Measurement Tomography

Measurement tomography is the reconstruction of an unknown measurement device from its responses to known probe states. In the simplest finite-dimensional setting, the unknown object is a POVM $\{F_m\}$, and the data are outcome frequencies obtained by sending calibrated states $\rho_k$ into the device.

The defining probability model is

$$p(m|k) = \operatorname{Tr}(\rho_k F_m).$$

State tomography and measurement tomography use the same trace rule in opposite directions. State tomography assumes the effects are known and estimates $\rho$. Measurement tomography assumes the probe states are known and estimates the effects $F_m$.

Helpful background. POVMs provide the forward probability rule; instruments describe the additional state-update information. Bayes’ rule is useful for statistical inference.

Detector effects as an inverse Born-rule problem

Suppose a detector has outcomes $m=1,\ldots,M$. Its POVM effects obey

$$F_m\ge0, \qquad \sum_{m=1}^M F_m=I.$$

For each trusted input state $\rho_k$, repeat the experiment $N_k$ times and record counts $n_{mk}$. The empirical frequency

$$\hat p(m|k) = \frac{n_{mk}}{N_k}$$

estimates

$$p(m|k) = \operatorname{Tr}(\rho_k F_m).$$

Assume independent repetitions with the same detector response and all outcomes, including failures, counted. If the probe states span the real vector space of Hermitian operators, ideal noiseless probabilities uniquely determine the effects. Finite counts add statistical error; imperfect probes add systematic error.

Operator-Basis Form

Choose a Hermitian operator basis $\{B_\alpha\}_{\alpha=0}^{d^2-1}$ for a $d$-dimensional Hilbert space. Write

$$F_m = \sum_\alpha f_{m\alpha}B_\alpha.$$

The known probe states define the design matrix

$$R_{k\alpha} = \operatorname{Tr}(\rho_k B_\alpha).$$

Then the Born rule becomes

$$p(m|k) = \sum_\alpha R_{k\alpha}f_{m\alpha}.$$

For each outcome $m$, measurement tomography is a linear inverse problem for the coefficient vector $f_{m\alpha}$. The physical constraints couple the outcomes:

$$\sum_m F_m=I, \qquad F_m\ge0.$$

Linear inversion is conceptually transparent, but finite data can make it return effects with negative eigenvalues or effects that do not sum exactly to $I$. Constrained reconstruction fits all outcomes jointly while enforcing positivity and completeness; it does not remove uncertainty or repair an incorrect probe model.

Informational Completeness

The probe set is informationally complete for detector tomography if the linear functionals

$$F\mapsto \operatorname{Tr}(\rho_k F)$$

separate all Hermitian operators $F$ on the chosen Hilbert space. Equivalently, the span of the probe states must be the full $d^2$-dimensional real vector space of Hermitian operators.

This is the dual of the usual state-tomography condition. For state tomography, a known measurement is informationally complete if its effects span the operator space. For measurement tomography, known probe states must span the operator space so that unknown effects can be identified.

Informational completeness is always relative to a model:

1. a Hilbert-space dimension or cutoff;
2. a trusted set of input states;
3. stable detector behavior;
4. known outcome labels and coarse graining;
5. an assumed count-noise model.

If the detector has memory, drift, leakage outside the modeled subspace, or state-preparation errors, the reconstructed POVM can be a faithful description of the calibration experiment while still failing in later use.

Constrained Estimation

A common statistical model treats each input setting $k$ as a multinomial experiment. Up to constants independent of the POVM, the likelihood is

$$\mathcal L(\{F_m\}) \propto \prod_k \prod_m \left[ \operatorname{Tr}(\rho_kF_m) \right]^{n_{mk}}.$$

The maximum-likelihood estimate solves

$$\text{maximize }\mathcal L(\{F_m\}) \quad \text{subject to} \quad F_m\ge0, \quad \sum_mF_m=I.$$

Bayesian detector tomography instead places a prior over valid POVMs and updates it with the same likelihood:

$$\pi(\{F_m\}\mid D) \propto \mathcal L(D\mid\{F_m\})\, \pi(\{F_m\}).$$

The Bayesian posterior is a state of knowledge about the detector model. It is not a post-measurement quantum state of a system. That distinction is reviewed in Bayes Rule.

