Phase-Space Distributions

A quantum state of one optical mode can be represented by functions or generalized distributions on a complex amplitude plane. The most common are the Glauber–Sudarshan $P$ distribution, the Wigner distribution $W$, and the Husimi $Q$ distribution. They contain the same density operator when used with their matching reconstruction rules, but they do not assign the same function to a state and they do not represent the same operator ordering.

That distinction is operational. In the conventions used here,

• $P$ converts normally ordered field moments into classical-looking averages and tests whether the state is a mixture of coherent states;
• $W$ converts symmetrically ordered moments into phase-space averages and has exact quadrature marginals;
• $Q$ is an everywhere nonnegative coherent-state measurement distribution, ideally sampled by heterodyne detection.

Calling all three “the phase-space probability” hides the physics. A one-photon state has a negative Wigner function, a nonnegative $Q$ function, and a singular nonclassical $P$ distribution. A squeezed vacuum has a nonnegative Gaussian Wigner function and a nonnegative $Q$ function but still lacks a positive Glauber–Sudarshan measure. Positivity therefore answers a different question in each representation.

This page develops optical mode amplitudes, ordering rules, and measurement access. The general Wigner transform and phase-space conventions provide the position–momentum formulation.

Required background. Quantized Modes and Density Operators define the state and operators. The examples use number states, coherent light, thermal light, and squeezed light.

Optical amplitudes and quadratures

A phase-space distribution belongs to a set of canonical mode operators, not to an unqualified beam. For one normalized temporal, spatial, spectral, and polarization mode $f$, write

$$[\hat a_f,\hat a_f^\dagger]=1.$$

Suppress the label $f$ only after that mode has been fixed. The dimensionless quadratures are

$$\hat X = \frac{ \hat a+\hat a^\dagger }{ \sqrt2 }, \qquad \hat P = \frac{ \hat a-\hat a^\dagger }{ i\sqrt2 },$$

so that

$$[\hat X,\hat P]=i.$$

The complex phase-space coordinate is

$$\begin{aligned} \alpha &= \frac{X+iP}{\sqrt2}, \\ X &= \sqrt2\operatorname{Re}\alpha, \\ P &= \sqrt2\operatorname{Im}\alpha. \end{aligned}$$

Its integration measure obeys

$$d^2\alpha \equiv d(\operatorname{Re}\alpha) d(\operatorname{Im}\alpha) = \frac12\,dX\,dP.$$

All three distributions on this page are normalized with respect to $d^2\alpha$:

$$\begin{aligned} \int_{\mathbb C}d^2\alpha\, P(\alpha) &=1, \\ \int_{\mathbb C}d^2\alpha\, W(\alpha) &=1, \\ \int_{\mathbb C}d^2\alpha\, Q(\alpha) &=1. \end{aligned}$$

where the first equality is understood distributionally when $P$ is not an ordinary function.

The factor $d^2\alpha=dX\,dP/2$ fixes both plot heights and marginal normalization. A density with respect to $dX\,dP$ is half the corresponding density written here; the physics is unchanged when the measure is changed consistently.

Displacements and Characteristic Functions

The displacement operator is

$$\hat D(\xi) = \exp\left( \xi\hat a^\dagger - \xi^*\hat a \right).$$

Its expectation value

$$\chi_0(\xi) = \operatorname{Tr} \left[ \rho\hat D(\xi) \right]$$

is the symmetrically ordered characteristic function. A compact family of ordered characteristic functions is

$$\chi_s(\xi) = \operatorname{Tr} \left[ \rho\hat D(\xi) \right] \exp\left( \frac{s}{2} \lvert\xi\rvert^2 \right),$$

where

$$s=1, 0, -1$$

correspond to normal, symmetric, and antinormal ordering. For positive $s$, $\chi_s$ may grow too rapidly for an ordinary integral or even a tempered distribution. The generalized representation must then be interpreted through its action on an appropriate test-function class. Formally, their Fourier transforms define the Cahill–Glauber distributions

$$W^{(s)}(\alpha) = \frac{1}{\pi^2} \int_{\mathbb C}d^2\xi\, \chi_s(\xi) \times \exp\left( \alpha\xi^* - \alpha^*\xi \right).$$

In the present notation,

$$\begin{aligned} P(\alpha) &= W^{(1)}(\alpha), \\ W(\alpha) &= W^{(0)}(\alpha), \\ Q(\alpha) &= W^{(-1)}(\alpha). \end{aligned}$$

This family shows that changing the ordering rule multiplies the characteristic function by a Gaussian, which becomes Gaussian smoothing or deconvolution in the amplitude plane.

