W state ($|W_n⟩=|D_1^n⟩$)

The $n$-qubit W state is the Dicke state with one excitation:For three qubits it is $(|001⟩+|010⟩+|100⟩)/\sqrt{3}$. Its reduced density matrix on any one qubit is $\mathrm{diag}((n-1)/n,1/n)$, from the Dicke state recursion. For $n>1$ both diagonal entries are nonzero, so that qubit is entangled with the rest: a pure product state would have a pure marginal. A symmetric-arm reduction of an exchange Hamiltonian supplies a dynamical preparation from one initial excitation.

Every rank-one projective measurement of one qubit of the three-qubit W state has at least one nonzero-probability outcome whose other two qubits remain entangled. To see this, contract the measured factor against $⟨u|$, where $|u⟩=a|0⟩+b|1⟩$. The unnormalized tensor-product coefficient matrix of the other pair isAt least one vector in a complete measurement basis has $a\ne 0$, giving nonzero determinant and an entangled conditional branch. Its probability $(2|a{|}^2+|b{|}^2)/3$ is positive. This property survives local unitary changes of basis. The GHZ state lacks it because computational-basis measurement of one qubit leaves a product state on each branch, so the property also separates the states under local-unitary equivalence.