Reconstructing a Quantum State

15.1 Measuring fresh copies across Pauli bases

15.1.1 Why a single measurement basis leaves relative phases hidden

Measuring a qubit solely in the computational basis \(\{|0\rangle, |1\rangle\}\) yields only the diagonal entries \(\rho_{00} = P(+Z)\) and \(\rho_{11} = P(-Z) = 1 - \rho_{00}\) of its density matrix \(\rho\). Every state of the form \((|0\rangle + e^{i\phi}|1\rangle)/\sqrt{2}\)—as well as the incoherent fifty-fifty mixture \(\frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1|\)—produces the exact same \(Z\)-basis statistics \(P(+Z) = P(-Z) = 1/2\). Reconstructing the off-diagonal coherence \(\rho_{01} = \frac{1}{2}(\langle X\rangle - i\langle Y\rangle)\) requires rotating the measurement axis into at least two complementary equatorial bases before projective readout.

15.1.2 Consuming identically prepared copies across orthogonal settings

Because a projective measurement collapses an individual copy and the no-cloning theorem forbids duplicating an unknown quantum state, quantum state tomography is inherently an ensemble protocol. The source must prepare \(N\) independent, identically prepared copies of \(\rho\), partitioned into three sub-ensembles that are measured along the Pauli axes \(X\), \(Y\), and \(Z\). In architectures where only \(Z\)-basis readout is hardware-native, measuring \(X\) is implemented by applying a pre-readout rotation \(R_Y(-\pi/2)\) (or Hadamard gate \(H\)), and measuring \(Y\) is implemented by applying \(R_X(+\pi/2)\) prior to \(Z\) readout.

15.1.3 Bloch-plane visualization of purity and orientation

The three measured expectation values \(\vec{r} = (r_x, r_y, r_z) = (\langle X\rangle, \langle Y\rangle, \langle Z\rangle)\) specify a point inside or on the unit Bloch ball. The direction of \(\vec{r}\) fixes the eigenbasis of \(\rho\), while its Euclidean length \(r = \|\vec{r}\| = \sqrt{r_x^2 + r_y^2 + r_z^2}\) determines the state’s purity: pure states lie on the spherical surface (\(r = 1\)), partially mixed states lie in the interior (\(0 < r < 1\)), and the maximally mixed state \(\frac{1}{2}I\) sits at the origin (\(r = 0\)).

15.2 Bloch vector inversion, positivity, and purity

15.2.1 Expanding the qubit density matrix in the Pauli basis

Because the four operators \(\{I, X, Y, Z\}\) form an orthogonal basis for \(2\times 2\) Hermitian matrices with \(\mathrm{Tr}(\sigma_j\sigma_k) = 2\delta_{jk}\), any unit-trace Hermitian matrix \(\rho\) has the unique linear tomographic expansion

where \(r_k = \mathrm{Tr}(\rho\sigma_k) = P(+\sigma_k) - P(-\sigma_k) = 2P(+\sigma_k) - 1\) for \(k \in \{x,y,z\}\).

15.2.2 Eigenvalue positivity and purity from Bloch-vector length

A valid physical density matrix must be positive semidefinite (\(\rho \succeq 0\)). Since the eigenvalues of \(\frac{1}{2}(I + \vec{r}\cdot\vec{\sigma})\) are

positivity \(\lambda_- \ge 0\) is equivalent to the Bloch-ball condition \(\|\vec{r}\| \le 1\). The purity of the reconstructed state is directly

15.2.3 Finite-sample overstepping of the Bloch sphere boundary

In an experiment with \(N_k\) shots along axis \(k\) and \(n_{k+}\) positive outcomes, the empirical estimator \(\hat{r}_k = (2n_{k+}/N_k) - 1\) has binomial variance \(\mathrm{Var}(\hat{r}_k) = (1 - r_k^2)/N_k\). When the true state lies on or very near the boundary (\(\|\vec{r}\| \approx 1\)), statistical fluctuations in the bases where \(r_k \approx 0\) add positively in quadrature:

Consequently, direct linear inversion of a high-purity state frequently produces \(\|\hat{\vec{r}}\| > 1\) and a slightly negative eigenvalue \(\hat{\lambda}_- < 0\). Constrained maximum-likelihood estimation (MLE) or least-squares projection onto the positive semidefinite cone \(\{\rho \succeq 0, \mathrm{Tr}\,\rho = 1\}\) restores physicality while reporting bootstrap or Bayesian confidence regions.