The Three Standard Representations

The same coherent state $\lvert\alpha_0\rangle$ in three representations. The arrow denotes the P delta measure and is not a finite-height density. The lower panel plots exact two-dimensional density cross-sections at $\operatorname{Im}(\alpha-\alpha_0)=0$; these are not one-dimensional probability marginals; $u=\operatorname{Re}(\alpha-\alpha_0)$. The Glauber–Sudarshan distribution is a delta measure, the Wigner function is a vacuum-width Gaussian, and the Husimi function is a broader positive Gaussian. The representations carry different ordering and measurement rules even though they encode the same density operator.

Glauber–Sudarshan P representation

The diagonal coherent-state representation is

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

If $P$ is a nonnegative normalized measure, the state is a statistical mixture of coherent states. This is the standard optical classicality criterion because each coherent component behaves like a classical complex amplitude for normally ordered photodetection correlations.

The word “measure” matters. For a coherent state,

$$P_{\alpha_0}(\alpha) = \delta^{(2)} \left( \alpha-\alpha_0 \right),$$

which is singular as a function but perfectly acceptable as a positive probability measure. A claim that every singular $P$ is nonclassical is therefore false. Nonvacuum number states and nontrivially squeezed states require more singular generalized distributions that cannot be interpreted as nonnegative measures.

When the indicated moments exist, the $P$ representation is adapted to normal ordering:

$$\left\langle (\hat a^\dagger)^m \hat a^n \right\rangle = \int d^2\alpha\, P(\alpha) (\alpha^*)^m \alpha^n.$$

This identity connects positive-$P$ classicality directly to counting statistics and intensity correlations. It does not mean that a singular $P$ can be sampled pointwise in an experiment.

Wigner representation

The Wigner distribution is the symmetrically ordered member of the family. In the optical amplitude convention used here, its displaced-parity form is

$$W(\alpha) = \frac{2}{\pi} \operatorname{Tr} \left[ \hat D^\dagger(\alpha) \rho \hat D(\alpha) \hat\Pi \right],$$

where

$$\hat\Pi = (-1)^{\hat a^\dagger\hat a}$$

is photon-number parity. The formula immediately gives the bound

$$-\frac{2}{\pi} \leq W(\alpha) \leq \frac{2}{\pi}$$

for a single mode, because parity has eigenvalues $\pm1$.

For a coherent state,

$$W_{\alpha_0}(\alpha) = \frac{2}{\pi} \exp\left[ -2\lvert\alpha-\alpha_0\rvert^2 \right].$$

Its center gives the mean complex amplitude, while its width is the vacuum uncertainty required by the commutator. The function is not an ordinary joint probability for sharp $X$ and $P$, even when it happens to be nonnegative.

Symmetric products are phase-space moments. For example,

$$\left\langle \frac{ \hat a^\dagger\hat a + \hat a\hat a^\dagger }{2} \right\rangle = \int d^2\alpha\, W(\alpha) \lvert\alpha\rvert^2.$$

Since

$$\frac{ \hat a^\dagger\hat a + \hat a\hat a^\dagger }{2} = \hat n+\frac12,$$

the mean photon number is

$$\langle\hat n\rangle = \int d^2\alpha\, W(\alpha) \left( \lvert\alpha\rvert^2 - \frac12 \right).$$

The subtraction is the vacuum half quantum. Omitting it is one of the most common ordering mistakes.

Husimi Q representation

Coherent states resolve the identity:

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

The Husimi function is

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

It obeys

$$Q(\alpha)\geq0, \qquad \int d^2\alpha\,Q(\alpha)=1.$$

These properties have a direct measurement interpretation: $Q$ is the outcome density of the coherent-state positive-operator-valued measure. An ideal heterodyne measurement realizes this POVM. The returned complex amplitude includes an irreducible vacuum-scale uncertainty, so $Q$ is not a joint distribution for two sharp noncommuting quadratures.

For a coherent state,

$$Q_{\alpha_0}(\alpha) = \frac{1}{\pi} \exp\left[ -\lvert\alpha-\alpha_0\rvert^2 \right].$$

Antinormally ordered moments follow from $Q$:

$$\left\langle \hat a^n (\hat a^\dagger)^m \right\rangle = \int d^2\alpha\, Q(\alpha) \alpha^n (\alpha^*)^m.$$

In particular,

$$\int d^2\alpha\, Q(\alpha) \lvert\alpha\rvert^2 = \langle\hat a\hat a^\dagger\rangle = \langle\hat n\rangle+1,$$

so

$$\langle\hat n\rangle = \int d^2\alpha\, Q(\alpha) \left( \lvert\alpha\rvert^2-1 \right).$$

The larger subtraction reflects the added heterodyne vacuum noise.

