Wigner Distribution in Quantum Mechanics

• Wigner Distribution is a quasi-probability function on phase space that encodes quantum state information and reproduces canonical position and momentum marginals.
• It exactly recovers quantum probabilities via integration and exhibits negativity that signals quantum interference and coherence.
• The distribution is central to quantum optics, statistical mechanics, and signal analysis, with extensions to discrete, finite, and composite systems.

The Wigner distribution is a quasi-probability distribution function on classical phase space $(q,p)$ that furnishes a full phase-space formulation of quantum mechanics. Originally introduced by Wigner in 1932, it has become a cornerstone in quantum statistical mechanics, quantum optics, signal processing, and multiple areas of field theory. Its primary distinguishing feature is its ability to reproduce the canonical quantum marginals for position and momentum while encoding the full quantum state in a rigorously defined functional form on phase space (O'Connell, 2010).

1. Formal Definitions and Mathematical Structure

Let $\psi(q)$ denote a normalized wavefunction in one dimension. The Wigner distribution $W(q,p)$ is defined as

$$W(q,p) \;=\;\frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dy\;e^{-\,\tfrac{i}{\hbar}p\,y}\; \psi^*\!\Bigl(q-\tfrac{y}{2}\Bigr)\, \psi\!\Bigl(q+\tfrac{y}{2}\Bigr)\,.$$

For a general density operator $\rho$, the definition generalizes to: $$W(q,p) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dy\; e^{-\,\tfrac{i}{\hbar}p\,y}\; \bigl\langle q+\tfrac{y}{2}\bigm|\rho\bigm|q-\tfrac{y}{2}\bigr\rangle\,.$$ This is the manifestly real-valued "Weyl-Wigner" transform, and extends naturally to multiple dimensions (O'Connell, 2010, Gosson et al., 2021).

2. Marginal Distributions and Quantum Probabilities

A defining property of the Wigner distribution is the precise recovery of quantum mechanical probabilities for $q$ and $p$ separately:

• Position marginal:

$\int_{-\infty}^{\infty}dp\;W(q,p) = |\psi(q)|^2\,,$

which is the position probability density.

• Momentum marginal:

If $\phi(p)$ is the momentum-space wavefunction,

$\int_{-\infty}^{\infty}dq\;W(q,p) = |\phi(p)|^2\,,$

thus reproducing the momentum probability density (O'Connell, 2010).

In general, for any tuple of observables $A_1,...,A_n$, the Wigner distribution $W_\rho(a)$ is uniquely characterized by the property that the marginal for any linear combination $\xi\cdot A$ coincides with its quantum probability, formalized by inverse Radon transforms (Schwonnek et al., 2018).

3. Expectation Values and Weyl Quantization

One of the chief analytical virtues of the Wigner distribution is the translation of quantum expectation values into phase-space integrals. For any operator $\hat{A}$ with Weyl symbol $A_W(q,p)$,

$$\langle\hat{A}\rangle = \mathrm{Tr}(\rho\,\hat{A}) = \int dq\,dp\,W(q,p)\,A_W(q,p)\,,$$

where

$$A_W(q,p) = \int dz\,e^{\,\tfrac{i}{\hbar}p\,z}\; \left\langle q-\tfrac{z}{2}\Bigm|\,\hat{A}\,\Bigm|q+\tfrac{z}{2}\right\rangle\,.$$

This "classical-looking" integral is exact for symmetrically-ordered quantum operators (O'Connell, 2010, Gosson et al., 2021).

