Marginals and Quasi-Probabilities

A marginal is obtained by integrating a joint distribution over variables that are not retained. For an ordinary classical density $f(x,p)$,

$$f_x(x) = \int dp\,f(x,p), \qquad f_p(p) = \int dx\,f(x,p).$$

The Wigner function has the same formal property: its position and momentum marginals are exactly the Born-rule distributions. Yet the full Wigner function can be negative and is not an ordinary joint probability for simultaneous sharp values of $x$ and $p$.

This page explains that apparent tension. Wigner Function is the canonical home for the definition and standard state examples. Here the focus is what the marginals mean, why negativity is allowed, and what changes in the Husimi $Q$ and Glauber–Sudarshan $P$ representations.

Position Marginal

Using the conventions fixed in Phase-Space Conventions, the Wigner function of a density operator $\rho$ is

$$\begin{aligned} W_\rho(x,p) &= \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dy\, e^{-ipy/\hbar} \\ &\quad\times \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle. \end{aligned}$$

Integrating over momentum uses the Fourier identity

$$\frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dp\, e^{-ipy/\hbar} = \delta(y).$$

Therefore

$$\begin{aligned} \int dp\,W_\rho(x,p) &= \int dy\, \delta(y) \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle \\ &= \langle x\rvert\rho\lvert x\rangle. \end{aligned}$$

For a pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$,

$$\int dp\,W_\psi(x,p) = |\psi(x)|^2.$$

The integral is a genuine probability density for a sharp position measurement. A horizontal slice $W(x,p_0)$ is not. Integrating over $p$ is what removes the off-diagonal separation variable $y$ and leaves the diagonal position-space density matrix.

For an interval $\Delta_x$,

$$\Pr(x\in\Delta_x) = \int_{\Delta_x}dx \int_{-\infty}^{\infty}dp\, W_\rho(x,p).$$

This probability is nonnegative even if the integrand has negative regions.

Momentum Marginal

The momentum marginal follows from the same definition. Integrate over $x$:

$$\begin{aligned} \int dx\,W_\rho(x,p) &= \frac{1}{2\pi\hbar} \int dx\,dy\, e^{-ipy/\hbar} \\ &\quad\times \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle. \end{aligned}$$

Introduce

$$x_1 = x+\frac{y}{2}, \qquad x_2 = x-\frac{y}{2}.$$

The Jacobian has unit absolute value, so $dx\,dy=dx_1\,dx_2$, and $y=x_1-x_2$. Hence

$$\begin{aligned} \int dx\,W_\rho(x,p) &= \frac{1}{2\pi\hbar} \int dx_1\,dx_2\, e^{-ip(x_1-x_2)/\hbar} \\ &\quad\times \langle x_1\rvert\rho\lvert x_2\rangle. \end{aligned}$$

Using

$$\langle x\rvert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar},$$

the right-hand side is precisely

$$\langle p\rvert\rho\lvert p\rangle.$$

Thus

$$\int dx\,W_\rho(x,p) = P_p(p).$$

Integrating either marginal once more gives

$$\int dx\,dp\,W_\rho(x,p) = \operatorname{Tr}\rho = 1$$

for a normalized state.

The two exact marginals do not make $W_\rho(x,p)$ a classical joint density. They say that the same quasiprobability packages the statistics of two incompatible sharp measurements. They do not say that each experimental run reveals a pre-existing pair $(x,p)$ sampled from $W_\rho$.

Other Quadrature Marginals

Position and momentum are two members of a continuous family of quadratures. In dimensionless oscillator variables satisfying $[\hat X,\hat P]=i$, define

$$\hat X_\theta = \hat X\cos\theta + \hat P\sin\theta.$$

A rotation in phase space gives coordinates $(X_\theta,P_\theta)$. The probability density for measuring $\hat X_\theta$ is the line marginal

$$P_\theta(X_\theta) = \int dP_\theta\, W_\theta(X_\theta,P_\theta),$$

where $W_\theta$ is the Wigner function expressed in the rotated coordinates.

This is the basis of optical homodyne tomography. Different local-oscillator phases select different quadratures, and their measured probability densities provide line projections of the Wigner function. Reconstructing $W$ from sufficiently many projections is an inverse Radon-transform problem. Individual detector outcomes are ordinary nonnegative frequencies; negative Wigner regions appear only in the reconstructed phase-space representation.

Why the Wigner Function Can Be Negative

The Wigner transform uses off-diagonal density-matrix elements:

$$\left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle.$$

These terms contain coherence and relative phase information. A Fourier transform of them need not be positive, just as an interference term in an amplitude need not behave like a classical probability contribution.