One State, Three Symbols

The same operator can require different c-number symbols depending on the distribution used for the state. For the number operator,

State representation	Matching symbol for $\hat n$	Average
$P(\alpha)$	$\lvert\alpha\rvert^2$	normal ordering
$W(\alpha)$	$\lvert\alpha\rvert^2-1/2$	symmetric ordering
$Q(\alpha)$	$\lvert\alpha\rvert^2-1$	antinormal ordering

Every row gives the same $\langle\hat n\rangle$ when the matching pair is used. Combining $Q$ with the normal symbol $\lvert\alpha\rvert^2$ would overestimate the occupation by one photon; combining $W$ with it would overestimate by one half.

The general rule is not “integrate the classical expression against whichever distribution is convenient.” It is “transform the state and observable using dual ordering conventions.”

Gaussian Smoothing Hierarchy

For $s'<s$, the less ordered distribution is a Gaussian convolution of the more ordered one:

$$W^{(s')}(\alpha) = \frac{2}{\pi(s-s')} \int d^2\beta\, W^{(s)}(\beta) \times \exp\left[ -\frac{ 2\lvert\alpha-\beta\rvert^2 }{ s-s' } \right].$$

Thus

$$P \longrightarrow W \longrightarrow Q$$

adds vacuum-scale Gaussian blur at each step. Directly,

$$Q(\alpha) = \frac{1}{\pi} \int d^2\beta\, P(\beta) e^{-\lvert\alpha-\beta\rvert^2}.$$

The smoothing hierarchy explains several facts at once:

• a singular $P$ can yield a regular Wigner function;
• Wigner interference fringes can disappear in $Q$;
• $Q$ is nonnegative for every state;
• reconstructing $P$ from noisy $Q$ data requires Gaussian deconvolution and is usually severely ill-conditioned.

Smoothing is not loss of information in the exact mathematical representation: each characteristic function determines the density operator. It is a practical loss of robustness because inversion amplifies high-frequency statistical and calibration noise.

State Atlas

The representation hierarchy becomes concrete by comparing standard optical states.

Vacuum and coherent states

Vacuum is the coherent state with $\alpha_0=0$:

$$P_0(\alpha)=\delta^{(2)}(\alpha),$$ $$W_0(\alpha) = \frac{2}{\pi} e^{-2\lvert\alpha\rvert^2},$$

and

$$Q_0(\alpha) = \frac{1}{\pi} e^{-\lvert\alpha\rvert^2}.$$

Displacement translates each representation without changing its shape:

$$F_{\hat D(\alpha_0) \rho \hat D^\dagger(\alpha_0)}(\alpha) = F_\rho(\alpha-\alpha_0),$$

for $F=P,W,Q$. This covariance is why the complex plane is so useful for laser-like fields and linear optical networks.

Thermal light

A one-mode thermal state with mean occupation $\bar n>0$ has the positive Glauber–Sudarshan density

$$P_{\mathrm{th}}(\alpha) = \frac{1}{\pi\bar n} \exp\left( -\frac{\lvert\alpha\rvert^2}{\bar n} \right).$$

Its Wigner and Husimi functions are

$$W_{\mathrm{th}}(\alpha) = \frac{2}{ \pi(2\bar n+1) } \exp\left[ -\frac{ 2\lvert\alpha\rvert^2 }{ 2\bar n+1 } \right],$$

and

$$Q_{\mathrm{th}}(\alpha) = \frac{1}{ \pi(\bar n+1) } \exp\left[ -\frac{ \lvert\alpha\rvert^2 }{ \bar n+1 } \right].$$

All are rotationally symmetric because the state has no phase preference. All are nonnegative, and the positive $P$ shows that ideal thermal light is classical under the coherent-mixture criterion despite its super-Poissonian counting and bunching.

Number states

For the number state $\lvert n\rangle$,

$$W_n(\alpha) = \frac{2}{\pi} (-1)^n L_n\left( 4\lvert\alpha\rvert^2 \right) e^{-2\lvert\alpha\rvert^2},$$

where $L_n$ is a Laguerre polynomial. At the origin,

$$W_n(0) = \frac{2}{\pi}(-1)^n,$$

which is the parity rule. Every odd number state is maximally negative at the origin in this normalization.

The corresponding Husimi function is

$$Q_n(\alpha) = \frac{ e^{-\lvert\alpha\rvert^2} \lvert\alpha\rvert^{2n} }{ \pi n! }.$$

It is nonnegative and has a zero of order $2n$ at the origin for $n>0$. The $P$ representation is a finite combination of derivatives of a delta distribution and, for $n>0$, is not a nonnegative measure.

Squeezed Gaussian states

A squeezed vacuum has an elliptical, nonnegative Gaussian Wigner function. In the quadrature convention used here, its covariance eigenvalues are

$$V_- = \frac12e^{-2r}, \qquad V_+ = \frac12e^{2r}.$$

The Husimi function is a broader positive ellipse because it includes the coherent-state measurement noise. The Glauber–Sudarshan representation is not a nonnegative measure whenever one quadrature variance falls below the vacuum value. This example is the standard warning that Wigner positivity does not imply optical classicality.

