Q037 · Rebuild a light state from measurements

Homodyne Quantum Tomography

Can you draw a quantum state from detector noise? Yes; here the noise is the measurement. Each phase of the reference beam, the local oscillator, samples one quadrature of the light, which is one projection of the state. Many projections together rebuild its Wigner function, a picture of the state in phase space. To trust the picture, check that it predicts measurements at a phase you left out.

Turn the local oscillator phase to measure the light along different directions, then rebuild the Wigner function from those readings alone. Check the result at a phase the reconstruction never used.

Recorded sample $\text{—}$

Measurement result $\text{—}$

01 / Turn noise into a picture

One measurement direction shows only a projection.

Scan the local oscillator. Reconstruct from the recorded projections, then test an unused phase. The picture stays empty until you collect data.

Signal and strong local oscillator meet at a balanced splitter. Two photodiodes feed a difference amplifier. Paths are guides, not photon trajectories.

Local oscillator phase

\theta/{}^\circ

33.75

Drag to choose a quadrature. Scanning records 24 phases; a probe reading never enters reconstruction.

Ready to record.

Make a comparison

Prepared statePhoton survival

\eta

0.9

Squeezing strength

r

0.65

State orientation

\varphi/{}^\circ

25

Coherent amplitude

|\alpha|

1.2

Sampling seed

s_0

37

Save before changing the prepared state or loss. Turning the local oscillator keeps the record.

The model has one optical mode, an ideal strong local oscillator and balanced detection, with a vacuum variance of one half. Loss happens before the detector. Each reading is a simulated, normalized difference signal. Filtered backprojection from a finite set of angles can add ripples and false negative regions, and it does not force the result to be a physical state.

02 / What did the detectors say?

A projection you can test.

Reveal ideal model

$q_\theta$ · 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

\text{—}

\text{—}

Latest records

03 / Reconstruct from the scan

Let the data draw the state.

Collect projections first.

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

\text{—}

24 recorded projections

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

Reconstruction bandwidth

k_c

4

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 / Save the evidence

Save a result, then change one thing

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

$q_\theta=(ae^{-i\theta}+a^\dagger e^{i\theta})/\sqrt2,\quad [q,p]=i$

$V_\theta=\tfrac12[1-\eta+\eta(e^{-2r}\cos^2(\theta-\varphi)+e^{2r}\sin^2(\theta-\varphi))]$

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

From this model to the evidence

Can the reconstruction predict a new projection?

Try: Scan 24 local-oscillator phases and reconstruct. Choose a phase between the scan settings, then collect 500 fresh probe readings.
Read: Compare the new histogram with the projection that the reconstruction predicts at that phase. The probe readings are never used in the reconstruction.
Scope: The article reconstructs polarization qubits; this Lab measures a continuous-variable field, so the two complement each other. The reconstruction uses filtered back-projection, not maximum likelihood. It recovers the state as detected, with loss left in. A negative patch on its own does not count as statistical certification.

q_\theta=\frac{ae^{-i\theta}+a^\dagger e^{i\theta}}{\sqrt{2}}

Model reference: Lvovsky & Raymer · Optical tomography, §§ II–III

Reconstructing a photon state from calibrated counts

Start with the essentials

Focus question: Can you draw a quantum state from detector noise?
One-sentence intuition: A reconstruction is trustworthy only if it predicts a measurement it was never shown.

Core mathematical model

A rotated quadrature

q_\theta=q\cos\theta+p\sin\theta,\quad [q,p]=i

The phase of a strong local oscillator picks the direction being measured. For ideal vacuum the variance is one half.

A projection of the Wigner function

p_\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

\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 keeps spatial frequencies up to a chosen cutoff. It uses only the scanned readings and knows nothing about which state was prepared.

Common difficulties

What a plot alone cannot prove

Typical misconceptionA negative pixel in the reconstruction proves the state is nonclassical.

Better mental modelA hard cutoff, a limited set of angles and finite data can all create negative ripples. Repeat the scan and vary the cutoff. The Lab gives no confidence level for nonclassicality.

Settings do not change saved data

Typical misconceptionMoving a slider improves data that was already recorded.

Better mental modelChanging the prepared state starts a new sample. Saved samples keep their original settings and raw readings.

Run the experiment

01
Predict
Try vacuum and squeezed vacuum. Predict how the width of the quadrature distribution changes as the local oscillator turns.
What to observe: Leave the optional model reference turned off for now.
02
Collect and save
Scan all 24 phases and reconstruct. Then record 500 readings at the selected phase, which the scan did not use.
What to observe: The histogram at the new phase is never used in the reconstruction.
03
Find where it fails
Choose one photon, reconstruct, then add loss. In a separate test, change the reconstruction cutoff without taking new data.
What to observe: Loss changes the state itself. The cutoff changes only how the state is estimated, and the limited number of readings adds sampling noise.

Homodyne Quantum Tomography

A projection you can test.

Let the data draw the state.

Save a result, then change one thing

From this model to the evidence