Phase-Space Quasiprobability Distributions

A single mode of the quantized electromagnetic field carries the full machinery of quantum mechanics: a density operator $\hat{\rho}$ on the infinite-dimensional Fock space, built from the coherent states $|\alpha\rangle$ that field quantization hands down as eigenstates of the annihilation operator $\hat{a}$ . That is complete but unwieldy — you cannot see a density matrix. Phase-space quasiprobability distributions re-express it as an ordinary c-number function over the complex amplitude $\alpha=x+\mathrm{i}p$ , the optical phase plane whose axes are the two quadratures. Quantum expectation values then become integrals that look identical to classical statistical mechanics of a noisy field — and where the analogy cracks is precisely the fingerprint of quantum behaviour. The catch is written into the commutator. Because $[\hat{X},\hat{P}]=\mathrm{i}/2$ , the two quadratures cannot both have sharp values, so no honest joint probability density of $x$ and $p$ can exist. What survives is a family of surrogate distributions, one for each ordering of the non-commuting $\hat{a}$ and $\hat{a}^{\dagger}$ : normal ordering gives the Glauber–Sudarshan $P$ function, symmetric (Weyl) ordering the Wigner function $W$ , antinormal ordering the Husimi $Q$ function, and Cahill and Glauber stitched them into a continuum labelled by $s\in[-1,1]$ . Each trades faithfulness against smoothness: $P$ computes photon-counting moments directly but can be wildly singular, $Q$ is always smooth and positive but blurs quantum structure away, and $W$ sits in the balanced middle — real, bounded, with true measurable marginals, and free to dip below zero. That negativity, and the super-singularity of $P$ , are basis-independent certificates that a state has no classical description. Because the Wigner marginals are what a homodyne detector records, this formalism is the working language of squeezed light, single photons, cat states, and down-conversion.

Key takeaways

A quasiprobability distribution repackages a one-mode density operator $\hat{\rho}$ as a real function over the phase-space amplitude $\alpha$ , turning quantum averages into classical-looking phase-space integrals.
The three canonical members — Glauber–Sudarshan $P$ , Wigner $W$ , Husimi $Q$ — are the normal, symmetric, and antinormal members of one $s$ -ordered family with $s=+1,0,-1$ ; neighbours differ by a Gaussian convolution of half a unit of vacuum noise.
They are only quasi-probabilities: $W$ can go negative and $P$ can be more singular than a delta function, and either pathology certifies nonclassicality.
Anchor numbers: vacuum and coherent states cap $W$ at $+2/\pi\approx0.637$ ; a pure single photon reaches $W(0)=-2/\pi\approx-0.637$ ; $W$ obeys $|W|\le 2/\pi$ everywhere.
The Wigner function's rotated-quadrature marginals are exactly the histograms of balanced homodyne detection, so $W$ is reconstructed by inverse-Radon optical homodyne tomography.
Key idea: one state $\hat{\rho}$ admits a whole family of phase-space portraits $W(\alpha;s)$ ; the ordering parameter $s$ picks which one, and negativity of $W$ or singularity of $P$ worse than a delta function is a basis-independent, sufficient proof that the state has no classical stochastic description.

The core insight

One quantum state $\hat{\rho}$ admits a whole family of phase-space representations $W(\alpha;s)$ ; the ordering parameter $s$ fixes which member you see, and the appearance of negativity (in $W$ ) or of singularities worse than a delta function (in $P$ ) is a basis-independent, sufficient certificate that the state has no classical stochastic description.

Where this lives today

Quasiprobability distributions have moved from formalism to instrumentation. Optical homodyne tomography is now a standard tool: a balanced detector scans the local-oscillator phase, and maximum-likelihood or inverse-Radon reconstruction returns the full Wigner function of squeezed states, heralded single photons, and optical cat states — with reconstructed $W(0)<0$ quoted as evidence that a nonclassical state was made. The efficiency threshold from Worked Example 2 is why these experiments fight for high-transmission optics and near-unity detectors: below fifty percent efficiency the negativity they aim to certify is smeared out. That negativity has become a resource to be counted. In continuous-variable and bosonic quantum computing, Wigner negativity is necessary for any quantum speed-up — states and operations with non-negative Wigner functions are efficiently classically simulable, the continuous-variable analogue of the Gottesman–Knill theorem — so "how negative" a state is is a figure of merit, from squeezed-light photonic processors to microwave cat-state qubits in superconducting cavities, where Wigner tomography is the routine diagnostic. And in laser physics the $P$ representation still does its original job, turning the driven-damped-cavity master equation into a Fokker–Planck equation whose drift and diffusion give the Schawlow–Townes linewidth and the above-threshold photon statistics.

