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Sandbox Physics

Q037 · Measure / reconstruct / test

Homodyne Quantum Tomography

Turn the local oscillator, record a different projection, then reconstruct the Wigner map from those readings. Test it at a phase the reconstruction has never seen.

Interactive modelHomodyne Quantum Tomography
Recorded sample—\text{—}
Measurement result—\text{—}

02 / WHAT DID THE DETECTORS SAY?

A projection you can test.

qθq_\theta · Blue: probe records at this phase. Orange: projection of your reconstruction. Dashed: optional ideal model. Filtered predictions may ring below zero; they are not sampling probabilities. Histogram tails remain in CSV.

HELD-OUT READINGS

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Latest records

03 / RECONSTRUCT FROM THE SCAN

Let the data draw the state.

Collect projections first.

(q,p)∈[−5,5]2(q,p)\in[-5,5]^2 · Horizontal: position quadrature. Vertical: momentum quadrature. Blue positive, red negative; common scale across samples.

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24 recorded projections

θ:0→172.5∘,qθ:−5→5\theta:0\to172.5^\circ,\quad q_\theta:-5\to5 · Phase increases rightward; quadrature increases upward. Each column uses 300 new preparations per scan.

Higher bandwidth resolves finer structure but amplifies noise. Change it and reconstruct the same data. Finite bandwidth blurs; finite angles and finite counts create ripples. Negative pixels alone are not certified nonclassicality.

04 / KEEP THE EVIDENCE

Save a result. Make one change.

Eight frozen samples. Restore for inspection; further acquisition starts a new run. Every saved raw record is exported once. Reusing a seed reproduces draws, not independent evidence.

Physical model, limits & primary sources

qθ=(ae−iθ+a†eiθ)/2,[q,p]=iq_\theta=(ae^{-i\theta}+a^\dagger e^{i\theta})/\sqrt2,\quad [q,p]=i

Vθ=12[1−η+η(e−2rcos⁡2(θ−φ)+e2rsin⁡2(θ−φ))]V_\theta=\tfrac12[1-\eta+\eta(e^{-2r}\cos^2(\theta-\varphi)+e^{2r}\sin^2(\theta-\varphi))]

Ideal vacuum calibration fixes the scale. The recorded number is the normalized photocurrent difference, not separate simulated diode counts. The local oscillator is taken infinitely strong; electronics noise, saturation, finite local oscillator and mode mismatch are omitted. Loss is not inverted: reconstruction describes the detected state.

A 61 by 61 grid, 24 equally spaced phases and a hard Fourier ramp cutoff. Filtered projections use raw samples with 0.04 interpolation spacing. Maps and held-out projections are neither clipped positive nor renormalized. No negative-event probabilities occur.

Lvovsky & Raymer · II A–B, III A 1, V A

Physics tutorial

Homodyne Quantum Tomography

BackgroundTurn the local oscillator, record a different projection, then reconstruct the Wigner map from those readings. Test it at a phase the reconstruction has never seen.

Why it mattersA quantum state is inferred from many preparations, never photographed in one shot.

Start with the essentials

Focus question
Can you draw a quantum state from detector noise?
One-sentence intuition
The same reconstruction must predict a measurement it was not given.

Core mathematical model

A rotated quadrature

qθ=qcos⁡θ+psin⁡θ,[q,p]=iq_\theta=q\cos\theta+p\sin\theta,\quad [q,p]=i

A strong local oscillator selects a direction. The ideal vacuum variance is one half.

A projection of the Wigner function

pθ(x)=∫W(xcos⁡θ−tsin⁡θ,xsin⁡θ+tcos⁡θ) dtp_\theta(x)=\int W(x\cos\theta-t\sin\theta,x\sin\theta+t\cos\theta)\,dt

Each projection is a nonnegative probability density even when the Wigner function is negative.

Finite-bandwidth inverse Radon transform

W^(q,p)=12πM∑j=1M1nj∑ℓ=1njKkc(qcos⁡θj+psin⁡θj−xjℓ)\widehat W(q,p)=\frac{1}{2\pi M}\sum_{j=1}^M\frac{1}{n_j}\sum_{\ell=1}^{n_j}K_{k_c}(q\cos\theta_j+p\sin\theta_j-x_{j\ell})

The kernel integrates spatial frequency up to a declared cutoff. It consumes scan records only, with no knowledge of the prepared state.

Common difficulties

A plot is not a certificate

Typical misconceptionA negative reconstructed pixel proves a nonclassical state.

Better mental modelHard cutoff, angular sampling and finite data can produce negative ripples. Repeat and vary the cutoff; no nonclassicality confidence certificate is supplied.

Model versus records

Typical misconceptionChanging a slider improves data already recorded.

Better mental modelPreparation changes start a new sample. Frozen samples preserve their original settings and raw readings.

Run the experiment

  1. 01

    Predict

    Try vacuum and squeezed vacuum. Predict how the quadrature width changes as the local oscillator turns.

    What to observe: Keep the optional model reference hidden.
  2. 02

    Collect and save

    Scan all 24 phases, reconstruct, then record 500 readings at the selected unused phase.

    What to observe: The validation histogram never enters the reconstruction.
  3. 03

    Find the failure

    Choose one photon, reconstruct, then introduce loss. Separately change the reconstruction cutoff without acquiring new data.

    What to observe: Optical loss changes the state. Cutoff changes the estimator. Finite counts add sampling noise.