Covariance formulas and multimode Gaussian transformations are developed in Gaussian States and Wigner Functions. Generation, calibration, and loss are treated in Squeezed Light.

Coherent superpositions

For an even or odd superposition

$$\lvert\psi_\pm\rangle \propto \lvert\alpha_0\rangle \pm \lvert-\alpha_0\rangle,$$

the Wigner function contains two positive lobes and an oscillatory interference term between them. Negative fringes encode the relative phase; the incoherent mixture of the two coherent states lacks those fringes.

The $Q$ function smooths the fringes and remains nonnegative. It can still distinguish the coherent superposition from the mixture when measured with enough precision, because positivity does not mean absence of phase information. The $P$ representation is highly singular and nonclassical.

How the Distributions Are Measured

A phase-space representation is not automatically a histogram of detector outcomes. The measurement map must be stated. Three standard routes have especially clean meanings.

Homodyne marginals and Wigner tomography

Balanced homodyne detection measures one rotated quadrature

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

selected by the local-oscillator phase. To write its marginal without hidden Jacobians, define the Wigner density normalized in the real quadratures by

$$W_{XP}(X,P) = \frac12 W\left( \frac{X+iP}{\sqrt2} \right).$$

Then

$$\int dX\,dP\,W_{XP}(X,P)=1.$$

With the rotated conjugate coordinate

$$P_\theta = -X\sin\theta + P\cos\theta,$$

the measured probability density is the line marginal

$$p_\theta(x)=\int_{-\infty}^{\infty}dy\, W_{XP}(x\cos\theta-y\sin\theta, x\sin\theta+y\cos\theta).$$

Here $y=P_\theta$; the arguments of $W_{XP}$ remain the original X,P coordinates. Every $p_\theta$ is nonnegative even when $W$ has negative regions. By collecting quadrature samples over $0\leq\theta<\pi$, one obtains the Radon projections needed to reconstruct the Wigner function or density matrix.

An inverse Radon transform is exact only for ideal continuous data. Real optical tomography must address:

• finite sample size and phase coverage;
• detector efficiency and electronic noise;
• local-oscillator mode matching;
• finite digitizer range and binning;
• filtering, regularization, or a finite reconstruction basis;
• positivity and normalization constraints on the inferred density operator.

Filtered back-projection can display the geometry directly, while maximum-likelihood or Bayesian state estimation can enforce a physical density operator. Different estimators can produce visibly different small features at finite data size. A negative patch is compelling only when it is stable against the stated uncertainty and reconstruction choices.

Homodyne and Heterodyne Detection derives the practical receiver, mode selection, vacuum calibration, and optical quadrature POVM. The open-systems Homodyne Detection page develops continuous stochastic records. The present page uses calibrated quadrature distributions as phase-space projections.

Heterodyne detection and Q sampling

The coherent-state POVM has elements

$$E(\alpha)\,d^2\alpha = \frac{d^2\alpha}{\pi} \lvert\alpha\rangle \langle\alpha\rvert.$$

Its outcome probability is

$$p(\alpha) = \operatorname{Tr} \left[ \rho E(\alpha) \right] =Q(\alpha).$$

Ideal heterodyne detection therefore samples $Q$ directly. Optical heterodyne receivers use a frequency-offset local oscillator; dual-homodyne receivers split the signal and measure orthogonal quadratures. In both pictures an unused port or image band introduces the vacuum noise required by simultaneous two-quadrature readout.

The raw histogram equals the ideal $Q$ function only after the complex gain, offset, phase convention, image-band response, and efficiency have been calibrated. Added amplifier noise in microwave receivers broadens the histogram beyond the ideal coherent-state POVM. Calling every IQ histogram a Husimi function without specifying that noise model is too strong.

Homodyne and Heterodyne Detection explains the frequency-offset receiver, image-band vacuum, calibration, and coherent-state POVM. The open-systems Heterodyne Detection develops the two-Wiener record and conditional-state description.

Displacement followed by parity

The displaced-parity identity provides a pointwise route to the Wigner function. Displace the state by $-\alpha$, resolve the photon-number probabilities $p_n(\alpha)$, and form

$$W(\alpha) = \frac{2}{\pi} \sum_{n=0}^{\infty} (-1)^n p_n(\alpha).$$

The procedure does not require an inverse Radon transform. It does require a calibrated displacement, photon-number resolution over enough of the distribution, and tight control of loss because parity is highly sensitive to missed photons.

Variants are used in cavity and circuit QED, trapped-ion motional-state measurements, and optical photon-counting experiments. The physical apparatus differs, but the mathematical observable is displaced parity.

