Quantum State Tomography of Light

Homodyne tomography is now standard laboratory instrumentation, and its most visible payoff is characterizing squeezed light for gravitational-wave detection: the squeezed vacuum injected into Advanced LIGO and Advanced Virgo to push noise below the shot-noise limit is validated by the very noise-ellipse reconstruction Smithey and Raymer first demonstrated. The same technique benchmarks single-photon and Fock-state sources — the depth of the reconstructed Wigner dip, after honest efficiency correction, is the accepted figure of merit for source purity — and verifies the two-mode entanglement of continuous-variable resources for teleportation and quantum key distribution. Where the light is not a propagating beam, its phase-space cousin takes over. In circuit and cavity QED the displaced-parity method of equation (6) reigns: a superconducting qubit or Rydberg atom displaces the field by a controlled $\alpha$ and reads its photon-number parity through a Ramsey sequence, mapping $W(\alpha)$ point by point with no inversion. This is how the celebrated Schrödinger-cat states of microwave fields were imaged, their fringes and negativities laid bare, and how error-correctable bosonic qubits are now diagnosed. Across both platforms the deliverable is what Fano asked for in 1957 — a complete, physical estimate of the state — with Wigner negativity as the cleanest, assumption-light certificate that the light is genuinely quantum.

Picture the Wigner function as a low relief over the phase plane $(x,p)$ — a mound for a coherent state, a tilted ellipse for squeezed light, and, for genuinely quantum states, a surface that dips below zero where no classical probability could. A homodyne measurement at phase $\theta$ does not sample a point of this relief; it records the shadow the relief casts onto a screen tilted at angle $\theta$ — the marginal $\mathrm{pr}(x,\theta)=\int W\,\mathrm{d}p_\theta$, the integral of $W$ along the line perpendicular to $\theta$. Rotate the local oscillator and you illuminate the same object from a new side; a 2D shape is fixed once you know its shadows from every direction. This is not a loose analogy — it is the identical mathematics. The local-oscillator phase is the CT gantry angle, the quadrature histogram the X-ray projection, and the inverse Radon transform the same kernel a hospital scanner runs.

Why does no single-shot trick collapse the ensemble into one measurement? Because complementarity is not negotiable: $\hat x$ and $\hat p$ cannot be co-measured on one copy, so each pulse yields one number along one direction. Tomography respects this exactly — it extracts one coordinate from each of very many photons and lets statistics do the rest. The price is copies; the payoff is total, since the reconstructed $W$, equivalently every density-matrix element $\rho_{mn}$, is the complete description of the mode. The reconstruction earns its cost by showing what nothing classical can. A Gaussian state — coherent, thermal, squeezed — has a strictly positive Wigner function whose every projection is a Gaussian, so its tomogram is a smooth blob fixed by a mean and covariance. Nonclassical states betray themselves: a single photon carves a hole through the center of phase space, a cat state stripes it with fringes, and their projections dip and wiggle in ways no product of Gaussians can imitate. The inversion resolves these into rings and troughs of negative Wigner density — the fingerprint the whole apparatus is built to expose, because no mixture of classical fields can fake it.

The starting point is the observable homodyne delivers. Rotating the local-oscillator phase $\theta$ rotates the measured quadrature, so one homodyne channel accesses any linear combination of the two canonical quadratures — the tomographic ray along direction $\theta$, whose eigenvalue each shot records.

where: $\hat x,\hat p$ — canonical quadratures with $[\hat x,\hat p]=i$ in the $\hbar\to1$ convention (vacuum variance $\tfrac12$); $\theta$ — local-oscillator phase, the projection angle; $\hat a,\hat a^\dagger$ — mode ladder operators. As $\theta$ runs over $[0,\pi)$, $\hat x_\theta$ sweeps every direction in the phase plane; $\theta$ and $\theta+\pi$ give the same axis with $x\to-x$, so a half-circle suffices. The central identity of the subject is that the probability density of homodyne outcomes at phase $\theta$ is a literal geometric projection of the Wigner function. It equals the diagonal element of $\hat\rho$ in the $\hat x_\theta$ eigenbasis, which is $W$ integrated along the conjugate direction — the Radon projection a CT scanner measures.

where: $\mathrm{pr}(x,\theta)$ — probability density of outcome $x$ at phase $\theta$; $|x_\theta\rangle$ — eigenstate of $\hat x_\theta$; $W$ — Wigner function. The identity holds because the projector $|x_\theta\rangle\langle x_\theta|$ has Weyl symbol $\delta(x'\cos\theta+p'\sin\theta-x)$, a delta ridge along the rotated axis, so tracing it against $\hat\rho$ integrates $W$ over the perpendicular line — the defining operation of the Radon transform. Because the marginals are Radon projections, the state is recovered by the inversion that reconstructs a CT slice: filtered back-projection. The ramp filter $|\xi|$ corrects the low-frequency oversampling of back-projected rays, and the angular integral stitches the filtered projections into the plane.

