# Bell States

**Bell states** are the four maximally entangled two-qubit states. They form a complete orthonormal basis for two-qubit Hilbert space and are fundamental to quantum communication, quantum cryptography, and quantum teleportation.

## The Four Bell States

1. **[[quantum-state-bell-00|Bell 00 (|Φ⁺⟩)]]**: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$
   - Correlated outcomes (00 or 11)
   - Eigenstate of $Z_1 Z_2$ with eigenvalue +1

2. **[[quantum-state-bell-11|Bell 11 (|Φ⁻⟩)]]**: $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$
   - Correlated outcomes (00 or 11)
   - Eigenstate of $Z_1 Z_2$ with eigenvalue +1

3. **[[quantum-state-bell-01|Bell 01 (|Ψ⁺⟩)]]**: $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$
   - Anti-correlated outcomes (01 or 10)
   - Eigenstate of $Z_1 Z_2$ with eigenvalue -1

4. **[[quantum-state-bell-10|Bell 10 (|Ψ⁻⟩)]]**: $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$
   - Anti-correlated outcomes (01 or 10)
   - Eigenstate of $Z_1 Z_2$ with eigenvalue -1

## Properties

All four Bell states are:
- Maximally entangled: non-factorable, represent maximum correlation
- Eigenstates of certain Pauli products: distinguishable via measurement operators
- Equal superposition: two basis states with equal amplitude
- Pure states: no mixing, zero purity loss

## Bell Measurement

A Bell measurement distinguishes all four states by measuring two-qubit observables. This requires entangling the qubits via gates before measurement or using post-selected measurement schemes in photonic systems.

## Applications

- **Quantum teleportation**: Bell pair enables teleportation of unknown qubit state
- **Superdense coding**: Bell pair allows sender to encode two classical bits by manipulating one qubit
- **Quantum cryptography**: Bell states used in E91 protocol for quantum key distribution
- **Entanglement swapping**: combining Bell states extends entanglement across quantum networks
- **Bell inequality tests**: Bell states violate classical inequalities demonstrating non-locality

## Creation

Create Φ⁺ (most common):
1. Apply Hadamard to first qubit of $|00\rangle$: $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes |0\rangle$
2. Apply CNOT with first as control: yields $|\Phi^+\rangle$

Other Bell states created by applying single-qubit rotations (Z or X gates) before or after CNOT.

## Relation to Multi-Qubit Entanglement

Bell states are building blocks for larger entangled states. Two Bell pairs can be swapped to extend entanglement; Bell states generalize to GHZ, W, and cluster states for multi-qubit systems.