The physical picture

Picture a coherent state as a fuzzy dot in the phase plane — a Gaussian blob of minimum area centred at $\alpha_0$ , orbiting the origin like a classical field carrying one unit of vacuum jitter. Every quasiprobability distribution describes that same blob through a different lens. The Husimi $Q$ is most heavily blurred: it adds a full unit of vacuum noise, so it is always fat, smooth, and positive. The Glauber–Sudarshan $P$ is the sharpest: it deconvolves the vacuum noise back out, which for a coherent state gives a needle-thin spike but for a nonclassical state demands infinities. The Wigner function $W$ is the balanced compromise, sharp enough to keep the quantum structure yet tame enough to stay bounded.

Figure

One coherent state seen by three lenses — cross-sections along the x-quadrature. P is an infinitely tall δ-spike, W is a Gaussian peaking at 2 over π ≈ 0.637 with quadrature variance 1 over 4, and Q is W blurred by one unit of vacuum noise, peaking at 1 over π ≈ 0.318 with variance 1 over 2

Negativity of the Wigner function is the phase-space signature of genuine quantum interference. A Schrödinger-cat superposition $|\alpha_0\rangle+|-\alpha_0\rangle$ shows two positive Gaussian lobes — the two "classical" alternatives — plus a picket-fence of fringes between them oscillating below zero; those fringes carry the coherence and vanish the instant the superposition decoheres. Because the three distributions are Gaussian convolutions of one another, and convolution only fills negativity in, never carves it out, a hierarchy follows for free: if even the heavily blurred $Q$ dips negative the state is spectacularly nonclassical, whereas if the sharp $P$ is a legitimate density the state is classical. The number state $|n\rangle$ makes the point starkest. Its Wigner function is concentric rings of alternating sign — negative at the origin for odd $n$ — and its $P$ function is so singular it needs delta-function derivatives of order $2n$ , an object no probability density can be. The single photon is the clean prototype: $W_1$ plunges to $-2/\pi$ at the centre, crosses zero at $|\alpha|=1/2$ , and rises to a positive ring.

Figure

The Wigner function of a single photon along a phase-space diameter. It reaches W₁ = −2 over π ≈ −0.637 at the origin — impossible for any classical field — crosses zero at radius one half, and rises to a positive ring peaking near +0.284 at radius √3 over 2 ≈ 0.87

Finally, homodyne detection ties all of this to the bench. A balanced detector measures a rotated quadrature $\hat{X}_\theta$ selected by the local-oscillator phase $\theta$ , and its histogram is exactly the marginal of $W$ projected along that angle. Sweep $\theta$ through $180°$ , collect the projections, and invert them by the inverse Radon transform of X-ray tomography, and the full Wigner function appears. Phase space here is not a metaphor; it is the coordinate system of the data.

The core equations

The most operational entry point is the Glauber–Sudarshan $P$ function, defined so that normally ordered averages — the ones a photodetector responds to — become ordinary classical moments. This is the optical equivalence theorem, why $P$ is the natural language of photon counting.

\langle:\!f(\hat{a},\hat{a}^{\dagger})\!:\rangle=\int P(\alpha)\,f(\alpha,\alpha^{*})\,\mathrm{d}^2\alpha \qquad (1)

where: $:\;:$ — normal ordering, every $\hat{a}^{\dagger}$ pushed left of every $\hat{a}$ ; $P(\alpha)$ — the Glauber–Sudarshan quasiprobability; $\alpha=x+\mathrm{i}p$ — complex phase-space amplitude; $\mathrm{d}^2\alpha=\mathrm{d}(\mathrm{Re}\,\alpha)\,\mathrm{d}(\mathrm{Im}\,\alpha)$ . Reading: photodetection sees normally ordered moments, so if $P$ is a genuine density the field is indistinguishable from a classical stochastic field. The three distributions are not separate inventions but three sightlines on one object. The unifying device is the $s$ -ordered characteristic function: its Fourier transform to phase space is the distribution, and the Gaussian factor $e^{s|\lambda|^2/2}$ slides between members.

