Photon tomography: a physical matrix still needs calibrated counts

The matrix is inferred from repeated preparations

James, Kwiat, Munro and White’s 2001 paper describes reconstructing two polarization qubits from a set of analyzer measurements. Different settings interrogate different members of an assumed consistently prepared ensemble. The apparatus does not photograph all entries of one photon pair’s density matrix.

The paper develops a sixteen-setting implementation, linear reconstruction and a constrained statistical fit. An arbitrary normalized two-qubit state has fifteen independent real parameters; an acquisition scheme must also deal with count scale. Sixteen settings are not a universal count of detector clicks or a guarantee that calibration is adequate.

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

A count needs a measurement operator

For a teaching count model, let each setting have a known positive measurement operator, an acquisition scale and an additive background. The predicted mean count follows the Born rule below. The state is unknown; analyzer response and normalization enter as separate model ingredients.

Assuming a projector when the detector actually has unequal efficiency can push a response error into the inferred state. Source drift between settings can imitate coherence or imbalance. Record raw counts, exposure, background, wave-plate calibration and the assumed repeated-preparation contract before interpreting a matrix plot.

\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 put linear inversion outside the state space

Our single-qubit example estimates the mean outcomes along three orthogonal spin axes as 0.80, 0.80 and zero. Each value separately is a valid binary average, but together their vector length is about 1.131. The corresponding trace-one Hermitian matrix has eigenvalues about 1.066 and minus 0.066.

Thus individually plausible count frequencies need not form a physical joint estimate after inversion. A constrained method must compromise among measurements. Simply shortening the vector to unit length gives one physical projection for this example, but that operation is not generally the maximum-likelihood solution for a specified count model.

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

Positivity is a constraint, not a calibration

The paper’s maximum-likelihood approach parameterizes a nonnegative matrix as an operator’s adjoint multiplied by itself, followed by trace normalization. Numerical optimization then compares predicted and recorded data using its chosen statistical model. Positivity follows from the parameterization rather than becoming an independent experimental discovery.

A physical result can still fit the wrong analyzer model. Boundary estimates can also have asymmetric uncertainty. Simulating repeated data or propagating errors is meaningful only with the declared noise, drift and calibration assumptions; attaching an error bar does not automatically include effects that were left outside that model.

Identical diagonal counts, different coherences

Compare a positive Bell superposition of both-horizontal and both-vertical polarization with their equal incoherent mixture. Their horizontal–vertical count tables agree. In the diagonal basis the pure Bell state has perfectly matching outcomes, while the mixture has four equally likely outcomes.

The mixture’s fidelity to that Bell target is exactly one half; the ideal pure state’s is one. Additional calibrated bases reveal the difference. A complete two-qubit reconstruction asks more than a single target witness, while a witness may establish entanglement without identifying every matrix parameter. Neither becomes a device-independent Bell test merely by using more settings.

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

Check predictions beyond the fitted picture

A useful reconstruction exercise reserves an additional setting or acquisition for comparison with the fitted prediction. Comparing predicted and acquired counts can expose some model failures and motivate recalibration. Success at one held-out setting still does not certify an arbitrary preparation and measurement device.

The existing Lab does this for a single qubit and separately explores channels. Its physical matrix projection is explicitly distinct from maximum likelihood, and it does not reproduce the historical two-photon sixteen-setting acquisition. Reading the original paper supplies that missing experimental context while preserving the Lab’s narrower inference contract.