The W State

The W state spreads a single excitation across many qubits and keeps its entanglement robust under particle loss.

Definition

The three-qubit W state is W = (|100> + |010> + |001>)/sqrt(3): exactly one qubit excited, in an equal superposition over which one. The n-qubit generalization is the symmetric single-excitation state. Unlike the GHZ state, the entanglement of the W state is distributed pairwise and is far more robust to the loss of any single party.

Robustness under loss

Trace out one qubit of a GHZ state and the rest become unentangled. Trace out one qubit of a W state and the remaining two are still entangled. This durability makes the W state a distinct entanglement class: no local operations with classical communication can turn a W state into a GHZ state or vice versa, even probabilistically. Three-qubit pure entanglement splits into exactly these two inequivalent families.

import numpy as np
w = np.zeros(8)
for i in (4,2,1): w[i] = 1/np.sqrt(3)  # |100>,|010>,|001>
# reduced 2-qubit state remains entangled after tracing one out
m = w.reshape(2,4)
rhoBC = m.conj().T @ m
print(np.round(np.diag(rhoBC),3))

Monogamy connection

The W state saturates the pairwise sharing allowed by the monogamy of entanglement. It maximizes the total two-qubit entanglement among three parties, distributing correlation as evenly as possible rather than concentrating it in genuine tripartite form. GHZ, by contrast, has zero pairwise concurrence but maximal three-way entanglement.

Uses

W states are useful where robustness matters: leader election and other distributed tasks, memory schemes tolerant of qubit loss, and single-photon-based network states where one excitation is shared across nodes. They arise naturally in systems with a conserved excitation number, such as coupled cavities or spin chains in the single-magnon sector, making them experimentally accessible in several platforms.