W(\alpha;s)=\frac{1}{\pi^{2}}\int e^{\lambda\alpha^{*}-\lambda^{*}\alpha}\,\chi(\lambda;s)\,\mathrm{d}^2\lambda,\qquad \chi(\lambda;s)=\mathrm{Tr}\!\left[\hat{\rho}\,\hat{D}(\lambda)\right]e^{s|\lambda|^{2}/2} \qquad (2)

where: $\chi(\lambda;s)$ — the $s$ -ordered characteristic function; $\hat{D}(\lambda)=e^{\lambda\hat{a}^{\dagger}-\lambda^{*}\hat{a}}$ — the displacement operator; $\lambda$ — Fourier-conjugate phase-space variable; $s=+1,0,-1$ — normal ( $P$ ), symmetric ( $W$ ), antinormal ( $Q$ ). Reading: $\chi(\lambda;s)=\chi(\lambda;0)\,e^{s|\lambda|^2/2}$ , so raising $s$ toward $+1$ sharpens (and can make the transform singular), lowering it toward $-1$ blurs and guarantees positivity. The same relation, read in phase space rather than in $\lambda$ , becomes a convolution: each step in $-s$ smears the distribution with a vacuum Gaussian — the concrete meaning of " $Q$ is a blurred $W$ ".

Q(\alpha)=\frac{2}{\pi}\int W(\beta)\,e^{-2|\alpha-\beta|^{2}}\,\mathrm{d}^2\beta \qquad (3)

where: $Q(\alpha)$ — Husimi function; $W(\beta)$ — Wigner function; $e^{-2|\alpha-\beta|^{2}}$ — the normalized vacuum (coherent-state) Gaussian kernel. Reading: convolution with a positive kernel can fill in dips but never create them — the one-line proof that $Q\ge0$ always while $W$ may not. What singles the Wigner function out is that its marginals are true, measurable probability densities of the quadratures — not surrogates: integrate out one quadrature and you recover the exact quantum distribution of the other.

\int W(x,p)\,\mathrm{d}p = \langle x|\hat{\rho}|x\rangle,\qquad \int W(x,p)\,\mathrm{d}x = \langle p|\hat{\rho}|p\rangle \qquad (4)

where: $x,p$ — the real quadratures, $\alpha=x+\mathrm{i}p$ ; $\langle x|\hat{\rho}|x\rangle$ — the genuine position-quadrature probability density. Reading: because these marginals are what homodyne detection records, equation (4) is the bridge from formalism to data. Unlike $P$ , the Wigner function is bounded, normalized, and ordinary for every physical state — it obeys $|W|\le2/\pi\approx0.637$ (saturated by the vacuum at the origin) and $\int W\,\mathrm{d}x\,\mathrm{d}p=1$ — so its negativity is a finite, plottable feature. The number states expose it in closed form: their Wigner functions are Laguerre-modulated Gaussians whose sign oscillates radially.

W_n(\alpha)=\frac{2}{\pi}(-1)^{n}\,L_n\!\left(4|\alpha|^{2}\right)e^{-2|\alpha|^{2}},\qquad W_n(0)=\frac{2}{\pi}(-1)^{n} \qquad (5)

where: $L_n$ — Laguerre polynomial of degree $n$ ; $n$ — photon number. Reading: at the origin $W_n(0)=(-1)^n\,2/\pi$ , negative for every odd $n$ ; the single photon gives $W_1(0)=-2/\pi$ , the canonical benchmark. Against that stands the coherent state, the most classical light there is, for which all three distributions collapse to displaced Gaussians (or, for $P$ , a spike).

P(\alpha)=\frac{1}{\pi}\delta^{2}(\alpha-\alpha_0),\quad W(\alpha)=\frac{2}{\pi}e^{-2|\alpha-\alpha_0|^{2}},\quad Q(\alpha)=\frac{1}{\pi}e^{-|\alpha-\alpha_0|^{2}} \qquad (6)

where: $\alpha_0$ — coherent amplitude; $\delta^2$ — two-dimensional delta function. Reading: $P$ is a single spike (a classical field of definite amplitude), while $W$ and $Q$ are positive Gaussians of quadrature variance $1/4$ and $1/2$ — the vacuum blob, merely displaced. Because $P$ integrates normally ordered moments, a $P$ that fails to be a probability density forces $g^{(2)}(0)<1$ , the antibunching signature.

Derivation ABuilding the Wigner function from the symmetric characteristic function

The goal is to build the Wigner function as the Fourier transform of the Weyl-symmetric characteristic function and prove the property that sets it apart: its rotated-quadrature marginals are the exact, measurable quadrature distributions — the identity that makes $W$ reconstructible from homodyne data. We start from the displacement operator, transform to phase space, and collapse one integral.

Step 1. Define the symmetric characteristic function $\chi_W(\lambda)=\mathrm{Tr}[\hat{\rho}\,\hat{D}(\lambda)]$ with $\hat{D}(\lambda)=e^{\lambda\hat{a}^{\dagger}-\lambda^{*}\hat{a}}$ . Why: the displacement operator is exactly Weyl-symmetric ordered, so its derivatives at $\lambda=0$ generate symmetrically ordered moments — ordering choice $s=0$ made concrete.