where: $\xi$ — Fourier variable conjugate to $x'$, with $|\xi|$ the ramp filter; $\theta\in[0,\pi)$ — the projection angles. Equation (3) is exact for noiseless, complete data, but the ramp filter amplifies high-frequency statistical noise — why raw inversion is treacherous on finite samples and why the field moved to constrained estimators. Often one wants the density matrix in the Fock basis directly. Pattern-function tomography does this in one linear pass over the same histograms: each element $\rho_{mn}$ is a weighted integral of the data against a precomputed kernel, sidestepping any image.

where: $\rho_{mn}=\langle m|\hat\rho|n\rangle$ — density-matrix element in the number basis; $f_{mn}(x,\theta)=e^{i(m-n)\theta}\,g_{mn}(x)$ — pattern functions built from the regular and irregular oscillator eigenfunctions. The factor $e^{i(m-n)\theta}$ shows the coherence $\rho_{mn}$ is read off the $(m-n)$-th Fourier component of the data's phase dependence. No detector is perfect, and loss is the dominant systematic. Modelling the overall efficiency $\eta$ as a beam splitter admixing vacuum, the state actually sampled is the true Wigner function blurred by a Gaussian that widens as $\eta$ drops — loss only smooths toward classicality, never sharpens.

where: $\eta$ — overall efficiency (photodiode quantum efficiency $\times$ mode overlap $\times$ transmission); $W_\eta$ — Wigner function sampled by the lossy detector; $\sigma^2$ — total added variance, $(1-\eta)/(2\eta)$ per quadrature. When this reaches a full vacuum unit $\tfrac12$ — at $\eta=\tfrac12$ — the convolution erases all negativity, leaving a Husimi-$Q$-like non-negative blob. Homodyne is not the only route. When the mode lives in a microwave cavity, one samples $W$ point by point with no inversion, using that $W(\alpha)$ is the expectation of photon-number parity after displacing the state by $-\alpha$.

where: $\hat D(\alpha)=e^{\alpha\hat a^\dagger-\alpha^*\hat a}$ — displacement operator; $\hat\Pi=(-1)^{\hat n}$ — parity operator; $\alpha$ — phase-space point being read. Royer's identity makes $W(\alpha)$ the parity expectation in the displaced frame, so displacing the field and measuring $P(n)$ yields $W(\alpha)=\tfrac{2}{\pi}\sum_n(-1)^nP(n)$ — the basis of circuit-QED cat-state imaging. Finally, since neither raw inversion nor a linear pattern-function sum guarantees a physical state, serious reconstruction is constrained statistical estimation. The maximum-likelihood estimate is the physical density matrix under which the data are most probable, reached by a fixed-point map that never leaves the physical set.

where: $(x_j,\theta_j)$ — the $j$-th quadrature value and its phase; $\hat\rho_{\mathrm{ML}}$ — the physical density matrix most likely to have produced the dataset. It is found by the iteration $\hat\rho^{(k+1)}=\mathcal N[\hat R\hat\rho^{(k)}\hat R]$ with $\hat R=\sum_j\hat\Pi_j/\mathrm{pr}(x_j,\theta_j|\hat\rho^{(k)})$; $\mathcal N$ restores unit trace each step, and the POVM elements $\hat\Pi_j$ can absorb the efficiency model so the estimate is loss-corrected.

Whether a reconstruction is trustworthy, and whether it can reveal nonclassicality at all, is governed by three knobs: the detection efficiency $\eta$, the number of copies $N$, and the phase-space extent of the state relative to the Hilbert-space cutoff.

1 · How deep is the single-photon dip, and how much loss can it survive? (confidence builder)

A heralded single photon has Wigner function $W(x,p)=\tfrac1\pi\big(2(x^2+p^2)-1\big)e^{-(x^2+p^2)}$. Find its value at the phase-space origin, then find the measured central value after passing through a detector of efficiency $\eta$, and determine the efficiency below which the negativity is no longer observable.

Step 1. Evaluate at the origin: $x=p=0$ gives $W(0,0)=\tfrac1\pi(0-1)=-\tfrac1\pi$. Numerically $W(0,0)=-1/\pi=\mathbf{-0.318}$, a deep hole at the center of phase space — the classically forbidden signature the whole measurement exists to see.

Step 2. Model loss. A single photon through a transmissivity-$\eta$ splitter becomes $\hat\rho_\eta=(1-\eta)\,|0\rangle\langle0|+\eta\,|1\rangle\langle1|$: lost with probability $1-\eta$ (to vacuum), surviving with probability $\eta$. Since $W$ is linear in $\hat\rho$, and $W_{|0\rangle}(0,0)=+\tfrac1\pi$, $W_{|1\rangle}(0,0)=-\tfrac1\pi$.

