

where $\widehat{x}$ and $\widehat{p}$ are the position and momentum operator, respectively.

The simple approach presented here can be extended in a variety of directions. Notable here is the development of relativistic Wigner functions, a topic that has been addressed in a variety of works [26,27,28,29,30,31,32]—even in complicated settings such as curved spacetimes [33,34]. Relativistic extensions of the Wigner have been developed in a variety of ways. Morgan [32], for instance, has derived a formalism for quasiprobability distributions over trajectories rather than phase space in order to ensure manifest covariance. Bialynicki-Birula et al. [28], in contrast, use an approach more analogous to the nonrelativistic case in that they assign phase space distributions to quantum states at a certain time. While they thereby sacrifice manifest relativistic covariance, their approach has the advantage that the results can be more easily interpreted intuitively (a typical reason why Wigner functions are used in the first place) and still allows for relativistic calculations to be performed.

Another issue that should be noted here is that, despite appearance, it is not in general possible to recover classical mechanics by simply taking the limit $\hslash \to 0$ in the Wigner description. A detailed discussion of issues arising in this approach is provided in Ref. [9].

4. Applications of Wigner Functions

An overview over theory and applications of Wigner functions is given in Refs. [10,11,12,13,14]. An important aspect here are applications in quantum technology and quantum information science [35]. A field where the Wigner function gained importance quite early is quantum optics [36,37,38,39,40], where it can be used to visualize coherent or squeezed states. (The Wigner function even has applications in classical optics [41].) Slightly later, the use of Wigner functions in solid-state physics gained traction [42,43,44,45]. Here, an important example is the application in semiconductor modeling [46,47,48]. A final notable area is the study of dissipative quantum systems and quantum Brownian motion [15,49,50,51,52,53,54]. In addition to these central fields, there are also more exotic applications such as quantum finance [55,56].

As an example, the author discusses here one application of Wigner functions, namely the study of the relation between quantum and classical systems. A standard route to investigate this transition is via the Ehrenfest theorem, which allows to write down for the expectation values of quantum-mechanical observables dynamic equations resembling classical ones [20]. This may give the impression that classical dynamics is recovered in case these equations correspond to the classical ones. The fact that this is not actually the case can already be seen from the harmonic oscillator, which would by this criterion be classified as classical despite the fact that the quantum harmonic oscillator shows non-classical behavior [57]. The Wigner function approach, on the other hand, correctly captures the quantum nature of the harmonic oscillator (see below). One reason for why using Wigner functions is helpful in studying the quantum–classical transition is that the dynamics of W may closely resemble that of a classical probability distribution even in cases where the microdynamics of individual particles is very non-classical. Wallace [18] suggests non-interacting dilute atomic gases as such a scenario.

A key topic in the use of Wigner functions for studying the quantum–classical transition are quantum systems in contact with an environment (open quantum systems). The correct way to model open quantum systems has been subject to much debate, so the author focuses here on the simplest case that was studied in the pioneering work of Caldeira and Leggett [53]. In the high-temperature limit, the dynamics of a quantum system coupled to a large environment is shown to be given by

$${\partial }_{t}W(x,p)={\{H(x,p),W(x,p)\}}_{★}+\gamma {\partial }_p\left(pW(x,p)\right)+\gamma m{k}_{\mathrm{B}}T{\partial }_p^{2}W(x,p),$$

(19)

where $\gamma$ is the friction coefficient, $k_{\mathrm{B}}$ is the Boltzmann constant, and T is the temperature. More realistic descriptions of quantum heat baths require a replacement of the last two terms in Equation (19) by more sophisticated expressions [58].

Notably, if one would replace the Moyal bracket in Equation (19) by a Poisson bracket (i.e., if one takes a naive classical limit), then one recovers the Fokker–Planck Equation [59], which describes the dynamics of a classical Brownian particle, i.e., a particle that is in contact with a heat bath consisting of a large number of particles that it is constantly colliding with. This equation is widely studied and well understood, and therefore the fact that Wigner functions allow to formulate the equation of motion for a quantum system coupled to a large environment in an analogous way allows to better understand the quantum case as well [15]. A major application of Equation (19) in the quantum case is the study of decoherence. This is a process were quantum coherence is lost by coupling to an external environment. In Ref. [60], this was investigated by studying the dynamics of the Wigner function for a system exhibiting classical chaos. Incorporating decoherence (represented by the dissipative terms in Equation (19)) allows to recover the classical chaotic behavior as a limit of the quantum dynamics, whereas without this term, classical and quantum dynamics (in this case, given by Equations (3) and (7), respectively) differ significantly.

