GHZ State

GHZ state $|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ is a three-qubit maximally entangled state. A generalization of the Bell state to three qubits, it exhibits global entanglement where no qubit can be separated as independent of the others.

Representation: GHZ state has two basis components with equal amplitude and zero amplitude for all others: $$|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$$

• Maximally entangled: all three qubits are entangled together
• All-or-nothing structure: measuring all three qubits yields either 000 or 111 with equal probability
• Symmetric: invariant under permutations of qubits
• Stabilized by $Z_1 Z_2$ and $Z_2 Z_3$ (and $X_1 X_2 X_3$)
• Fragile: loss of one qubit destroys all three-party entanglement

Create a Bell pair on first two qubits, then apply CNOT from first to third:

1. Apply $H \otimes I \otimes I$ to prepare superposition
2. Apply CNOT (first to second)
3. Apply CNOT (first to third) Result: $|GHZ\rangle$

• Z basis: all three measured simultaneously yields 000 or 111 with probability $1/2$ each
• Parity measurement: measuring $Z_1 Z_2$ or $Z_2 Z_3$ yields +1 (both 000 and 111 have even parity)
• Single qubit measurement: measuring one qubit gives 0 or 1 with equal probability, but projects the other two into a Bell state

• Quantum error correction: GHZ-type states detect certain errors
• Quantum metrology: GHZ state enables enhanced phase measurement precision
• Entanglement verification: GHZ states violate Bell inequalities more strongly than Bell pairs
• Quantum computing: used in quantum algorithms for error correction and quantum teleportation

Unlike the W state, which distributes entanglement evenly, the GHZ state is “all or nothing”—measuring one qubit collapses the entire state. The W state is more robust to loss of a single qubit.