Step 3. Combine: $W_\eta(0,0)=(1-\eta)\tfrac1\pi-\eta\tfrac1\pi=\dfrac{1-2\eta}{\pi}$. At $\eta=0.6$, $(1-1.2)/\pi=\mathbf{-0.064}$ — still negative, still certifiable. At $\eta=0.45$, $(1-0.9)/\pi=+\mathbf{0.032}$ — positive, the hole filled in. The sign flips exactly at $\eta=\tfrac12$.

Step 4. Conclusion: negativity is observable only for $\mathbf{\eta>1/2}$, in exact agreement with the general loss-convolution threshold. The pure dip is $-1/\pi\approx-0.318$; at $\eta=0.6$ it reads $-0.064$; below $\eta=1/2$ it vanishes.

2 · Reading a squeezed-vacuum tomogram in decibels

A homodyne tomogram of squeezed vacuum returns a minimum quadrature variance $V_{\min}=0.125$ (shot noise is $\tfrac12$). Express the squeezing in decibels, find the minimum possible anti-squeezed variance for a pure state, and say what a measured $V_{\max}=3.0$ would imply.

Step 1. Squeezing in dB compares the variance to the vacuum floor: $S=10\log_{10}(V_{\text{vac}}/V_{\min})=10\log_{10}(0.5/0.125)=10\log_{10}4=\mathbf{6.02\ dB}$ of noise reduction below shot noise at the squeezed phase.

Step 2. The uncertainty product $V_{\min}V_{\max}\ge\tfrac14$ (from $[\hat x,\hat p]=i$) forces $V_{\max}\ge0.25/0.125=\mathbf{2.0}$ for a minimum-uncertainty (pure) state, i.e. exactly $10\log_{10}(2.0/0.5)=6.02$ dB of anti-squeezing — a symmetric $6.02$ dB up and down.

Step 3. A measured $V_{\max}=3.0$ gives $V_{\min}V_{\max}=0.125\times3.0=0.375>\tfrac14$, so the state is not minimum uncertainty — the excess $0.375-0.25=0.125$ is added technical or thermal noise, and the tomogram is flagging an impure source. (For scale, a realistic budget of photodiode $0.99\times$ visibility$^2\,(0.98)^2\times$ transmission $0.95$ gives $\eta\approx0.90>\tfrac12$, so negativity stays certifiable.) 6.02 dB squeezed; a pure state needs $V_{\max}=2.0$; $V_{\max}=3.0$ signals excess noise.

Optical homodyne tomography is where a piece of medical-imaging mathematics, a quantum-optics detector, and a decades-old question about measuring a quantum state converge into a single working camera for light.

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 6 (homodyne detection and quadratures) · Leonhardt, Measuring the Quantum State of Light (the dedicated monograph on optical tomography) · Saleh and Teich, Fundamentals of Photonics, 3rd ed. (quantum optics chapters) · Foot, Atomic Physics, Ch. 7 (coherent states and field measurement) · Demtröder, Laser Spectroscopy, Vol. 1 (homodyne and heterodyne detection). Papers. Vogel and Risken, Phys. Rev. A 40, 2847 (1989) · Smithey, Beck, Raymer and Faridani, Phys. Rev. Lett. 70, 1244 (1993) · Lvovsky and Raymer, Rev. Mod. Phys. 81, 299 (2009).

Provenance. All physics uses the $\hbar\to1$, $[\hat x,\hat p]=i$ convention (vacuum quadrature variance $\tfrac12$); rescale for $[\hat x,\hat p]=i/2$. A verifier should recompute: (1) $\hat n_-=\sqrt2\,|\alpha_L|\,\hat x_\theta$ (Loudon Ch. 6; Saleh and Teich); (2) $\langle\Delta\hat x_\theta^2\rangle=\tfrac12$ from $[\hat x,\hat p]=i$; (3) the marginal/Radon identity $\mathrm{pr}(x,\theta)=\int W\,\mathrm{d}p_\theta$ and its inverse (Vogel and Risken 1989); (4) $W(x,p)=\pi^{-1}(2(x^2+p^2)-1)e^{-(x^2+p^2)}$ with $W(0,0)=-1/\pi\approx-0.318$, normalization to unity, and lossy value $W_\eta(0,0)=(1-2\eta)/\pi$; (5) the loss convolution and the $\eta>\tfrac12$ threshold (Lvovsky and Raymer, RMP 81, 299, 2009); (6) Royer's displaced-parity formula. No non-standard numerical values are asserted.

Prerequisites. Quadrature Operators and the Uncertainty Relation · Phase-Space Quasiprobability Distributions · Coherent States · Fock States and the Photon-Number Basis. Leads to. Squeezed States of Light · Single-Photon Sources · Photon Statistics and the Mandel Q Parameter.

Ask the tutor: Ask me to walk through reconstructing a specific state — squeezed vacuum, a heralded single photon, or a cat — from its quadrature histograms, and we will trace every projection into the final Wigner image.