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.

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

• GHZ state: $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$: all qubits perfectly correlated, fragile
• W state: $\frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)$: distributed entanglement, robust

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

• Example: 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

• Example: 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

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.

• 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

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

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

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

• 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

Three-qubit states serve as building blocks for larger systems:

• Multi-qubit states generalize GHZ and W structures to arbitrary $n$
• 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
• Dicke states generalize symmetric superposition patterns to arbitrary $n$

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 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.