Why P is rarely plotted from raw samples

The Glauber–Sudarshan representation is most naturally accessed through normally ordered moments and inequalities rather than pointwise sampling. Photocount correlations estimate quantities such as

$$\left\langle (\hat a^\dagger)^m \hat a^m \right\rangle,$$

which are ordinary moments of a positive $P$ if such a measure exists. Violations of inequalities obeyed by every classical random intensity then witness a nonclassical $P$.

In principle, complete tomographic data determine $P$. In practice, obtaining it from $W$ or $Q$ reverses Gaussian smoothing and can turn small high-frequency noise into enormous oscillations. A regularized surrogate must not be described as the unique raw $P$ distribution without its bandwidth or prior.

Loss, Efficiency, and Added Noise

Loss does more than lower a plotted amplitude. It mixes the selected mode with an environmental mode and changes the state.

Pure-loss channel

A pure-loss channel of transmissivity $\eta$ is modeled by

$$\hat a_{\mathrm{out}} = \sqrt\eta\, \hat a_{\mathrm{in}} + \sqrt{1-\eta}\, \hat v,$$

where $\hat v$ is an independent environmental vacuum mode. It transforms the first moment as

$$\langle\hat a\rangle_{\mathrm{out}} = \sqrt\eta\, \langle\hat a\rangle_{\mathrm{in}},$$

and a one-mode quadrature covariance matrix as

$$V_{\mathrm{out}} = \eta V_{\mathrm{in}} + (1-\eta) \frac{I_2}{2}.$$

For the ordered characteristic functions,

$$\chi_{s,\mathrm{out}}(\xi) = \chi_{s,\mathrm{in}} \left( \sqrt\eta\,\xi \right) \times \exp\left[ -\frac{ (1-s)(1-\eta) }{2} \lvert\xi\rvert^2 \right].$$

The $P$ characteristic function with $s=1$ receives no extra Gaussian factor; a positive coherent-state mixture remains a positive coherent-state mixture with contracted amplitudes. The Wigner and $Q$ functions acquire vacuum smoothing as well as contraction. Nonclassical structure can therefore become invisible after enough loss.

Example: one photon under loss

An incident one-photon state becomes

$$\rho_{\mathrm{out}} = \eta \lvert1\rangle\langle1\rvert + (1-\eta) \lvert0\rangle\langle0\rvert.$$

At the origin,

$$\begin{aligned} W_{\mathrm{out}}(0) &= \frac{2}{\pi} \left[ (1-\eta)-\eta \right] \\ &= \frac{2}{\pi} (1-2\eta). \end{aligned}$$

The measured state retains Wigner negativity at the origin only when

$$\eta>\frac12.$$

For $0<\eta\le1/2$ this output has an everywhere nonnegative Wigner function, yet its nonzero-mean factorial correlation remains $g^{(2)}=0$: the output itself is still optically nonclassical. More generally, the characteristic-function formula at half transmission gives

$$W_{\rm out}(\alpha)=2Q_{\rm in}(\sqrt2\alpha)\ge0 \qquad (\eta=1/2).$$

Any further pure loss preserves Wigner positivity. Thus transmission at or below one half removes Wigner negativity for every input state, without universally removing optical nonclassicality. At $\eta=0$ the output is vacuum and normalized photon correlations are undefined.

Detector correction versus measured state

There are two legitimate but different reporting choices:

1. reconstruct the effective state after propagation and the loss channel representing detector inefficiency;
2. infer an upstream source state by inverting a calibrated channel model.

The second can be scientifically useful, but it is model dependent and often ill-conditioned. The distinction requires labeling raw and corrected quantities, propagates efficiency uncertainty, states the regularization, and verifies the inferred density operator against independent observables.

An efficiency correction that merely sharpens a plot until negativity appears is not evidence by itself.

Mode mismatch is loss plus possible contamination

Homodyne and heterodyne receivers project the signal onto the local-oscillator mode. If the normalized signal mode is $f$ and the analyzed mode is $g$, the overlap

$$\mu = \langle g,f\rangle$$

sets an effective efficiency $\lvert\mu\rvert^2$ only when unmatched modes are equivalent to vacuum and are not otherwise detected. Background light, orthogonal populated modes, phase drift, or correlated technical noise can add contamination rather than pure vacuum loss.

The quoted phase-space distribution always belongs to the analyzed mode $g$. This is why mode declarations are part of the state definition, not merely an experimental footnote.