4. Key Properties: Reality, Normalization, Negativity, and Limits

The Wigner distribution exhibits several crucial, nontrivial mathematical features:

Property	Mathematical Expression	Physical/Interpretive Significance
Realness	$W(q,p)\in\mathbb{R}$	Direct from symmetric complex conjugation
Normalization	$\int dq\,dp\,W(q,p)=1$	Ensures correct total probability
Quasi-probability	$W(q,p)$ can be negative	Encodes quantum interference/fringes
Marginals	$\int dp\,W(q,p)=|\psi(q)|^2$, $\int dq\,W(q,p)=|\phi(p)|^2$	Exact quantum probabilities
Classical limit	$W(q,p)\to$ positive, sharply peaked for $\hbar\to 0$ or highly excited states	Approaches classical phase-space density

Negativity in $W(q,p)$ serves as a signature of quantum coherence; for harmonic oscillator ground and coherent states, $W$ is strictly positive and Gaussian, while for "Schrödinger cat" states, $W$ develops alternating positive/negative interference stripes. In the semiclassical limit, $W$ approximates the classical Liouville density, but sign-changing regions persist where quantum effects survive (O'Connell, 2010).

5. Generalizations, Discretizations, and Extensions

• Arbitrary observables: The Wigner distribution can be constructed for any tuple of Hermitian operators $(A_1, ..., A_n)$ via a unique joint distribution $W_\rho(a)$ on $\mathbb{R}^n$ whose marginals correspond to quantum probability distributions of all linear combinations. For $n=2$, $(Q,P)$, this definition reduces to the standard phase-space Wigner function. In finite dimensions, $W_\rho(a)$ is supported on the convex hull of expectation value tuples, with singularities at boundary degeneracies and (unless all $A_i$ commute) displays nontrivial negativity (Schwonnek et al., 2018).
• Finite oscillator systems: The Wigner distribution can be discretized for quantum systems with finite spectra, defining a real quasiprobability $W(n;p_k,q_\ell)$ on a $(N+1)\times(N+1)$ grid. Proper combinations of Vandermonde matrices recover phase-space moments, marginals, and normalization analogous to the continuous case. As system size $N\to\infty$, the discrete model converges to the canonical Wigner function (Jeugt, 2013).
• Reduced density matrices and bipartite systems: In composite systems, the Wigner function for the reduced density operator is obtained by integrating the total Wigner function over the phase-space variables of the traced-out subsystem. For Gaussian states, the reduced Wigner function remains Gaussian with the marginal mean and block covariance (Gosson et al., 2021).
• Generalized stochastic processes: The Wigner distribution extends to time-frequency analysis of generalized stochastic processes (GSPs), including stationary and Gaussian cases, with explicit formulae for covariance in terms of Weyl symbols. In the deterministic case, the standard Wigner formula is recovered (Wahlberg, 4 Apr 2025).

6. Physical Interpretation and Applications

The Wigner distribution maps quantum states to (generally non-positive) functions on phase space, transferring statistical information from the density operator to the Wigner function. It offers an intuitive phase-space landscape for quantum phenomena, allowing direct visualization of localization, delocalization, and quantum coherence.

• Quantum optics: Used extensively in the characterization of quantum states of light, analysis of nonclassicality, and quantum tomography.
• Quantum statistical mechanics: Facilitates semiclassical expansions and exploration of the quantum-classical transition.
• Signal analysis: Fundamental in time-frequency representations, including generalizations like Cohen's class (convolutions of $W$ with phase-space kernels) (Cordero et al., 2018).
• Quantum information: The Wigner function's negativity is exploited in quantum computational speedups, and forms the basis for state reconstruction in high-dimensional protocols.

7. Limitations and Alternative Representations

Despite its utility, the Wigner distribution is not a true probability density:

• Local negativity precludes classical probabilistic interpretation.
• In settings involving spin or relativistic kinematics, generalizations are nontrivial.
• For certain tasks (e.g., extracting entanglement, requiring manifest positivity), alternative quasiprobability distributions such as the Husimi Q-function or the Glauber–Sudarshan P-representation are preferred.

Furthermore, some quantum features are less directly accessible in phase-space and may require careful analysis of negative regions or moments of the Wigner distribution (O'Connell, 2010).

---

In summary, the Wigner distribution $W(q,p)$ provides a mathematically rigorous, physically transparent, and broadly applicable phase-space formulation of quantum mechanics. It reproduces quantum statistics, encodes both classical and quantum features, admits generalizations to arbitrary observables and discrete systems, and is foundational to theory and application across quantum science (O'Connell, 2010).