= Solution
{c}
The <Pauli multiplication identity> gives $(\mathbf s\cdot\boldsymbol\sigma)^2=|\mathbf s|^2I$. Its <eigenvalues> are therefore $\pm|\mathbf s|$, and those of the <density matrix> are
$$
\lambda_\pm=\frac{1\pm|\mathbf s|}{2}.
$$
Hermiticity and trace one are automatic; <positive semidefiniteness> holds exactly for $|\mathbf s|\leq1$. Thus physical qubit states form the <Bloch ball>, with <pure states> on the <Bloch sphere> and mixed states inside.
For the mixture of the two standard-basis states,
$$
\boxed{\rho=\begin{pmatrix}p_0&0\\0&p_1\end{pmatrix},\qquad
\mathbf s=(0,0,p_0-p_1),\quad p_0+p_1=1.}
$$
These form the entire $z$-axis diameter, shown in the left panel of the diagram in part <1/b>. Generate the state by selecting $|0\rangle$ with classical <probability> $p_0$ and $|1\rangle$ with probability $p_1$, then discarding the classical selection record. A coherent <superposition> is generally a different state: the forgotten classical record removes off-diagonal coherence.
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