Multimode and Pulsed Fields

For $M$ declared modes, collect the amplitudes into

$$\boldsymbol\alpha = (\alpha_1,\ldots,\alpha_M)^{\mathsf T}.$$

A multimode distribution is normalized over

$$d^{2M}\boldsymbol\alpha = \prod_{j=1}^{M}d^2\alpha_j.$$

The coherent-state representation becomes

$$\rho = \int d^{2M}\boldsymbol\alpha\, P(\boldsymbol\alpha) \lvert\boldsymbol\alpha\rangle \langle\boldsymbol\alpha\rvert.$$

Passive linear optics applies a unitary mode transformation

$$\boldsymbol{\hat a}_{\mathrm{out}} = U \boldsymbol{\hat a}_{\mathrm{in}},$$

and rotates coherent amplitudes according to

$$\boldsymbol\alpha_{\mathrm{out}} = U \boldsymbol\alpha_{\mathrm{in}}.$$

The full phase-space distribution is transformed by the corresponding linear change of coordinates. A reduced one-mode distribution is obtained by integrating over all unobserved mode amplitudes. This marginalization can hide correlations, entanglement, and joint negativity.

Pulsed homodyne modes

A pulsed local oscillator defines a temporal supermode

$$\hat a_g = \int dt\, g^*(t) \hat a(t), \qquad \int dt\,\lvert g(t)\rvert^2=1.$$

The measured quadrature samples and reconstructed Wigner function refer to that operator. Changing the pulse shape $g(t)$ changes the mode and therefore the reduced state. Principal-component or mode-function analyses can identify the supermodes carrying the nonclassical structure, but the resulting basis must be reported.

Continuous-wave sidebands

A continuous-wave homodyne record analyzed near a radio-frequency offset can involve upper and lower optical sidebands jointly. Depending on the demodulation phase and filter, the effective observables may be quadratures of a symmetric or antisymmetric sideband supermode rather than of one isolated frequency mode.

An ellipse plotted from a noise spectrum is not automatically the Wigner function of a single carrier mode. The temporal filter, sideband pairing, and normalization determine which canonical mode is represented.

Classicality and Negativity

The implications among the three common positivity conditions are one-way:

$$\begin{gathered} \text{positive }P \Longrightarrow W\geq0, \\ W\geq0 \Longrightarrow Q\geq0. \end{gathered}$$

The final condition holds for every quantum state, so it is not a classicality test. Neither converse is valid.

Positive P is the optical classical set

If $P$ is a nonnegative measure, then

$$\rho = \int d^2\alpha\, P(\alpha) \lvert\alpha\rangle \langle\alpha\rvert$$

is a convex mixture of coherent states. Normally ordered moments then equal moments of an ordinary classical random complex amplitude. Classical inequalities for intensity fluctuations follow.

For example, every positive-$P$ one-mode state satisfies

$$\operatorname{Var}(X_\theta) \geq \frac12,$$

because each coherent component contributes vacuum variance $1/2$ and classical mixing can only add variance of the component means. Quadrature squeezing below vacuum therefore certifies failure of a positive Glauber–Sudarshan measure.

Likewise, under the stated single-mode and detector assumptions,

$$g^{(2)}(0)<1$$

or a sub-Poissonian Fano factor certifies nonclassicality. These witnesses can detect states whose Wigner functions remain nonnegative.

Wigner negativity is sufficient, not necessary

If $W$ is negative anywhere, the state cannot have a positive $P$, because Gaussian smoothing of a positive measure cannot create negative values. Wigner negativity is therefore a sufficient witness of optical nonclassicality.

It is not necessary. Squeezed Gaussian states and many entangled Gaussian states have nonnegative Wigner functions. For pure continuous-variable states, Hudson’s theorem makes the boundary sharp: under its regularity assumptions, a pure state has a nonnegative Wigner function only if it is Gaussian. Mixed non-Gaussian states can nevertheless have positive Wigner functions.

A common negativity volume is

$$\mathcal N_W = \frac12 \left[ \int d^2\alpha\, \lvert W(\alpha)\rvert -1 \right].$$

When the absolute integral is finite, its numerical value depends on the normalized state but not on a harmless linear rescaling of coordinates when the integration measure is transformed consistently. Its uncertainty can be strongly estimator dependent in finite tomography.

Q positivity is not classicality

Every density operator gives

$$Q(\alpha) = \frac{1}{\pi} \langle\alpha\rvert\rho\lvert\alpha\rangle \geq0.$$

The positivity reflects positivity of a POVM outcome probability. Number states, coherent superpositions, and squeezed states all have positive $Q$ functions. Nonclassical information remains encoded in their shapes, zeros, moments, and incompatibility with any positive-$P$ model, but not in negative $Q$ values.

Nonclassicality is not the same as entanglement

A single-mode squeezed state is nonclassical without being entangled across the declared one-mode tensor factor. Sending it and vacuum through a beam splitter can produce two-mode entanglement. Conversely, whether a multimode state is separable depends on the subsystem split, while positive-$P$ classicality asks whether it is a mixture of multimode coherent products in a specified mode basis.

The concepts are related through optical transformations but should not be used interchangeably.

