W state (three-qubit)

The three-qubit $W$ state is

$|W\rangle = \frac{1}{\sqrt{3}}\big(|100\rangle + |010\rangle + |001\rangle\big).$

Generalizing to $n$ qubits, $|W_n\rangle = \frac{1}{\sqrt n}\sum_{i=1}^{n} |0\cdots 0\,1_i\,0\cdots 0\rangle$ places the single excitation in a uniform superposition over all $n$ positions.

Entanglement structure. The $W$ state and the GHZ state $\frac{1}{\sqrt2}(|00\cdots0\rangle + |11\cdots1\rangle)$ are the two SLOCC-inequivalent classes of genuine tripartite entanglement identified by Dür, Vidal, and Cirac (2000): no local operations and classical communication, even applied probabilistically, can convert one into the other. Operationally this shows up in robustness — tracing out any single qubit of $|W\rangle\langle W|$ leaves the remaining two qubits in a mixed state with nonzero concurrence (residual pairwise entanglement survives), whereas tracing out one qubit of the GHZ state leaves the rest in a separable classical mixture $\frac12(|00\rangle\langle00| + |11\rangle\langle11|)$.

Construction. The state is prepared with a chain of controlled rotations rather than a single Hadamard: an $X$ gate flips the last qubit to $|1\rangle$, and a cascade of controlled-$R_y$ rotations with angles $\theta_k = 2\arccos(1/\sqrt{n-k+1})$ (for $n=3$: $\theta_1 = 2\arccos(1/\sqrt3)$, $\theta_2 = 2\arccos(1/\sqrt2) = \pi/2$) redistributes the excitation probability evenly across the remaining qubits, followed by CNOTs that copy the "excitation present" flag outward. Because every rotation angle and CNOT target is fixed by the recursion, the resulting amplitudes are exact closed-form values rather than estimates.

Significance. $W$ states appear as the natural output of single-photon-loss-tolerant encodings, in quantum secret sharing, and as the ground state of the single-excitation sector of the ferromagnetic Heisenberg model (the state is a highest-weight total-spin eigenstate), making it a bridge between circuit-model entanglement and condensed-matter spin physics.