Computational Basis States

Computational basis states $|0\rangle$ and $|1\rangle$ are the two orthogonal eigenstates of the Pauli Z operator. They represent the classical outcomes 0 and 1 and form the default measurement basis on most quantum computers.

• Zero state (|0⟩): eigenstate with $Z|0\rangle = |0\rangle$ (eigenvalue +1)
• One state (|1⟩): eigenstate with $Z|1\rangle = -|1\rangle$ (eigenvalue -1)

Both basis states are:

• Orthonormal: $\langle 0|1\rangle = 0$, $\langle 0|0\rangle = \langle 1|1\rangle = 1$
• Pure states: $\rho = |i\rangle\langle i|$ for $i \in \{0, 1\}$
• Eigenstates of Z: measurement in Z basis always yields definite outcome

• Initial state: qubits default to $|0\rangle$ on most platforms
• Measurement outcome: quantum states collapse to $|0\rangle$ or $|1\rangle$ when measured in the computational basis
• Classical simulation: any state without superposition of computational basis states is trivially classical
• Basis decomposition: any $n$-qubit state is $|\psi\rangle = \sum_{x \in \{0,1\}^n} \alpha_x |x\rangle$

The computational basis is the platform for understanding superposition:

• Single-qubit superposition: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$
• Multi-qubit superposition: $|\psi\rangle = \sum_x \alpha_x |x_1 x_2 \ldots x_n\rangle$
• Entanglement: multi-qubit states that cannot factor into products of computational basis eigenstates

Measurement in the computational basis is the most direct; measurement in other bases (X or Y) requires rotating the state first via single-qubit gates before measurement.