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.

Eigenstates of the Pauli Z operator:

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

Eigenstates of the Pauli X operator:

• Plus state (|+⟩): $X|+\rangle = |+\rangle$ (eigenvalue +1)
  • $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$
• Minus state (|-⟩): $X|-\rangle = -|-\rangle$ (eigenvalue -1)
  • $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$

Eigenstates of the Pauli Y operator:

• Plus-i state (|+i⟩): $Y|+i\rangle = |+i\rangle$ (eigenvalue +1)
  • $|+i\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$
• Minus-i state (|-i⟩): $Y|-i\rangle = -|-i\rangle$ (eigenvalue -1)
  • $|-i\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)$

Each Pauli basis is a valid measurement basis. Rotating from one basis to another requires single-qubit gates:

• Z ↔ X: apply Hadamard
• Z ↔ Y: apply $S^\dagger H$ or $H S^\dagger$
• X ↔ Y: apply $S^\dagger$ or $S$

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

On the 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)

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