A Qubit Two-Outcome Detector

For a qubit detector with outcomes $+$ and $-$, write

$$F_+ = \frac{1}{2} \left( \alpha I+\boldsymbol\beta\cdot\boldsymbol\sigma \right), \qquad F_-=I-F_+.$$

For a probe state

$$\rho(\mathbf r) = \frac{1}{2} \left( I+\mathbf r\cdot\boldsymbol\sigma \right),$$

the probability of the $+$ outcome is

$$p(+|\mathbf r) = \operatorname{Tr}[\rho(\mathbf r)F_+] = \frac{1}{2} \left( \alpha+\boldsymbol\beta\cdot\mathbf r \right).$$

If one can prepare the maximally mixed state and the three Pauli-axis states with Bloch vectors $\mathbf e_x,\mathbf e_y,\mathbf e_z$, then

$$\alpha=2p(+|\mathbf 0),$$

and

$$\beta_j = 2p(+|\mathbf e_j)-\alpha, \qquad j=x,y,z.$$

The effect is physical only if

$$0\le F_+\le I.$$

In Bloch parameters this means

$$\lvert\boldsymbol\beta\rvert \le \alpha \le 2-\lvert\boldsymbol\beta\rvert.$$

Finite-count estimates can violate this inequality. A constrained maximum-likelihood fit optimizes the likelihood over valid effects; it is not generally the same as clipping negative eigenvalues after inversion.

For a concrete example, use probabilities $0.50,0.65,0.40,0.80$ for the mixed, $+x$, $+y$, and $+z$ probes. They give $\alpha=1$ and $\boldsymbol\beta=(0.3,-0.2,0.6)$. Since $|\boldsymbol\beta|=0.7$, the eigenvalues of $F_+$ are $0.85$ and $0.15$. An additional $-y$ probe should give $p(+|-y)=(1-\beta_y)/2=0.60$; disagreement beyond counting error tests whether the fitted model predicts a new preparation.

If each of the first four probabilities came from $N=1000$ independent trials, the direct estimate $\widehat\beta_x=2\widehat p_x-2\widehat p_0$ would have standard deviation

$$\sqrt{\frac{4}{N}\{0.65(0.35)+0.50(0.50)\}}\approx0.0437.$$

The estimates of different $\beta_j$ are correlated because they reuse $\widehat p_0$. Positivity constraints can further change the sampling distribution, especially near a boundary.

Detector Characterization

A detector model may include more than ideal outcome effects. Depending on the platform, calibration may include:

1. efficiency and loss;
2. dark counts and false positives;
3. outcome misassignment;
4. saturation and dead time;
5. crosstalk between channels;
6. finite resolution or binning;
7. leakage outside the computational subspace.

Some of these imperfections are naturally represented by POVM effects. Others require a larger instrument model, a classical readout model, or a time-dependent detector model. The right level of description depends on how the calibrated device will be used.

For example, an inefficient two-outcome photon detector may be modeled by a no-click effect that includes both vacuum and missed photons. If later predictions depend only on click probabilities, a POVM may suffice. If later predictions depend on what state remains after a click or no-click result, a Quantum Instrument is needed.

Instrument Tomography

POVM tomography reconstructs probabilities. It does not determine the post-measurement state. To reconstruct an instrument, one must estimate the outcome-resolved completely positive maps

$$\mathcal I_m.$$

For each input $\rho_k$ and outcome $m$, the unnormalized output state is

$$\mathcal I_m(\rho_k),$$

with probability

$$p(m|k) = \operatorname{Tr}\mathcal I_m(\rho_k).$$

Instrument tomography therefore requires both outcome probabilities and conditional output information. With calibrated output effects $G_\ell$, the measured joint probabilities are

$$p(m,\ell|k)=\operatorname{Tr}[G_\ell\mathcal I_m(\rho_k)].$$

Both the input probes and output effects must span their respective operator spaces for unrestricted finite-dimensional reconstruction. Normalized conditional output states alone omit the branch probabilities and are insufficient.

For the input-first Choi convention

$$J_m=\sum_{i,j}|i\rangle\langle j|\otimes \mathcal I_m(|i\rangle\langle j|),$$

the same data satisfy

$$p(m,\ell|k)=\operatorname{Tr}[(\rho_k^T\otimes G_\ell)J_m], \qquad J_m\ge0, \qquad \sum_m\operatorname{Tr}_{\mathrm{out}}J_m=I.$$

The transpose is in the input basis used to define $J_m$; omitting it reverses imaginary coherences. The Choi-matrix page explains other ordering conventions. Reconstructing a map still does not pick a unique set of Kraus operators.

