# Three-Qubit Entangled States

**Three-qubit entangled states** are fundamental examples of multi-qubit entanglement, exhibiting global correlations that cannot be reduced to two-qubit interactions. Three qubits represent the smallest system showing genuinely multi-partite entanglement, with distinct entanglement classes and rich measurement properties.

## Overview

The two main three-qubit entangled state families have very different structures:

- **[[quantum-state-ghz|GHZ state]]**: $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ — all qubits perfectly correlated, fragile
- **[[quantum-state-w|W state]]**: $\frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)$ — distributed entanglement, robust

## Entanglement Classification

Three-qubit entangled states fall into two inequivalent classes under local unitaries (LOCC):

### GHZ-Type Entanglement
- Example: [[quantum-state-ghz|GHZ state]] $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$
- Structure: all-or-nothing correlation; measuring any qubit determines all others
- Fragility: loss of one qubit breaks all entanglement
- Use: quantum metrology, quantum error correction, non-locality demonstrations

### W-Type Entanglement
- Example: [[quantum-state-w|W state]] $\frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)$
- Structure: distributed, symmetric superposition of single excitations
- Robustness: removing one qubit leaves two qubits entangled
- Use: quantum networks, distributed quantum computing

## Key Differences

| Property | GHZ | W |
|----------|-----|---|
| Correlations | All-or-nothing | Distributed |
| Measurement outcome | All 0s or all 1s | Exactly one 1 |
| Robustness to loss | Very fragile | Robust |
| Entanglement entropy (reduced) | Maximum (1 ebit) | Less than maximum |

## Measurement and Distinguishability

All three-qubit entangled states can be distinguished via collective measurements on all three qubits simultaneously. Local measurements (on individual qubits) cannot distinguish all entangled three-qubit states.

## Applications

- **Quantum teleportation**: GHZ state enables teleportation with only one qubit of entanglement per pair
- **Quantum error correction**: both GHZ and W patterns appear in stabilizer codes
- **Quantum networks**: W states distribute entanglement; multiple network nodes share one excitation
- **Bell test violations**: three-qubit GHZ states violate Mermin inequalities
- **Quantum metrology**: GHZ states enable $\sqrt{3}$ sensitivity improvement over W states for phase estimation

## Generation Methods

### GHZ State
1. Initialize $|000\rangle$
2. Apply Hadamard to first qubit: $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes |00\rangle$
3. Apply CNOT (first → second), then CNOT (first → third): yields GHZ state

### W State
1. Initialize $|001\rangle$
2. Apply controlled-X gate (first two qubits): spreads excitation to first two qubits
3. Apply additional controlled interactions to create equal superposition of all single-excitation states

### Dicke States
Use symmetric state preparation protocols or direct state synthesis via gate sequences designed for fixed Hamming weight.

## Practical Considerations

- **Gate depth**: GHZ requires 2 CNOTs; W requires additional gates for equal superposition
- **Error sensitivity**: GHZ fragile to amplitude damping; W more robust to single-qubit errors
- **Measurement fidelity**: distinguishing three-qubit states requires three-qubit measurement correlations
- **Scalability preview**: patterns here generalize to $n$ qubits; understanding three-qubit cases guides multi-qubit design

## Relation to Multi-Qubit States

Three-qubit states serve as building blocks for larger systems:
- [[quantum-state-multiqubit|Multi-qubit states]] generalize GHZ and W structures to arbitrary $n$
- [[quantum-state-graph|Graph states]] on three vertices embed GHZ-like or W-like entanglement patterns depending on connectivity
- Generalized GHZ and W states for $n > 3$ follow same structure as three-qubit cases
- [[quantum-state-dicke|Dicke states]] generalize symmetric superposition patterns to arbitrary $n$

## Bell Inequalities and Non-Locality

Three-qubit states violate Bell-type inequalities (Mermin, GHZ-Mermin):
- GHZ state violates most strongly; measurement in specific bases yields perfect correlations
- W state violates some inequalities; more robust under local noise
- These violations certify genuine multi-partite entanglement

## Entanglement Witnesses and Detection

Entanglement witnesses are observables that detect entanglement:
- For GHZ: witness operators based on $XXX$ and $ZZZ$ correlations
- For W: witnesses based on single-excitation patterns

Measuring correlations in the correct bases reveals entanglement without full state tomography.