Exercises

1. Coordinate measure and vacuum marginal

Using

$$\alpha=\frac{X+iP}{\sqrt2},$$

show that $d^2\alpha=dX\,dP/2$. Convert

$$W_0(\alpha) = \frac{2}{\pi} e^{-2\lvert\alpha\rvert^2}$$

to the normalized real-quadrature density and recover the vacuum probability density for $X$.

Solution

The real and imaginary parts are

$$\operatorname{Re}\alpha = \frac{X}{\sqrt2}, \qquad \operatorname{Im}\alpha = \frac{P}{\sqrt2}.$$

The Jacobian is therefore $1/2$:

$$d^2\alpha = d(\operatorname{Re}\alpha) d(\operatorname{Im}\alpha) = \frac12\,dX\,dP.$$

Since

$$\lvert\alpha\rvert^2 = \frac{X^2+P^2}{2},$$

the density normalized with respect to $dX\,dP$ is

$$\begin{aligned} W_{XP,0}(X,P) &= \frac12W_0(\alpha) \\ &= \frac{1}{\pi} e^{-(X^2+P^2)}. \end{aligned}$$

Integrating over $P$ gives

$$\begin{aligned} p_0(X) &= \int_{-\infty}^{\infty} dP\,W_{XP,0}(X,P) \\ &= \frac{e^{-X^2}}{\pi} \int_{-\infty}^{\infty} dP\,e^{-P^2} \\ &= \frac{1}{\sqrt\pi} e^{-X^2}. \end{aligned}$$

This Gaussian has variance $1/2$, as required by the chosen quadrature normalization.

2. A rotated Gaussian marginal

Let $W_{XP}$ be a centered Gaussian with covariance $V=\operatorname{diag}(1/8,2)$. Use the rotated-coordinate marginal to find $p_{\pi/4}(x)$. Why would substituting $W_{XP}(x,y)$ without rotating its arguments fail?

Solution

The measured coordinate is $x=(X+P)/\sqrt2$, so its variance is $(1/8+2)/2=17/16$. Consequently

$$p_{\pi/4}(x)=\frac{e^{-x^2/[2(17/16)]}}{\sqrt{2\pi(17/16)}}.$$

Integrating $W_{XP}(x,y)$ with unrotated arguments would give variance $1/8$, the X marginal at phase zero. Naming the integration variable $P_\theta$ does not rotate the function itself.

3. Mean occupation in all three representations

For a coherent state $\lvert\alpha_0\rangle$, evaluate the phase-space formulas for $\langle\hat n\rangle$ using $P$, $W$, and $Q$. Explain why the three different symbols give one answer.

Solution

The positive delta measure gives immediately

$$\int d^2\alpha\, P(\alpha) \lvert\alpha\rvert^2 = \lvert\alpha_0\rvert^2.$$

The coherent-state Wigner Gaussian has

$$\int d^2\alpha\, W(\alpha) \lvert\alpha\rvert^2 = \lvert\alpha_0\rvert^2 + \frac12.$$

Therefore its symmetric symbol gives

$$\int d^2\alpha\, W(\alpha) \left( \lvert\alpha\rvert^2-\frac12 \right) = \lvert\alpha_0\rvert^2.$$

Similarly, the $Q$ Gaussian has

$$\int d^2\alpha\, Q(\alpha) \lvert\alpha\rvert^2 = \lvert\alpha_0\rvert^2+1,$$

so

$$\int d^2\alpha\, Q(\alpha) \left( \lvert\alpha\rvert^2-1 \right) = \lvert\alpha_0\rvert^2.$$

The offsets are the vacuum corrections associated with normal, symmetric, and antinormal ordering. Each distribution must be paired with its matching operator symbol.

4. A zero in a nonnegative distribution

Show that a classical state with a normalized nonnegative P measure has $Q(\alpha)>0$ at every finite $\alpha$. What does an exact zero of Q imply?

Solution

The convolution kernel $\pi^{-1}e^{-|\alpha-\beta|^2}$ is strictly positive for every finite pair. Its integral against a probability measure is strictly positive. Thus an exact zero rules out every classical P measure even though Q is never negative. For example, a nonvacuum number state has $Q_n(0)=0$. Experimentally, an empty finite histogram bin is not an exact zero of the underlying density; uncertainty and detector smoothing still matter.

5. Parity at the origin

Use displaced parity to show that

$$W_n(0) = \frac{2}{\pi}(-1)^n$$

for a number state. Why does this make the origin especially useful for odd states?