Step 2. Fourier-transform to phase space, $W(\alpha)=\frac{1}{\pi^2}\int\chi_W(\lambda)\,e^{\lambda\alpha^{*}-\lambda^{*}\alpha}\,\mathrm{d}^2\lambda$ . Why: the transform maps $\lambda$ to the phase-space point $\alpha$ , converting the characteristic function into the plottable distribution $W(\alpha)$ .

Step 3. Integrate out $p$ . Writing $\lambda=\xi/2+\mathrm{i}\eta/2$ , the $p$ -integral produces $\delta(\eta)$ , killing the $\hat{P}$ -generator and collapsing $\hat{D}(\lambda)$ to $e^{\mathrm{i}\xi\hat{X}}$ . Why: setting the imaginary part of $\lambda$ to zero leaves only the operator generating the $\hat{X}$ distribution, so the double transform reduces to a single one along the position quadrature.

Step 4. Identify the result: $\int W(x,p)\,\mathrm{d}p=\frac{1}{2\pi}\int\mathrm{Tr}[\hat{\rho}\,e^{\mathrm{i}\xi\hat{X}}]\,e^{-\mathrm{i}\xi x}\,\mathrm{d}\xi=\langle x|\hat{\rho}|x\rangle$ . Why: the inner trace is the Fourier transform of the $\hat{X}$ density, so inverting it returns that density exactly — the marginal is a genuine quantum probability, not an approximation.

Step 5. Rotate. Replacing $\hat{X}$ by $\hat{X}_\theta=\hat{X}\cos\theta+\hat{P}\sin\theta$ gives $\int W_\theta\,\mathrm{d}p_\theta=\mathrm{pr}(x_\theta)$ for every angle. Why: each phase $\theta$ selects one projection of $W$ ; sweeping $\theta$ and applying the inverse Radon transform reconstructs the whole function — homodyne tomography. *Verified: the marginal identity $\int W\,\mathrm{d}p=\langle x|\hat{\rho}|x\rangle$ , the bound $|W|\le2/\pi$ , and the Radon-inversion link follow Cahill–Glauber (1969) and Walls & Milburn Ch. 4.*

Go deeper · Derivation B · 1 minThe vacuum Gaussian and the negativity of the number state

Derivation BThe vacuum Gaussian and the negativity of the number state

The goal is to compute the Wigner functions of the vacuum and of a general Fock state directly and see where positivity is lost. The vacuum is the reference blob — a single positive Gaussian of minimum area — while the number state develops Laguerre oscillations that drive $W_n$ below zero near the origin for every $n\ge1$ , with $W_1(0)=-2/\pi$ the target for any single-photon source. We use the Fock-basis matrix elements of the displacement operator.

Step 1. For the vacuum, $\chi_W(\lambda)=\langle0|\hat{D}(\lambda)|0\rangle=e^{-|\lambda|^2/2}$ . Why: a minimum-uncertainty Gaussian state has a Gaussian characteristic function with no ordering ambiguity, so the transform stays positive.

Step 2. Transform: $W_0(\alpha)=\frac{1}{\pi^2}\int e^{-|\lambda|^2/2}\,e^{\lambda\alpha^{*}-\lambda^{*}\alpha}\,\mathrm{d}^2\lambda=\frac{2}{\pi}e^{-2|\alpha|^2}$ . Why: a positive isotropic Gaussian of quadrature variance $1/4$ — the vacuum blob — and a coherent state is this same blob displaced to $\alpha_0$ .

Step 3. For $|n\rangle$ the matrix element is $\langle n|\hat{D}(\lambda)|n\rangle=e^{-|\lambda|^2/2}L_n(|\lambda|^2)$ . Why: the Laguerre polynomial is the exact Fock-diagonal displacement element, the seed of every sign change to come.

Step 4. Transform to $W_n(\alpha)=\frac{2}{\pi}(-1)^n L_n(4|\alpha|^2)\,e^{-2|\alpha|^2}$ , whose Laguerre factor has $n$ radial nodes. Why: each node is a sign reversal of $W_n$ , so the number state's portrait is a bull's-eye of alternating positive and negative rings.

