# Quantum state

**Quantum states** describe the complete information content of a quantum system. Like classical bits, qubits occupy definite states; unlike classical bits, qubits can exist in superposition—a linear combination of basis states. A quantum state of $n$ qubits is represented as a vector $|\psi\rangle$ in a complex Hilbert space $\mathbb{C}^{2^n}$, normalized so $\langle\psi|\psi\rangle = 1$.

A single-qubit state can be written $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $|\alpha|^2 + |\beta|^2 = 1$. Multi-qubit states live in tensor products of qubit spaces. Entangled states cannot be factored into independent qubits—they exhibit non-local correlations measured by entropy or purity.

States can be pure (perfect knowledge, represented by a state vector) or mixed (statistical mixture, represented by a density matrix). Measurement collapses a superposition to a definite basis state with probability determined by the state's amplitudes.

## Concepts

 1. [[quantum-state-bloch-sphere|Bloch sphere representation]]
 2. [[quantum-state-superposition|Superposition]]
 3. [[quantum-state-entanglement|Entanglement]]
 4. [[quantum-state-basis|Basis states and measurement]]
 5. [[quantum-state-density-matrix|Density matrices and mixed states]]
 6. [[quantum-state-stabilizer|Stabilizer states]]
 7. [[quantum-state-tomography|State tomography]]
 8. [[quantum-state-fidelity|Fidelity and purity]]

## States

- [[quantum-state-single-qubit|Single-qubit states]]
  - [[quantum-state-computational|Computational basis]]
    - [[quantum-state-0|Zero state (|0⟩)]]
    - [[quantum-state-1|One state (|1⟩)]]
  - [[quantum-state-pauli-eigenstates|Pauli eigenstates]]
    - [[quantum-state-plus|Plus state (|+⟩)]]
    - [[quantum-state-minus|Minus state (|-⟩)]]
    - [[quantum-state-plus-i|Plus-i state (|+i⟩)]]
    - [[quantum-state-minus-i|Minus-i state (|-i⟩)]]
- [[quantum-state-two-qubit|Two-qubit states]]
  - [[quantum-state-computational-two-qubit|Two-qubit computational basis]]
    - [[quantum-state-computational-00|Computational 00 (|00⟩)]]
    - [[quantum-state-computational-01|Computational 01 (|01⟩)]]
    - [[quantum-state-computational-10|Computational 10 (|10⟩)]]
    - [[quantum-state-computational-11|Computational 11 (|11⟩)]]
  - [[quantum-state-bell|Bell states]]
    - [[quantum-state-bell-00|Bell 00 (|Φ⁺⟩)]]
    - [[quantum-state-bell-01|Bell 01 (|Ψ⁺⟩)]]
    - [[quantum-state-bell-10|Bell 10 (|Ψ⁻⟩)]]
    - [[quantum-state-bell-11|Bell 11 (|Φ⁻⟩)]]
- [[quantum-state-three-qubit|Three-qubit entangled states]]
  - [[quantum-state-ghz|GHZ state]]
  - [[quantum-state-w|W state]]
- [[quantum-state-multiqubit|Multi-qubit states (n-qubit general)]]
  - [[quantum-state-dicke|Dicke state]]
  - [[quantum-state-graph|Graph state]]
  - [[quantum-state-cluster|Cluster state]]
  - [[quantum-state-maximally-entangled|Maximally entangled state]]
  - [[quantum-state-choi|Choi state]]