= Solution
{c}
Use the source's north-pole assignment $|1\rangle\leftrightarrow(0,0,1)$. Fix the ordered <basis> as $(|1\rangle,|0\rangle)$ and use the standard <Pauli matrices> in that order, so
$$
\rho=\tfrac12(I+\mathbf r\cdot\boldsymbol\sigma),\qquad
\rho_{11}=\tfrac12(1+r_z).
$$
A <pure state> has $|\mathbf r|=1$: examples are $|1\rangle$ at the north pole and $(|1\rangle+|0\rangle)/\sqrt2$ at $(1,0,0)$. An incoherent <mixed state>
$$
\rho=p|1\rangle\langle1|+(1-p)|0\rangle\langle0|
$$
has $\mathbf r=(0,0,2p-1)$ inside the <Bloch sphere>. At $p=1/2$ it is the origin, distinct from an equatorial <pure state> despite equal populations. The distinction is the off-diagonal <quantum coherence> of the <density matrix>.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2021-bloch-trajectories.png]
{title=<Pure states>, an incoherent <mixed state>, and two complete <Ramsey interferometry> trajectories}
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