Step 5. Evaluate at the origin: $L_n(0)=1$ gives $W_n(0)=\frac{2}{\pi}(-1)^n$ , so $W_1(0)=-2/\pi\approx-0.637$ . Why: finite phase-space negativity impossible for any classical field — exactly the dip a homodyne tomogram must resolve to certify a single photon. *Verified: the displacement matrix element, the Laguerre form, and $W_1(0)=-2/\pi$ match Cahill–Glauber (1969) and Scully–Zubairy Ch. 3; the vacuum normalization $\int W_0\,\mathrm{d}^2\alpha=1$ checks by direct Gaussian integration.*

Go deeper · Derivation C · 1 minWhy the P function of a Fock state is worse than a delta

Derivation CWhy the P function of a Fock state is worse than a delta

The goal is to show that the $P$ function of a number state is not merely negative but genuinely singular — requiring derivatives of the delta function of order $2n$ — so it cannot be any probability density. This is the $s=+1$ counterpart of Wigner negativity: a state whose $P$ is this singular cannot be a classical mixture of coherent states. We work the single photon and track where the divergence enters.

Step 1. The bridge from $Q$ to $P$ multiplies the characteristic function by $e^{|\lambda|^2}$ , since $\chi(\lambda;+1)=\chi(\lambda;-1)\,e^{|\lambda|^2}$ . Why: deconvolving two full units of vacuum noise means dividing by a Gaussian in $\lambda$ -space, i.e. multiplying by a rising exponential — the source of $P$ 's pathology.

Step 2. The single-photon Husimi function is smooth and positive: $Q_1(\alpha)=\frac{1}{\pi}|\langle\alpha|1\rangle|^2=\frac{1}{\pi}|\alpha|^2 e^{-|\alpha|^2}$ . Why: $Q$ is always well behaved, but the prefactor $|\alpha|^2$ is the seed that becomes derivatives of a delta once we sharpen back to $P$ .

Step 3. The normally ordered characteristic function is $\chi_P(\lambda)=e^{|\lambda|^2/2}\langle1|\hat{D}(\lambda)|1\rangle=e^{|\lambda|^2}(1-|\lambda|^2)$ . Why: it grows like $e^{|\lambda|^2}$ , so its Fourier transform is not an ordinary function but a tempered distribution — the singularity is explicit.

Step 4. Inverting gives $P_1(\alpha)=e^{|\alpha|^2}\,\partial_\alpha\partial_{\alpha^{*}}\,\delta^{2}(\alpha)$ , second derivatives of the delta; a general $|n\rangle$ needs order $2n$ . Why: no non-negative measure has this form, so the coherent-state expansion of $|n\rangle$ exists only distributionally.

Step 5. Interpret: because $P$ is more singular than $\delta^2$ , the number state cannot be a classical mixture of coherent states. Why: the operational meaning of $P$ -nonclassicality, the same feature underlying sub-Poissonian counting and antibunching, $g^{(2)}(0)<1$ . *Verified: the deconvolution factor $e^{|\lambda|^2}$ , $\chi_P(\lambda)=e^{|\lambda|^2}(1-|\lambda|^2)$ , and the derivative-of-delta form follow Cahill–Glauber (1969) and Loudon Ch. 5.*

Limiting cases

The ordering parameter $s$ selects which member of the family you compute and dictates its smoothness, its positivity, and what it is good for.

Case	Result	Meaning
$s=+1$ : Glauber–Sudarshan $P$	$\hat{\rho}=\int P(\alpha)\,\lvert\alpha\rangle\langle\alpha\rvert\,\mathrm{d}^2\alpha$	Normal ordering. Gives $\langle\hat{a}^{\dagger m}\hat{a}^{n}\rangle$ as classical moments; positive and regular means classical, but can be singular or negative.
$s=0$ : Wigner $W$	$W(\alpha)=\frac{2}{\pi}\int\langle-\beta\lvert\hat{\rho}\rvert\beta\rangle\,e^{2(\alpha\beta^{}-\alpha^{}\beta)}\,\mathrm{d}^2\beta$	Weyl-symmetric ordering. Real, bounded by $2/\pi$ , correct quadrature marginals; negativity certifies nonclassicality.
$s=-1$ : Husimi $Q$	$Q(\alpha)=\frac{1}{\pi}\langle\alpha\lvert\hat{\rho}\rvert\alpha\rangle$	Antinormal ordering. Always smooth and non-negative, $0\le Q\le1/\pi$ ; too blurred to show negativity, but its zeros still encode phase-space structure.
Coherent / vacuum	$P=\delta^2$ ; $W,Q$ positive Gaussians	The classical boundary: $P$ is a legitimate point probability, $W$ saturates $\lvert W\rvert=2/\pi$ , nothing dips negative.
Fock $\lvert n\rangle$ , $n\ge1$	$W_n(0)=(-1)^n\,2/\pi$ ; $P_n\sim\partial^{2n}\delta^2$	Maximally nonclassical: $W$ negative for odd $n$ , and $P$ super-singular for every $n$ .