It should be pointed out here that the Wigner function, while presenting a different mathematical formulation of quantum mechanics, does not provide an ontologically different framework (or at least is not usually interpreted this way). This is an important difference to the otherwise related formalism of Bohmian mechanics [4] mentioned above. Bohmian mechanics also offers a solution to the quantum measurement problem [6]; in other words, it is able to explain why, if we perform a measurement on a quantum system in a superposition, we only measure one possible outcome rather than both (the Bohmian solution consists of postulating that the measurement outcome depends on hidden variables). Since superpositions are found in quantum but not in classical mechanics, resolving the measurement problem is in principle an important element of resolving the quantum–classical transition [15].

Still, the fact that the Wigner approach is simply a mathematical reformulation of the standard quantum formalism implies that it is perfectly compatible with all typical interpretations of quantum mechanics that are employed to resolve the measurement problem. Take, as an example, the Ghirardi–Rimini–Weber (GRW) theory [61], which postulates that quantum-mechanical particles do, in addition to unitary time evolution, undergo random stochastic collapses. Denoting the localization rate for the i-th particle by ${\lambda }_i$, this gives, for an N-particle system, the dynamic equation

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\widehat{\rho }=-{\displaystyle \frac{\mathrm{i}}{\hslash }}[\widehat{H},\widehat{\rho }]-\sum _{i=1}^N{\lambda }_i(\widehat{\rho }-{\widehat{T}}_i\left(\widehat{\rho }\right)),$$

(20)

with the localization operator

$${\widehat{T}}_i\left(\widehat{\rho }\right)=\sqrt{{\displaystyle \frac{\alpha }{\pi }}}{\int }_{-\infty }^{\infty }\mathrm{d}x{e}^{-{\displaystyle \frac{\alpha }{2}}{({\widehat{x}}_i-x)}^2}\widehat{\rho }{e}^{-{\displaystyle \frac{\alpha }{2}}{({\widehat{x}}_i-x)}^2},$$

(21)

where $\alpha$ is a constant and ${\widehat{x}}_i$ is the i-th particle’s position operator. In Ref. [15], it was shown that (assuming ${\lambda }_i=\lambda \forall i$) Equation (20) can be reformulated as an equation for Wigner functions, giving

$${\partial }_{t}W(x,p)={\{H(x,p),W(x,p)\}}_{★}-\lambda \left(W(x,p)-{\displaystyle \frac{1}{\sqrt{\pi \alpha {\hslash }^2}}}\int \mathrm{d}{p}^{\prime }{e}^{-{\displaystyle \frac{1}{\alpha {\hslash }^2}}{\left(p^{\prime }\right)}^2}W(x,p-p^{\prime })\right).$$

(22)

Another interesting question in the context of the classical–quantum transition is the significance of negative values of the Wigner function. The fact that W can become negative is one of the main aspects that sets it apart from the classical distribution function $\rho$, and thus, it is plausible to assume that negative values of W are a signature of non-classical behavior [62]. For pure states, it turns out that Gaussian wavefunctions are the only ones with a positive wavefunction everywhere [63]. Moreover, for quantum circuits operating with states that have a positive Wigner function, an efficient classical simulation is possible, whereas negative Wigner functions are what allows quantum computers to outperform classical ones [64]. This is why Wigner function negativity is important in quantum computing [65,66]. However, quantum behavior can also be exhibited by systems with purely positive Wigner functions. An obvious example (see Ref. [9]) is the ground state (11) of the harmonic oscillator; its ground state energy is, using Equation (13), found to be

$$\langle \widehat{H}\rangle =\int \mathrm{d}x\int \mathrm{d}p{\displaystyle \frac{1}{\pi \hslash }}{e}^{-{\displaystyle \frac{a^2{p}^2}{{\hslash }^2}}-{\displaystyle \frac{x^2}{a^2}}}\left({\displaystyle \frac{p^2}{2m}}+{\displaystyle \frac{m{\omega }^2{x}^2}{2}}\right)={\displaystyle \frac{\hslash \omega }{2}},$$

(23)

where the Weyl symbol of the Hamiltonian of the harmonic oscillator is $p^2/\left(2m\right)+m{\omega }^2{x}^2/2$ (as one would expect). Thus, Equation (11), despite being perfectly positive, represents the quantum effect of zero-point energy. Moreover, quantum nonlocality also turns out to be possible in systems whose Wigner function is positive [67].

5. Conclusions

In this entry, the author has provided a brief introduction to the Wigner function, which provides a phase space representation of the state of a quantum system. In particular, the author has emphasized the way in which it can be used to study the classical limit of quantum mechanics, and highlighted some subtleties in this context. The author hopes that this entry provides a helpful starting point for graduate students aiming to learn about the exciting fields of quantum foundations and open quantum systems.