Single-Qubit States

Single-qubit states are quantum states of a single qubit, normalized vectors in a two-dimensional Hilbert space. All single-qubit states can be represented on the Bloch sphere and are characterized by two real parameters (polar angles).

Single-qubit states form the foundation of quantum computing. Two main classification schemes organize them:

All pure single-qubit states are eigenstates of some Pauli operator. The six Pauli eigenstates form three orthogonal pairs:

• Computational basis: Z eigenstates
  • Zero state (|0⟩)
  • One state (|1⟩)
• Pauli eigenstates: X, Y, Z bases
  • Plus state (|+⟩) and Minus state (|-⟩) (X eigenstates)
  • Plus-i state (|+i⟩) and Minus-i state (|-i⟩) (Y eigenstates)
  • Zero and One states (Z eigenstates)

• Product states: $|0\rangle$ and $|1\rangle$ — definite classical outcomes
• Equal superposition: $|+\rangle$ and $|-\rangle$ — equal-amplitude combinations
• Complex superposition: arbitrary $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$

Any pure single-qubit state maps to a point on the unit Bloch sphere via:

$$|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$$

where:

• $\theta \in [0, \pi]$ is the polar angle from north pole ($|0\rangle$) to south pole ($|1\rangle$)
• $\phi \in [0, 2\pi)$ is the azimuthal angle determining global phase

The six Pauli eigenstates correspond to the poles of three perpendicular axes on the sphere.

Measuring a single-qubit state in the computational basis yields $|0\rangle$ with probability $|\alpha|^2$ and $|1\rangle$ with probability $|\beta|^2$. Measuring in another basis (X or Y) requires rotating the state first via single-qubit gates before measurement.

Quantum gates transform single-qubit states via unitary rotations:

• Pauli gates (X, Y, Z): $\pi$ rotations about Bloch sphere axes
• Hadamard: maps $|0\rangle \leftrightarrow |+\rangle$ and $|1\rangle \leftrightarrow |-\rangle$
• Phase gates (S, T): rotations about Z axis with specific phases
• Rotation gates (RX, RY, RZ): arbitrary rotations by angle $\theta$ about each axis
• Universal rotations (U gate): arbitrary single-qubit unitary via three rotations

See single-qubit gates for details on each gate.

Pure single-qubit states are points on the Bloch sphere. Mixed states (statistical mixtures) are represented by density matrices and correspond to points inside the sphere. Decoherence moves states from the surface toward the center via:

• Depolarization: random flips to orthogonal state
• Dephasing: loss of phase coherence in superposition
• Amplitude damping: decay to ground state $|0\rangle$

• Qubit initialization: prepare known initial state (usually $|0\rangle$)
• Basis rotation: rotate to different measurement basis for state characterization
• Quantum gates: inputs and outputs of single-qubit operations
• Quantum error correction: stabilizer codes measure Pauli eigenstates
• Variational algorithms: single-qubit states parameterize ansätze in VQE, QAOA

The six Pauli eigenstates satisfy:

• Orthonormal within each basis: e.g., $\langle 0|1\rangle = 0$
• Cross-basis overlaps: e.g., $\langle 0|+\rangle = 1/\sqrt{2}$
• All overlaps are real and symmetric

These inner products determine measurement probabilities when state is prepared in one basis but measured in another.

For pure states:

• Purity: $\text{Tr}(\rho^2) = 1$ (maximum)
• Fidelity: $F(\rho, \sigma) = \text{Tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}) \leq 1$, with equality only for identical states

Fidelity between two single-qubit states is computed via their Bloch vectors or density matrices.

Single-qubit states extend to multi-qubit systems via tensor products:

• Two-qubit: $|\psi\rangle_A \otimes |\phi\rangle_B$
• $n$-qubit: $|\psi_1\rangle \otimes |\psi_2\rangle \otimes \cdots \otimes |\psi_n\rangle$

Entangled states cannot be factored as products of single-qubit states.