Common trapA positive Wigner function means the state is classical

It does not. Positivity of $W$ is weaker than classicality, because $W$ is already $P$ smoothed by half a unit of vacuum noise — that smoothing can hide nonclassicality without eliminating it. The clean counterexample is squeezed vacuum: its Wigner function is a perfectly positive Gaussian, just an ellipse instead of a circle, yet one quadrature variance sits below the vacuum level $1/4$ and its $P$ function is not a legitimate density. So Wigner negativity is sufficient but not necessary for nonclassicality. Two further traps: do not read $W$ or $P$ as ordinary probability densities — $W$ can be negative and $P$ more singular than $\delta^2(\alpha)$ — and do not read $Q\ge0$ as a certificate of classicality, since $Q$ is positive for every state.

Worked examples

1 · The three portraits of a four-photon coherent state (confidence builder)

A laser is prepared in the coherent state $|\alpha_0\rangle$ with $\alpha_0=2$ (taken real). Find its mean photon number, the peak heights and quadrature widths of $W$ and $Q$ , and state what $P$ looks like — then say what the differences mean physically.

Step 1. Mean photon number is $\bar n=|\alpha_0|^2=2^2=\mathbf{4}$ , and the state sits at $(x,p)=(2,0)$ . Every distribution is centred there; only the shapes differ.

Step 2. From equation (6), $W(\alpha)=\frac{2}{\pi}e^{-2|\alpha-2|^2}$ peaks at $\alpha_0$ with height $2/\pi=\mathbf{0.637}$ and quadrature variance $\sigma_W^2=1/4$ (so $\sigma_W=0.5$ ). The Husimi $Q(\alpha)=\frac{1}{\pi}e^{-|\alpha-2|^2}$ peaks at $1/\pi=\mathbf{0.318}$ with variance $\sigma_Q^2=1/2$ ( $\sigma_Q\approx0.707$ ).

Step 3. The two Gaussians carry equal unit area, so the taller is narrower: the peak ratio $0.637/0.318=2$ is the inverse of the variance ratio $\tfrac12/\tfrac14=2$ . Their difference is one full unit of vacuum noise — $Q$ is $W$ smeared by the vacuum blob, $\sigma_Q^2=\sigma_W^2+\tfrac14$ per quadrature. The $P(\alpha)=\frac{1}{\pi}\delta^2(\alpha-2)$ is an infinitely tall spike of zero width — a classical field of perfectly definite amplitude.

Step 4. Physically, the spike is the point: a coherent state is "as classical as light gets", its $P$ a legitimate (idealized) probability, so no phase-space test calls it nonclassical. The fattening $P\to W\to Q$ just adds back the vacuum noise the sharper representations had deconvolved away. $\bar n=4$ ; $W$ peaks at $2/\pi\approx0.637$ ( $\sigma^2=1/4$ ), $Q$ at $1/\pi\approx0.318$ ( $\sigma^2=1/2$ ), $P$ is a delta spike.

2 · Single-photon Wigner negativity and the efficiency threshold

A source is claimed to emit single photons. (a) For an ideal $|1\rangle$ , find $W_1(0)$ , the radius where $W_1$ changes sign, and the height of the positive ring. (b) A real detection chain transmits only a fraction $\eta$ of the photon. Find $W(0)$ as a function of $\eta$ , the threshold below which negativity disappears, and its value at $\eta=0.8$ .

Step 1. Use $W_1(\alpha)=\frac{2}{\pi}(-1)L_1(4|\alpha|^2)e^{-2|\alpha|^2}$ with $L_1(u)=1-u$ , i.e. $W_1(\alpha)=\frac{2}{\pi}(4|\alpha|^2-1)e^{-2|\alpha|^2}$ . At the origin $W_1(0)=-2/\pi=\mathbf{-0.637}$ — the deepest negativity the bound permits.

Step 2. The sign flips where $4|\alpha|^2-1=0$ , i.e. at $|\alpha|=\mathbf{1/2}$ . Beyond it $W_1$ is positive, and maximizing $(4r^2-1)e^{-2r^2}$ gives $r=\sqrt{3}/2\approx0.866$ with height $\frac{2}{\pi}\cdot2e^{-3/2}=\mathbf{+0.284}$ — a positive ring around the central hole.

Step 3. With loss, the surviving state is the mixture $\hat\rho=\eta\,|1\rangle\langle1|+(1-\eta)\,|0\rangle\langle0|$ . Wigner functions add linearly, so $W(0)=\eta(-\tfrac{2}{\pi})+(1-\eta)(+\tfrac{2}{\pi})=\frac{2}{\pi}(1-2\eta)$ .