15.3 Detector calibration and ensemble ambiguity

15.3.1 Systematic axis miscalibration and cross-validation

Statistical uncertainty decreases as \(1/\sqrt{N}\), but systematic errors in the pre-readout rotations \(R_Y(-\pi/2)\) and \(R_X(+\pi/2)\) do not average away with more shots. If the \(X\) and \(Y\) analysis pulses suffer from an under-rotation angle \(\epsilon\) or an axis tilt, linear inversion maps a pure state into a distorted ellipsoid and misreports both its Bloch angles and its purity. Overcomplete measurement sets—such as measuring along all six Pauli eigenstates \(\{\pm X, \pm Y, \pm Z\}\)—combined with detector tomography or gate-set tomography cross-check the measurement axes independently of state preparation.

15.3.2 Why identical density matrices hide distinct mixing histories

Tomography reconstructs the density operator \(\rho\), which encodes the probabilities of all possible measurements on the system, but it cannot recover the specific preparation ensemble that produced a mixed state. For instance, an equal classical mixture of \(|0\rangle\) and \(|1\rangle\), an equal mixture of \(|+\rangle\) and \(|-\rangle\), and one half of a maximally entangled Bell pair \((|00\rangle + |11\rangle)/\sqrt{2}\) traced over its partner all yield the exact same reduced density matrix \(\rho = \frac{1}{2}I\) (\(\vec{r} = \vec{0}\)). No local measurement in any basis can distinguish those preparations.

15.3.3 Reporting raw counts, positivity constraints, and confidence regions

Rigorous tomographic reporting never publishes a constrained MLE state alone without the raw count pairs \((n_{k+}, N_k)\) and the unconstrained linear-inversion vector \(\hat{\vec{r}}\). When \(\|\hat{\vec{r}}\| \gg 1\) beyond what binomial sampling noise \(\sim \sqrt{2/N}\) can explain, clamping the vector onto the unit sphere masks a calibration drift (such as dark-count subtraction bias or pulse-area error) that should instead be diagnosed and corrected at the hardware level.

Explore the Apparatus

Set the target Bloch vector components \((r_x, r_y, r_z)\) and the number of shots \(N\) per Pauli basis, then compare the true density matrix against the empirical linear-inversion estimate \(\hat{\rho}_{\mathrm{lin}}\) and the positivity-projected physical estimate \(\hat{\rho}_{\mathrm{phys}}\). Observe how near-pure states frequently produce \(\|\hat{\vec{r}}\| > 1\) at small \(N\).

Worked Check

Problem: A qubit ensemble is measured with \(N = 500\) shots in each Pauli basis, yielding positive-outcome counts \(n_{x+} = 400\), \(n_{y+} = 150\), and \(n_{z+} = 250\). Compute the linear-inversion Bloch vector \(\hat{\vec{r}}\), the reconstructed density matrix \(\hat{\rho}\), its eigenvalues \(\hat{\lambda}_{\pm}\), and its purity \(\hat{\gamma}\).

Practice Problems

Key Takeaways

• Reconstructing a single-qubit density matrix \(\rho = \frac{1}{2}(I + \vec{r}\cdot\vec{\sigma})\) requires measuring identically prepared copies across three complementary Pauli bases (\(X, Y, Z\)).
• Positivity \(\rho \succeq 0\) is equivalent to \(\|\vec{r}\| \le 1\), and the state’s purity is \(\mathrm{Tr}(\rho^2) = (1 + \|\vec{r}\|^2)/2\).
• For states near the surface of the Bloch sphere, finite-sample binomial fluctuations in orthogonal bases frequently push linear-inversion estimates slightly outside the sphere (\(\|\hat{\vec{r}}\| > 1\)), motivating constrained maximum-likelihood estimation alongside raw count reporting.
• The density matrix predicts all measurement statistics on the system but does not distinguish different physical ensembles that share the same \(\rho\).

Sources and Further Reading

• D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Physical Review A 64, 052312 (2001).
• M. Paris and J. Řeháček (eds.), Quantum State Estimation, Lecture Notes in Physics 649, Springer (2004).
• R. Blume-Kohout, Optimal, reliable estimation of quantum states, New Journal of Physics 12, 043034 (2010).