# State Tomography

**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.

## Principle

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.

## Single-Qubit Tomography

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

## Multi-Qubit Tomography

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.

## Fidelity Benchmarking

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

## Limitations

- **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