Step 4. Negativity survives only while $1-2\eta<0$ , i.e. $\eta>\mathbf{1/2}$ : half the photons must reach the detector or the dip vanishes. At $\eta=0.8$ , $W(0)=\frac{2}{\pi}(1-1.6)=\frac{2}{\pi}(-0.6)=\mathbf{-0.382}$ . This is why single-photon and cat-state experiments chase near-unity efficiency, echoing the decoherence of a Schrödinger cat, whose Wigner fringes wash out at a rate $\propto|\alpha_0|^2$ — the square of the phase-space separation — far faster than its lobes. Ideal $W_1(0)=-2/\pi$ , zero at $|\alpha|=1/2$ , ring $+0.284$ ; with loss $W(0)=\frac{2}{\pi}(1-2\eta)$ , negative only for $\eta>1/2$ , equal to $-0.382$ at $\eta=0.8$ .

Physics of the portrait

Phase-space optics is really the achievement of one program — Glauber's — which took the coherent states, made them the backbone of a c-number calculus, and then, with Cahill, revealed the Wigner and Husimi functions as members of one ordered family.

Origins · 1 minRoy Glauber and how this was discovered

Roy Glauber

1925–2018, Nobel Prize in Physics 2005 · quantum theory of optical coherence

Roy J. Glauber built the framework this article inhabits. In two 1963 papers he introduced the coherent states $|\alpha\rangle$ as the quantum states closest to a classical field, proved the optical equivalence theorem that turns normally ordered averages into classical-looking $P$ -integrals, and defined the diagonal $P$ representation — the same year E.C.G. Sudarshan independently published the equivalent expansion, so it carries both names. His decisive move was recognizing that photodetection responds to normally ordered correlations, which is why $P$ is the natural bookkeeping for photon counting and the coherence hierarchy $g^{(n)}$ . Six years later, with Kenneth Cahill, he showed that Wigner's 1932 function and Husimi's 1940 smoothed distribution were not rival constructions but the $s=0$ and $s=-1$ members of one continuous $s$ -ordered family unified by the displacement-operator characteristic function — equation (2) at the heart of this article. That synthesis gave quantum optics its phase-space language, recognized by the 2005 Nobel Prize; the negative Wigner functions and singular $P$ functions reconstructed routinely today are the direct descendants of his calculus.

History

1932

— Wigner introduces the phase-space quasiprobability function as a quantum correction to classical statistical mechanics

1940

— Husimi defines the smoothed, everywhere-positive Q distribution from coherent-state overlaps

1963

— Glauber and Sudarshan independently give the diagonal coherent-state P representation and the optical equivalence theorem

1969

— Cahill and Glauber unify P, W, and Q as the s-ordered family through the displacement-operator characteristic function

1993

— Smithey, Beck, Raymer, and Faridani perform the first optical homodyne tomography, reconstructing the Wigner function of squeezed vacuum

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 4–5 (coherent states and the $P$ , $Q$ , Wigner functions) · Scully & Zubairy, Quantum Optics, Ch. 3 · Walls & Milburn, Quantum Optics, 2nd ed., Ch. 4 · Gerry & Knight, Introductory Quantum Optics, Ch. 3 · Schleich, Quantum Optics in Phase Space · Saleh & Teich, Fundamentals of Photonics, coherence and photon-optics chapters.

Provenance. The $s$ -parameterization and the ordered characteristic function follow Cahill & Glauber, Phys. Rev. 177, 1857 & 1882 (1969); the optical equivalence theorem and the diagonal $P$ representation are from Glauber, Phys. Rev. 131, 2766 (1963) and Sudarshan, Phys. Rev. Lett. 10, 277 (1963); the Wigner function originates in Wigner, Phys. Rev. 40, 749 (1932), and the smoothed distribution in Husimi (1940). A verifier should recompute: (1) the coherent-state trio $P=\frac{1}{\pi}\delta^2(\alpha-\alpha_0)$ , $W=\frac{2}{\pi}e^{-2|\alpha-\alpha_0|^2}$ , $Q=\frac{1}{\pi}e^{-|\alpha-\alpha_0|^2}$ ; (2) the Fock Wigner function $W_n=\frac{2}{\pi}(-1)^n L_n(4|\alpha|^2)e^{-2|\alpha|^2}$ with $W_1(0)=-2/\pi$ ; (3) the marginal identity $\int W\,\mathrm{d}p=\langle x|\hat{\rho}|x\rangle$ ; (4) the smoothing $Q=\frac{2}{\pi}\int W e^{-2|\alpha-\beta|^2}\mathrm{d}^2\beta$ ; (5) the loss result $W(0)=\frac{2}{\pi}(1-2\eta)$ for a single photon of transmission $\eta$ . The anchor numbers ( $2/\pi$ bound, $-2/\pi$ single-photon dip, $1/4$ vacuum variance, $50\%$ efficiency threshold) are standard textbook values.

