# Pauli Eigenstates

**Pauli eigenstates** are the +1 and -1 eigenstates of the three Pauli operators X, Y, Z. Each Pauli has two orthogonal eigenstates spanning the two-dimensional single-qubit Hilbert space.

## Z Basis (Computational)

Eigenstates of the Pauli Z operator:
- **[[quantum-state-0|Zero state (|0⟩)]]**: $Z|0\rangle = |0\rangle$ (eigenvalue +1)
- **[[quantum-state-1|One state (|1⟩)]]**: $Z|1\rangle = -|1\rangle$ (eigenvalue -1)

## X Basis

Eigenstates of the Pauli X operator:
- **[[quantum-state-plus|Plus state (|+⟩)]]**: $X|+\rangle = |+\rangle$ (eigenvalue +1)
  - $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$
- **[[quantum-state-minus|Minus state (|-⟩)]]**: $X|-\rangle = -|-\rangle$ (eigenvalue -1)
  - $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$

## Y Basis

Eigenstates of the Pauli Y operator:
- **[[quantum-state-plus-i|Plus-i state (|+i⟩)]]**: $Y|+i\rangle = |+i\rangle$ (eigenvalue +1)
  - $|+i\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$
- **[[quantum-state-minus-i|Minus-i state (|-i⟩)]]**: $Y|-i\rangle = -|-i\rangle$ (eigenvalue -1)
  - $|-i\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)$

## Measurement and Basis Rotation

Each Pauli basis is a valid measurement basis. Rotating from one basis to another requires single-qubit gates:
- Z ↔ X: apply [[quantum-gate-h|Hadamard]]
- Z ↔ Y: apply $S^\dagger H$ or $H S^\dagger$
- X ↔ Y: apply $S^\dagger$ or $S$

## Orthonormality

All six Pauli eigenstates are mutually orthogonal except within each basis. For example, $\langle 0|1\rangle = 0$ but $\langle 0|+\rangle = 1/\sqrt{2}$.

## Bloch Sphere

On the [[quantum-state-bloch-sphere|Bloch sphere]], the six Pauli eigenstates are the poles of three perpendicular axes:
- Z axis poles: $|0\rangle$ (north), $|1\rangle$ (south)
- X axis poles: $|+\rangle$ (east), $|-\rangle$ (west)
- Y axis poles: $|+i\rangle$ (up), $|-i\rangle$ (down)

## Importance in Quantum Computing

Pauli eigenstates are fundamental to:
- Quantum error correction: stabilizer codes use Pauli operators and their eigenstates
- Quantum algorithms: many algorithms prepare and measure in non-computational bases
- Quantum metrology: measuring different Pauli observables reveals different information about the state