Tomography Simulation

Tomography simulation separates a known generating state, finite measurement counts, and a reconstruction algorithm that does not receive the hidden state. This separation allows quantitative tests of bias, uncertainty, and physical constraints. State tomography explains the operational theory.

Helpful background. Measurement sampling supplies finite-shot statistics; density-matrix diagnostics supplies physicality checks.

Measurement design as a linear map

For POVM effects $E_{m|s}$ in setting $s$, generate probabilities

$$p_{m|s}=\operatorname{Tr}(\rho E_{m|s}), \qquad \sum_mE_{m|s}=I.$$

Choose a real basis $\{B_a\}$ for Hermitian matrices and expand $\rho=\sum_a x_aB_a$. Then $p=Mx$ with $M_{(m,s),a}=\operatorname{Tr}(E_{m|s}B_a)$.

Informational completeness means the measurement map determines all trace-one Hermitian states. Check the rank and singular values of the design matrix after accounting for normalization constraints. Near linear dependence amplifies count noise even when the matrix has full rank.

Generate independent multinomial counts for each setting with its declared shot allocation. Sampling all settings from one joint multinomial generally describes a different experiment.

Qubit linear inversion

For $\rho=(I+xX+yY+zZ)/2$, measure $X,Y,Z$ separately. If $n_{+a}$ and $n_{-a}$ are counts for axis $a$,

$$\hat r_a=\frac{n_{+a}-n_{-a}}{N_a}, \qquad \hat\rho_{\rm lin}=\frac12(I+\hat{\boldsymbol r}\cdot\boldsymbol\sigma).$$

Finite sampling can produce $|\hat{\boldsymbol r}|>1$ and a negative eigenvalue. This does not imply the true state was unphysical; it shows that unconstrained inversion does not enforce positivity.

For the maximally mixed state, each axis has mean zero and variance $1/N_a$. For $|0\rangle$, the ideal $Z$ outcome is deterministic while $X$ and $Y$ remain random. These cases test the measurement rotations and the distinction between state purity and outcome uncertainty.

Constrained likelihood

For counts $n_{m|s}$, minimize

$$-\log\mathcal L(\rho) =-\sum_{s,m}n_{m|s}\log\operatorname{Tr}(\rho E_{m|s})$$

subject to $\rho\geq0$ and $\operatorname{Tr}\rho=1$. A factorization $\rho=TT^\dagger/\operatorname{Tr}(TT^\dagger)$ enforces physicality but introduces redundant parameters and a nonlinear optimization landscape. Convex optimization in $\rho$ exposes the convex constraint directly.

A physical result is not automatically statistically unbiased. Near rank-deficient states, positivity constraints create boundary effects and non-Gaussian uncertainty. Clipping eigenvalues of linear inversion is a different estimator from maximum likelihood and should be named as such.

For a concrete comparison, take counts $(90,10)$ for $X$, $(9,1)$ for $Y$, and $(5,5)$ for $Z$. Linear inversion gives $(0.8,0.8,0)$. Its Euclidean projection onto the Bloch ball is $(1/\sqrt2,1/\sqrt2,0)$, regardless of shot allocation. The binomial likelihood instead gives approximately $(0.78572,0.61859,0)$: the more accurately sampled $X$ expectation moves less.

To reproduce this constrained example, the optimum has $z=0$ and lies on the boundary. Write $x=\cos\theta$, $y=\sin\theta$ with $0<\theta<\pi/2$, and maximize

$$\ell(\theta)={}90\log\frac{1+x}{2} +10\log\frac{1-x}{2} +9\log\frac{1+y}{2} +\log\frac{1-y}{2}.$$

The omitted $Z$ term is constant at $z=0$. A bracketed zero of $d\ell/d\theta$ supplies a transparent one-parameter reference for a general optimizer. At probabilities equal to zero, a zero count contributes zero to the negative log likelihood, while a positive count contributes infinity; silently flooring every probability changes the objective.

Synthetic-data validation

Repeat the entire data-generation and reconstruction pipeline for many independent data sets. Evaluate trace distance, infidelity under an explicitly stated convention, observable errors, and interval coverage. An optimizer’s convergence tolerance is not an uncertainty estimate.

Model mismatch matters: generate a small readout bias while reconstructing with ideal effects to show that increasing shots can converge more precisely to a wrong state. If the effects are calibrated, propagate their uncertainty or state that they are treated as exact.

For process tomography, prepare an informationally complete set of input states and reconstruct a Choi matrix. Enforce complete positivity and the appropriate partial-trace constraint, not merely unit trace; see channel numerics.

In the input-first unnormalized convention of that page, the synthetic probability map is

$$p(m|s,a)=\operatorname{Tr}\!\left[ J_{\mathcal E}(\rho_a^{\mathsf T}\otimes E_{m|s}) \right],\qquad \operatorname{Tr}_{\rm out}J_{\mathcal E}=I_{\rm in}.$$

The transpose acts in the declared input basis. Test a complex input, such as a $Y$ eigenstate: real inputs cannot reveal an omitted transpose. Known input states and measurement effects are assumptions of ordinary process tomography; preparation and readout errors otherwise become part of the inferred process.

Exercise

A reconstructed qubit Bloch vector is $(0.8,0.8,0)$. Find the density-matrix eigenvalues and explain the failure.

Solution

Its length is $\sqrt{1.28}\approx1.1314$, so the eigenvalues are $(1\pm1.1314)/2$, approximately $1.0657$ and $-0.0657$. Each individual expectation lies between $-1$ and 1, but the combined vector violates positivity. Marginal range checks alone are insufficient.

References

• James, D. F. V., Kwiat, P. G., Munro, W. J., and White, A. G., “Measurement of qubits,” Physical Review A 64, 052312 (2001), arXiv:quant-ph/0103121.
• Watrous, J., The Theory of Quantum Information, Cambridge University Press (2018), author’s text — channel-state constraints.