= Conditional entanglement of the three-qubit W state
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 $\langle u|$, where $|u\rangle=a|0\rangle+b|1\rangle$. The unnormalized <tensor-product coefficient matrix> of the other pair is
$$
C_u=\frac1{\sqrt3}\begin{pmatrix}\overline b&\overline a\\\overline a&0\end{pmatrix},\qquad\det C_u=-\frac{\overline a^{\,2}}3.
$$
At least one <vector> in a complete measurement basis has $a\ne0$, 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>.
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