= Solution
{c}
The six <Pauli eigenstates> have <Bloch vectors> $\pm\mathbf e_x$, $\pm\mathbf e_y$, $\pm\mathbf e_z$:
$$
\begin{aligned}
X:&\quad |\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2,\\
Y:&\quad |\pm i\rangle=(|0\rangle\pm i|1\rangle)/\sqrt2,\\
Z:&\quad |0\rangle,\ |1\rangle.
\end{aligned}
$$
Classical <convex combinations> of <density operators> produce the corresponding <convex combinations> of their <Bloch vectors>. Their <convex hull> is the <stabilizer octahedron>:
$$
\boxed{|s_x|+|s_y|+|s_z|\leq1.}
$$
The six vertices are pure Pauli states; the faces and interior represent their mixtures. To verify the reverse inclusion, give the vertex in the sign of each coordinate weight $|s_j|$ and use the leftover weight at the origin, itself the equal mixture of either opposite pair.
\Image[/past-exam-of-the-introduction-to-quantum-information-course-of-the-university-of-oxford/2021-bloch-mixtures.png]
{title=The standard-basis mixture diameter and the <stabilizer octahedron> inside the <Bloch ball>}
{description=The red diameter shows all mixtures of $|0\rangle$ and $|1\rangle$. The six Pauli vertices generate the blue octahedron. The gold point is the T-gate resource state: it lies on the Bloch sphere but outside this octahedron.}
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