For a coherent superposition

$$\lvert\psi\rangle = c_1\lvert\psi_1\rangle + c_2\lvert\psi_2\rangle,$$

the density operator contains diagonal pieces and cross terms:

$$\begin{aligned} \rho &= |c_1|^2 \lvert\psi_1\rangle\langle\psi_1\rvert + |c_2|^2 \lvert\psi_2\rangle\langle\psi_2\rvert \\ &\quad+ c_1c_2^* \lvert\psi_1\rangle\langle\psi_2\rvert + c_1^*c_2 \lvert\psi_2\rangle\langle\psi_1\rvert. \end{aligned}$$

Linearity gives

$$W_\psi = |c_1|^2W_1 + |c_2|^2W_2 + W_{\mathrm{int}}.$$

The cross-term contribution $W_{\mathrm{int}}$ produces oscillatory fringes. Some fringes can be negative, while integration over $p$ or $x$ still returns a nonnegative Born distribution.

Negativity does not violate normalization. Decompose a real Wigner function into positive and negative parts,

$$W = W_+ - W_-, \qquad W_\pm\geq0.$$

Normalization constrains the signed difference

$$\int W_+ - \int W_- = 1,$$

not each part separately.

Negativity as a Nonclassicality Preview

Wigner negativity is a strong witness that no ordinary probability density with the Wigner representation’s exact rules is being used. A common quantitative measure is the negative volume

$$\mathcal N_W = \frac12 \left[ \int dx\,dp\, |W(x,p)| - 1 \right].$$

Equivalently,

$$\mathcal N_W = \int dx\,dp\, \max\bigl(0,-W(x,p)\bigr).$$

It vanishes exactly when $W$ is nonnegative almost everywhere.

For pure continuous-variable states under the standard regularity assumptions, Hudson’s theorem says that a nonnegative Wigner function must be Gaussian. The restriction to pure states matters. Mixed non-Gaussian states can have positive Wigner functions.

Negativity is not a universal definition of nonclassicality:

• coherent states have positive Gaussian Wigner functions;
• squeezed Gaussian states also have positive Wigner functions, yet display nonclassical noise relative to coherent states;
• entanglement can occur in multimode Gaussian states with a positive Wigner function;
• every Husimi $Q$ function is nonnegative, including those of highly nonclassical states.

The operational meaning depends on the allowed states, measurements, transformations, and task. A resource theory may treat Wigner negativity as useful under a specified set of operations, but that statement should not be promoted into a representation-independent definition of “quantum.”

Other Quasiprobability Distributions

Phase-space distributions form a family tied to operator ordering. In oscillator variables, three standard members are:

Distribution	Ordering naturally represented	Positivity and regularity
Glauber–Sudarshan $P$	normal ordering	can be singular or fail to be nonnegative
Wigner $W$	symmetric ordering	regular in many cases, but can be negative
Husimi $Q$	antinormal ordering	smooth and nonnegative

Moving from $P$ toward $W$ and then $Q$ adds Gaussian smoothing. Smoothing removes fine oscillations and can remove negativity, but it also reduces resolution and changes which operator moments are obtained by ordinary integration.

No member is simply “the true distribution.” Each combines a state representation with a matching symbol or ordering rule for observables. Mixing a $Q$ function with a Weyl symbol, or a $P$ function with an antinormally ordered moment formula, produces incorrect expectation values.

Husimi Q Preview

Oscillator coherent states obey the resolution of identity

$$\int_{\mathbb C} \frac{d^2\alpha}{\pi} \lvert\alpha\rangle\langle\alpha\rvert = I.$$

Define

$$Q(\alpha) = \frac{1}{\pi} \langle\alpha\rvert\rho\lvert\alpha\rangle.$$

Because $\rho$ is positive,

$$Q(\alpha)\geq0,$$

and the coherent-state resolution gives

$$\int_{\mathbb C}d^2\alpha\, Q(\alpha) = 1.$$

The $Q$ function is the outcome density of the coherent-state POVM, as realized ideally by heterodyne-type measurements. Its nonnegativity therefore has a direct measurement meaning. It is not an exact joint density for sharp $x$ and $p$: the coherent-state POVM is unsharp, and its distribution includes the corresponding vacuum-scale smoothing.

Since every state has a nonnegative $Q$, $Q$ positivity alone cannot diagnose classicality. Zeros, shape constraints, or comparison with a specified classical set can carry information, but those are different criteria.

Glauber–Sudarshan P Preview

The Glauber–Sudarshan representation writes a density operator formally as

$$\rho = \int_{\mathbb C}d^2\alpha\, P(\alpha) \lvert\alpha\rangle\langle\alpha\rvert.$$

If $P(\alpha)$ is an ordinary nonnegative normalized density, then $\rho$ is a classical statistical mixture of coherent states. This gives a particularly strong optical notion of classicality.

Many quantum states do not admit such a regular positive $P$. Number states, squeezed states, and superpositions of coherent states require a distribution that is negative, more singular than an ordinary function, or both. A coherent state itself has

$$P(\alpha) = \delta^{(2)}(\alpha-\alpha_0),$$

which is singular as a function but is still a nonnegative probability measure. Singularity alone must therefore be interpreted with care; the relevant question is whether $P$ defines a nonnegative measure.

The $P$ representation is especially useful for normally ordered moments in quantum optics. Its apparent simplicity is paired with the possibility that the representing object is a generalized distribution rather than a plot-ready function.