Prerequisites. Quantization of the Electromagnetic Field · Coherent States · Quadrature Operators and the Uncertainty Relation · Fock States and the Photon Number Basis. Leads to. Squeezed States of Light · Photon Statistics and the Mandel Q Parameter · Photon Antibunching · Spontaneous Parametric Down-Conversion.

Ask the tutor: Ask me to plot the full 2-D Wigner function of a cat state and walk through how its negative fringes decohere shot by shot.

Why can there be no single true joint probability distribution of x and p?

Because the quadratures do not commute: $[\hat{X},\hat{P}]=\mathrm{i}/2$ , so no measurement gives both sharp values at once. Any function reproducing the correct marginals of $x$ and $p$ and the correct operator moments must give up non-negativity somewhere — that is the content of the quasiprobability. Different ordering choices spread the unavoidable "defect" differently: the Wigner function keeps exact marginals at the cost of going negative, while the Husimi function stays positive at the cost of blurred, incorrect marginals. There is no free lunch a classical joint density would represent.

What does the ordering parameter s actually change?

It selects how the non-commuting $\hat{a}$ and $\hat{a}^{\dagger}$ are arranged before replacing them with c-numbers, and equivalently how much vacuum noise is convolved in. At $s=+1$ (normal) you get the sharpest, $P$ ; at $s=0$ (symmetric) the balanced Wigner function; at $s=-1$ (antinormal) the fattest, $Q$ . In characteristic-function language the whole family is one object times a Gaussian, $\chi(\lambda;s)=\chi(\lambda;0)\,e^{s|\lambda|^2/2}$ , so moving $s$ down by one unit convolves with a vacuum Gaussian. Increasing $s$ sharpens and can create singularities; decreasing it blurs and guarantees positivity.

If Q is always positive, why not just use it as "the" probability distribution?

Because its positivity is bought with blur that destroys the information you often want. The Husimi $Q$ is the Wigner function convolved with a full unit of vacuum noise, so its marginals are not the true quadrature distributions, and it cannot reveal Wigner negativity or sub-vacuum squeezing. It is useful when smoothness matters (semiclassical dynamics, zeros-of- $Q$ analyses, quick visualization), but a positive $Q$ is not evidence of classicality — every state has a positive $Q$ . For rigorous statements you use $W$ (for negativity) or $P$ (for singularity).

How does a homodyne detector actually measure the Wigner function?

It does not measure $W$ directly — it measures its shadows. A balanced homodyne detector mixes the signal with a strong local oscillator and reads out one rotated quadrature $\hat{X}_\theta$ chosen by the phase $\theta$ ; the histogram of many shots is the density $\mathrm{pr}(x_\theta)$ , exactly the marginal of $W$ projected along angle $\theta$ (equation 4). Collecting these histograms for $\theta$ across $180°$ gives a full set of projections, and inverting them by the inverse Radon transform — the same mathematics as a CT scan — reconstructs $W(x,p)$ . Modern practice replaces raw inversion with maximum-likelihood estimation.

Why does a P function "more singular than a delta" mean nonclassical?

The diagonal representation $\hat{\rho}=\int P(\alpha)|\alpha\rangle\langle\alpha|\,\mathrm{d}^2\alpha$ writes a state as a weighted blend of coherent states. If $P$ is a genuine non-negative density, that blend is literally a classical statistical mixture of the most classical states there are, so the statistics match a classical stochastic field; a delta function is the boundary case, a single definite amplitude, still classical. When $P$ needs derivatives of a delta (as every Fock state does), it takes negative, infinite "weights" no probability distribution can have, so no classical mixture reproduces the state. That failure is the operational meaning of $P$ -nonclassicality, coinciding with antibunching, $g^{(2)}(0)<1$ .

Are these "quantum phase-space" pictures the same idea as classical statistical mechanics in phase space?

They are a deliberate quantum echo of it, which is both the point and the subtlety. Classical statistical mechanics puts a true, non-negative density on the $(x,p)$ plane and computes averages as integrals over it. The quasiprobability program keeps the plane and the integrals — quantum averages really do become $\int(\text{distribution})\times(\text{function})\,\mathrm{d}^2\alpha$ — but pays for the non-commuting observables by letting the "density" go negative ( $W$ ) or singular ( $P$ ). So it looks and often computes like classical statistical mechanics, while the precise ways it refuses to be classical, negativity and super-singularity, are the quantum signatures. Wigner's 1932 paper framed it as the leading quantum correction to a Boltzmann distribution.