Relation to State and Process Tomography

The three standard inverse problems differ by which object is unknown:

Task	Unknown object	Trusted ingredients	Probability model
State tomography	$\rho$	effects $F_m$	$\operatorname{Tr}(\rho F_m)$
Measurement tomography	effects $F_m$	states $\rho_k$	$\operatorname{Tr}(\rho_kF_m)$
Process tomography	channel $\Phi$	states and effects	$\operatorname{Tr}[F_m\Phi(\rho_k)]$

Process Tomography is a channel-reconstruction problem. In finite dimensions, it is often expressed through the Choi matrix of $\Phi$ and physical constraints such as complete positivity and trace preservation. Measurement tomography is narrower: it calibrates the measurement apparatus itself.

In practice, no experiment has perfectly trusted states, gates, and measurements. Self-consistent protocols estimate state-preparation-and-measurement errors together, but they introduce gauge freedoms and belong to the broader benchmarking and verification toolkit.

SPAM errors explains why imperfect preparation can be confused with detector bias. After calibration, measurement-error mitigation uses a justified response model to correct terminal outcome estimates; it cannot undo physical backaction on a surviving system.

Check whether the fitted model predicts new data

Repeat some probe settings at different times to detect drift, and use additional preparations to test predictions beyond the fitted settings. Compare discrepancies with a multinomial noise model or a suitable resampling analysis. A positive, normalized POVM can still fit the wrong model poorly.

For optical and motional systems, vary the reconstruction cutoff or bound the omitted input population. Stable results within a truncated space do not prove that leakage or detector memory is absent. Keep no-click outcomes in the counts unless the stated model explicitly conditions on detection.

Exercises

Qubit coefficient reconstruction

For the qubit detector above, suppose the measured probabilities are

$$p(+|\mathbf 0)=0.45, \qquad p(+|\mathbf e_x)=0.60, \qquad p(+|\mathbf e_y)=0.40, \qquad p(+|\mathbf e_z)=0.70.$$

Find $\alpha$ and $\boldsymbol\beta$.

Solution

From the maximally mixed probe,

$$\alpha=2p(+|\mathbf 0)=0.90.$$

Then

$$\beta_x=2(0.60)-0.90=0.30,$$ $$\beta_y=2(0.40)-0.90=-0.10,$$

and

$$\beta_z=2(0.70)-0.90=0.50.$$

Thus

$$\boldsymbol\beta=(0.30,-0.10,0.50).$$

Positivity check

For the coefficients in the previous exercise, check whether $F_+$ is a valid effect.

Solution

Compute

$$\lvert\boldsymbol\beta\rvert = \sqrt{0.30^2+(-0.10)^2+0.50^2} = \sqrt{0.35} \approx 0.592.$$

The positivity constraints are

$$\lvert\boldsymbol\beta\rvert \le \alpha \le 2-\lvert\boldsymbol\beta\rvert.$$

Here

$$0.592\le0.90\le1.408,$$

so $F_+$ and $F_-=I-F_+$ are both positive.

State versus measurement tomography

Explain why the same data table $p(m|k)$ can be read as state tomography or measurement tomography only after declaring which side of the trace rule is trusted.

Solution

The Born rule

$$p(m|k)=\operatorname{Tr}(\rho_kF_m)$$

is bilinear in $\rho_k$ and $F_m$. If $F_m$ is trusted, the data can estimate unknown states $\rho_k$. If $\rho_k$ is trusted, the data can estimate unknown effects $F_m$. If neither side is trusted, the problem is underdetermined without additional assumptions, reference devices, or self-consistency constraints.

References

• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer (2016). Publisher.
• G. M. D’Ariano, L. Maccone, and P. Lo Presti, “Quantum calibration of measurement instrumentation,” Physical Review Letters 93, 250407 (2004). Article.
• J. Fiurášek, “Maximum-likelihood estimation of quantum measurement,” Physical Review A 64, 024102 (2001). Article.
• A. Luis and L. L. Sánchez-Soto, “Complete characterization of arbitrary quantum measurement processes,” Physical Review Letters 83, 3573–3576 (1999). Article.
• J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nature Physics 5, 27–30 (2009). Article.