= Solution
The normalized <singlet state> has column vector $(0,1,-1,0)^{\mathsf T}/\sqrt2$ in the specified product basis. Its <outer product> is
$$
\boxed{\rho_D=\frac12\begin{pmatrix}0&0&0&0\\0&1&-1&0\\0&-1&1&0\\0&0&0&0\end{pmatrix}.}
$$
It obeys $\rho_D^2=\rho_D$ and has <eigenvalues> $(1,0,0,0)$, hence <purity of a density operator> one. The joint two-atom system is therefore a <pure state>, even though the <quantum entanglement> prevents assigning a separate pure spin state to each atom.
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