State tomography is an experimental procedure to reconstruct the quantum state $\rho$ of a system by performing many measurements in different bases and using classical processing to infer the state from outcome statistics.

Measuring a qubit multiple times in the same basis gives outcome probabilities but not the quantum state itself. Measuring in the Z basis yields probabilities for $|0\rangle$ and $|1\rangle$; measuring in the X basis yields probabilities for $|+\rangle$ and $|-\rangle$. Combining results from measurements in three orthogonal bases (X, Y, Z) over many copies of the state allows reconstruction of the full density matrix.

A single-qubit state has 3 real parameters (up to global phase). Measuring in the Z basis gives the Z expectation value; measuring in X and Y bases gives X and Y expectation values. These three values determine the density matrix uniquely.

Procedure:

1. Prepare the state many times
2. Measure 1/3 of copies in Z basis
3. Measure 1/3 of copies in X basis (rotate by Hadamard, then measure Z)
4. Measure 1/3 of copies in Y basis (rotate by $S^\dagger H$, then measure Z)
5. Use classical post-processing to infer density matrix from outcome statistics

An $n$-qubit state requires $3^n$ measurements bases to fully characterize. This grows exponentially, making full tomography impractical for many qubits. Partial tomography focuses on subsystems or specific observables.

State tomography is used to verify that quantum circuits prepare intended states. Comparing the measured state to the target state quantifies preparation fidelity.

• Exponential scaling: full tomography requires exponentially many measurements for $n$ qubits
• Statistical noise: finite measurement statistics introduce errors
• Entanglement signature: tomography cannot distinguish locally indistinguishable entangled states