Solution

At $\alpha=0$, no displacement is applied. Since

$$\hat\Pi\lvert n\rangle = (-1)^n\lvert n\rangle,$$

the displaced-parity identity gives

$$\begin{aligned} W_n(0) &= \frac{2}{\pi} \langle n\rvert \hat\Pi \lvert n\rangle \\ &= \frac{2}{\pi}(-1)^n. \end{aligned}$$

Every odd number state reaches the minimum allowed one-mode value $-2/\pi$ at the origin. A calibrated parity measurement there is therefore a direct negativity test, although loss can rapidly reduce the contrast.

6. Positive P forbids quadrature squeezing

Prove that every state with a nonnegative Glauber–Sudarshan measure satisfies

$$\operatorname{Var}(X_\theta)\geq\frac12.$$

Solution

For a coherent component $\lvert\alpha\rangle$,

$$\operatorname{Var}_\alpha(X_\theta) = \frac12.$$

Its mean is the real classical quantity

$$x_\theta(\alpha) = \sqrt2 \operatorname{Re} \left( \alpha e^{-i\theta} \right).$$

For a positive mixture $P(\alpha)$, the law of total variance gives

$$\begin{aligned} \operatorname{Var}(X_\theta) &= \int d^2\alpha\, P(\alpha) \operatorname{Var}_\alpha(X_\theta) + \operatorname{Var}_P \left[ x_\theta(\alpha) \right] \\ &= \frac12 + \operatorname{Var}_P \left[ x_\theta(\alpha) \right] \geq \frac12. \end{aligned}$$

Therefore any verified variance below vacuum rules out every nonnegative coherent-state mixture, even though the squeezed state’s Wigner function can remain positive.

7. Pair moments need a different symbol

Find the P, Wigner, and Q symbols of $n(n-1)$ in terms of $z=|\alpha|^2$. Check them for vacuum.

Solution

The normal symbol is $z^2$. Symmetrizing four ladder operators gives $\langle z^2\rangle_W=\langle n^2+n+1/2\rangle$, while antinormal ordering gives $\langle z^2\rangle_Q=\langle(n+1)(n+2)\rangle$. Using the first-moment offsets,

$$f_P=z^2,\qquad f_W=z^2-2z+\frac12,\qquad f_Q=z^2-4z+2.$$

Vacuum has $(\langle z\rangle,\langle z^2\rangle)=(1/2,1/2)$ in Wigner and $(1,2)$ in Q, giving zero in both cases. Squaring the symbol of n would give a wrong answer: operator multiplication is not ordinary multiplication of these symbols.

References

• K. Banaszek and K. Wódkiewicz, “Direct Probing of Quantum Phase Space by Photon Counting,” Physical Review Letters 76, 4344–4347 (1996), doi:10.1103/PhysRevLett.76.4344.
• K. E. Cahill and R. J. Glauber, “Density Operators and Quasiprobability Distributions,” Physical Review 177, 1882–1902 (1969), doi:10.1103/PhysRev.177.1882.
• K. E. Cahill and R. J. Glauber, “Ordered Expansions in Boson Amplitude Operators,” Physical Review 177, 1857–1881 (1969), doi:10.1103/PhysRev.177.1857.
• R. J. Glauber, “Coherent and Incoherent States of the Radiation Field,” Physical Review 131, 2766–2788 (1963), doi:10.1103/PhysRev.131.2766.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution Functions in Physics: Fundamentals,” Physics Reports 106, 121–167 (1984), doi:10.1016/0370-1573(84)90160-1.
• R. L. Hudson, “When Is the Wigner Quasi-Probability Density Non-Negative?” Reports on Mathematical Physics 6, 249–252 (1974), doi:10.1016/0034-4877(74)90007-X.
• K. Husimi, “Some Formal Properties of the Density Matrix,” Proceedings of the Physico-Mathematical Society of Japan, 3rd Series 22, 264–314 (1940), doi:10.11429/ppmsj1919.22.4_264.
• U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press (1997).
• A. I. Lvovsky and M. G. Raymer, “Continuous-Variable Optical Quantum-State Tomography,” Reviews of Modern Physics 81, 299–332 (2009), doi:10.1103/RevModPhys.81.299.
• W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001), doi:10.1002/3527602976.
• D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner Distribution and the Density Matrix of a Light Mode Using Optical Homodyne Tomography: Application to Squeezed States and the Vacuum,” Physical Review Letters 70, 1244–1247 (1993), doi:10.1103/PhysRevLett.70.1244.
• E. C. G. Sudarshan, “Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams,” Physical Review Letters 10, 277–279 (1963), doi:10.1103/PhysRevLett.10.277.
• K. Vogel and H. Risken, “Determination of Quasiprobability Distributions in Terms of Probability Distributions for the Rotated Quadrature Phase,” Physical Review A 40, 2847–2849 (1989), doi:10.1103/PhysRevA.40.2847.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer (2008), doi:10.1007/978-3-540-28574-8.
• E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Physical Review 40, 749–759 (1932), doi:10.1103/PhysRev.40.749.