Basis states form an orthonormal set that spans the Hilbert space of a quantum system. A qubit has two computational basis states $|0\rangle$ and $|1\rangle$; any qubit state can be written as a superposition of these. Measurement projects a state onto one basis state, collapsing superposition to a definite outcome.

The computational (Z) basis $\{|0\rangle, |1\rangle\}$ is the default measurement basis. Measuring a qubit yields eigenvalues 0 or 1 corresponding to eigenstates of the Pauli Z operator. Most quantum devices measure in this basis.

The X basis $\{|+\rangle, |-\rangle\}$ corresponds to eigenstates of the Pauli X operator. The Y basis $\{|+i\rangle, |-i\rangle\}$ corresponds to eigenstates of Y. A state can be expressed in any orthonormal basis; the outcome probabilities depend on the chosen basis.

Applying a Hadamard gate rotates between the Z and X bases. More generally, single-qubit rotations change the measurement basis. This is crucial for quantum algorithms that need to measure in non-computational bases.

If a state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ is measured in the computational basis, the probability of outcome 0 is $|\alpha|^2$ and outcome 1 is $|\beta|^2$. For measurement in a different basis $|+\rangle, |-\rangle$, first express $|\psi\rangle$ in that basis, then apply the same rule.

Measuring a qubit in basis $\mathbf{n}$ projects it onto one of the two eigenstates of the measurement operator. The superposition collapses irreversibly. Measuring twice in the same basis yields the same outcome; measuring in different bases can yield different results.

An $n$-qubit system has $2^n$ computational basis states $|x_1 x_2 \ldots x_n\rangle$. Measuring all qubits yields one of $2^n$ classical bitstrings with probabilities determined by the state's amplitudes.