# W State

**W state** $|W\rangle = \frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle)$ is a three-qubit maximally entangled state where the entanglement is distributed evenly. Unlike the [[quantum-state-ghz|GHZ state]], the W state is robust to loss of a single qubit.

Representation: W state has three equal-amplitude basis components:
$$|W\rangle = \frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle)$$

Each term has exactly one qubit in state $|1\rangle$ and two in state $|0\rangle$.

## Properties

- Maximally entangled: all three qubits are entangled
- Equal superposition of three basis states (not two like GHZ)
- Robust: measuring and losing one qubit still leaves the other two in an entangled state
- Symmetric: invariant under permutations of qubits
- Hamming weight conservation: always exactly one $|1\rangle$ among the three qubits

## Creation

One method:
1. Prepare $|001\rangle$
2. Apply a superposition operator to the first two qubits that creates equal amplitude for having $|1\rangle$ in any position

Alternatively: use a sequence of controlled rotations to create the three-term superposition.

## Measurement

- Z basis: measuring all three qubits yields exactly one 1 and two 0s (outcome from {001, 010, 100})
- Probability: each of the three outcomes has probability $1/3$
- Loss robustness: if one qubit is lost or measured, the remaining two-qubit state $\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ is still entangled (a Bell state)

## Applications

- **Quantum networks**: W state structure suits distributed quantum computing where qubit loss is realistic
- **Quantum error correction**: W-like states used in some code constructions
- **Bell inequality tests**: W states violate Bell inequalities differently than GHZ or Bell pairs
- **Cluster states**: W state is one pattern in graph states used for measurement-based quantum computing

## Comparison to GHZ

- **GHZ**: all-or-nothing, fragile to qubit loss (loses all entanglement)
- **W**: distributed, robust to qubit loss (retains some entanglement)
- **Entanglement type**: GHZ has tripartite entanglement; W has a different entanglement structure