Common Mistakes

• Calling $W(x,p)$ a probability density because its two primary marginals are probabilities.
• Confusing a slice of a Wigner function with a marginal.
• Interpreting a negative Wigner value as a negative detector count.
• Assuming every positive Wigner function represents a classical state.
• Using Hudson’s pure-state theorem for mixed states.
• Treating Husimi $Q$ as an exact sharp joint distribution of position and momentum.
• Declaring a state nonclassical merely because its coherent-state $P$ is a delta distribution.
• Combining a quasiprobability with an observable symbol from a different ordering convention.
• Comparing negativities before matching quadrature scales and normalization conventions.

Cross-Links

• Optical Phase-Space Distributions for explicit $\alpha$-plane normalizations, optical state examples, tomography, and detector loss.
• Why Phase Space in Quantum Mechanics?
• Wigner Function
• Weyl Transform
• Gaussian States and Wigner Functions
• Coherent States in Phase Space
• Coherent States
• Phase-Space Conventions
• Classical vs Quantum Probability
• Quantum Optics
• Wigner Function Notebook

References

• E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review 40, 749–759 (1932).
• K. Husimi, “Some formal properties of the density matrix,” Proceedings of the Physico-Mathematical Society of Japan 22, 264–314 (1940).
• E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Physical Review Letters 10, 277–279 (1963).
• R. J. Glauber, “Coherent and incoherent states of the radiation field,” Physical Review 131, 2766–2788 (1963).
• K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Physical Review 177, 1857–1881 (1969).
• R. L. Hudson, “When is the Wigner quasi-probability density non-negative?” Reports on Mathematical Physics 6, 249–252 (1974).
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Physics Reports 106, 121–167 (1984).
• U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press, 1997.

Exercises

1. Derive the position marginal directly from the Wigner definition.

Solution

Integrate over $p$:

$$\begin{aligned} \int dp\,W_\rho(x,p) &= \int dy\, \left[ \frac{1}{2\pi\hbar} \int dp\, e^{-ipy/\hbar} \right] \\ &\quad\times \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle. \end{aligned}$$

The expression in brackets is $\delta(y)$. It sets $y=0$, leaving

$$\int dp\,W_\rho(x,p) = \langle x\rvert\rho\lvert x\rangle.$$

2. Show that the momentum marginal equals $\langle p\rvert\rho\lvert p\rangle$.

Solution

After integrating over $x$, change variables to

$$x_1=x+\frac{y}{2}, \qquad x_2=x-\frac{y}{2}.$$

Then

$$\begin{aligned} \int dx\,W_\rho(x,p) &= \frac{1}{2\pi\hbar} \int dx_1\,dx_2\, e^{-ipx_1/\hbar} \\ &\quad\times \langle x_1\rvert\rho\lvert x_2\rangle e^{ipx_2/\hbar}. \end{aligned}$$

Using

$$\langle p\rvert x_1\rangle = \frac{e^{-ipx_1/\hbar}}{\sqrt{2\pi\hbar}}, \qquad \langle x_2\rvert p\rangle = \frac{e^{ipx_2/\hbar}}{\sqrt{2\pi\hbar}},$$

the integral is $\langle p\rvert\rho\lvert p\rangle$.

3. Prove the two expressions for Wigner negative volume are equal.

Solution

For real $W$,

$$|W|-W = 2\max(0,-W).$$

Integrating and using $\int W=1$ gives

$$\begin{aligned} \frac12 \left( \int |W|-1 \right) &= \frac12 \int \left( |W|-W \right) \\ &= \int \max(0,-W). \end{aligned}$$

4. Use the coherent-state resolution of identity to prove that $Q(\alpha)$ is normalized.

Solution

By definition,

$$Q(\alpha) = \frac{1}{\pi} \langle\alpha\rvert\rho\lvert\alpha\rangle.$$

Therefore

$$\begin{aligned} \int d^2\alpha\,Q(\alpha) &= \operatorname{Tr} \left[ \rho \int\frac{d^2\alpha}{\pi} \lvert\alpha\rangle\langle\alpha\rvert \right] \\ &= \operatorname{Tr}(\rho I) = 1. \end{aligned}$$

Positivity follows from $\langle\alpha\rvert\rho\lvert\alpha\rangle\geq0$.

5. Compare the signs of $W$, $Q$, and $P$ for a coherent state, a one-photon number state, and a squeezed vacuum.

Solution

A coherent state has a positive Gaussian Wigner function, a positive Gaussian $Q$ function, and a $P$ representation given by a nonnegative delta measure at its coherent amplitude.

A one-photon number state has a Wigner function that is negative near the origin. Its $Q$ function is nevertheless nonnegative, as every $Q$ function must be. Its $P$ representation is nonclassical and highly singular.

A squeezed vacuum has a positive Gaussian Wigner function and a positive $Q$ function. Its $P$ representation is not an ordinary nonnegative measure. This example shows why positive Wigner and $Q$ functions do not imply optical classicality under the coherent-state $P$ criterion.