# 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 [[quantum-state-bell-00|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:
$$|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$$

## Properties

- 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

## Creation

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: $|GHZ\rangle$

## Measurement

- 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 Information Applications

- **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

## Relationship to W State

Unlike the [[quantum-state-w|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.