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 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 $|0\rangle$.

• 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

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.

• 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)

• 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

• 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