Reconstructing a photon state from calibrated counts

The matrix is inferred from repeated preparations

Open James, Kwiat, Munro and White’s 2001 paper, On the Measurement of Qubits, arXiv:quant-ph/0103121. It develops two routes from polarization counts to a two-qubit density matrix, the array that predicts the probabilities of different quantum measurements. Start with the sixteen analyzer settings, then compare the linear reconstruction with the constrained statistical fit. Each setting uses different photon pairs, assumed to come from the same repeated preparation.

The paper develops a scheme with sixteen settings, a linear reconstruction and a constrained statistical fit. A general normalized two-qubit state has fifteen independent real parameters, and the measurement scheme must also fix the overall count scale. Sixteen counts settings, each of which needs many detector clicks; calibration is an additional requirement.

Read the evidence: D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White (2001), On the Measurement of Qubits — author manuscript, arXiv:quant-ph/0103121 D. F. V. James et al. (2001), Measurement of qubits, PRA 64, 052312 — author-hosted journal copy

Each count needs a model of the measurement

For each setting, identify the measurement operator, which predicts the detector response to a given state, the exposure scale and the additive background. In the teaching equation, rho is the unknown density matrix, E the known measurement operator, s the count scale, and b the mean background; their subscripts j identify the setting. Tr denotes the matrix trace. The Born rule predicts the mean count, written mu with the same subscript. An inferred state is meaningful only together with this response model.

If the detector’s efficiencies are unequal but the analysis assumes an ideal projector, the response error ends up in the inferred state. Source drift between settings can mimic coherence or imbalance. Before reading a matrix plot, keep the raw counts, exposure times, background, wave-plate calibration and the assumption that every preparation is the same.

\mu_j=s_j\operatorname{Tr}(\rho E_j)+b_j,\qquad \rho\succeq0,\quad\operatorname{Tr}\rho=1

Read the evidence: D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White (2001), On the Measurement of Qubits — author manuscript, arXiv:quant-ph/0103121 Kwiat Quantum Information Group, Quantum state tomography — experimental and reconstruction resources

Noise can push a linear inversion outside the physical states

In our single-qubit example, the mean outcomes along three orthogonal spin axes are estimated as 0.80, 0.80 and zero. Each value on its own is a valid average of a two-valued outcome, but together they make a vector of length about 1.131. The matching Hermitian matrix with trace one has eigenvalues of about 1.066 and minus 0.066.

Frequencies that each look plausible can still invert to an unphysical joint estimate. A constrained method has to compromise among the measurements. Shrinking the vector to unit length gives one physical projection. Maximum likelihood instead compares how well each candidate state accounts for all the observed counts, and generally gives a different answer.

Read the evidence: D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White (2001), On the Measurement of Qubits — author manuscript, arXiv:quant-ph/0103121

Built-in positivity does not check the calibration

The paper’s maximum-likelihood method writes the matrix as an operator’s adjoint times the operator, then normalizes the trace, so the matrix is nonnegative by construction. A numerical optimization then compares predicted and measured counts under the chosen statistical model. Positivity comes from this parameterization; it is not a separate experimental finding.

A physical result can still come from fitting the wrong analyzer model, and estimates near the boundary can have lopsided uncertainties. Simulated repeat data or propagated errors mean something only under stated assumptions about noise, drift and calibration. The scope of the error bar depends on which effects the model actually includes.

Same populations, different coherence

Compare the positive Bell superposition of both-horizontal and both-vertical polarization with an equal incoherent mixture of the two. Their count tables in the horizontal–vertical basis are identical. In the diagonal basis, the pure Bell state always gives matching outcomes, while the mixture gives all four outcomes equally often.

The mixture’s fidelity to the Bell state is exactly one half; the ideal pure state’s is one. Measuring in more calibrated bases reveals the difference. A full two-qubit reconstruction demands more than a single witness for one target state, while a witness can establish entanglement without pinning down every matrix entry. Both methods rely on calibration; a device-independent Bell test has its own trial design and assumptions.

Read the evidence: D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White (2001), On the Measurement of Qubits — author manuscript, arXiv:quant-ph/0103121 X.-S. Ma et al. (2012), Experimental delayed-choice entanglement swapping — author manuscript, arXiv:1203.4834

Test the fit on data it has not seen

A good reconstruction exercise holds back one extra setting or run and compares it with the fitted prediction. A mismatch between predicted and measured counts can expose some model failures and prompt recalibration. Success at one held-out setting still does not certify an arbitrary preparation or measurement device.

For the next reconstruction, retain a short checklist: the raw counts and exposure for every setting; the calibrated detector response; the noise model and physical-state constraint; the uncertainty procedure; and a prediction tested on unused data. A projection onto physical matrices and a maximum-likelihood fit are distinct procedures. Specify which was used, since the difference can affect the estimate.

Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

Homodyne quantum tomography

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

Sources and reading notes

We read James et al.’s 2001 manuscript, especially the 16-setting two-qubit scheme, positivity, the maximum-likelihood parameterization and the error analysis, and checked the journal’s publication details. We also read the Kwiat group’s tomography resources. The small single-qubit and Bell-pair examples are our own calculations. The single-qubit Lab uses physical projection, not the paper’s maximum-likelihood fit. The linked model reconstructs one qubit by physical-matrix projection and explores channels separately; it does not implement the paper’s two-photon sixteen-setting maximum-likelihood analysis.