Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 3 / a / Solution

Use the source's north-pole assignment $|1⟩\leftrightarrow (0,0,1)$. Fix the ordered basis as $(|1⟩,|0⟩)$ and use the standard Pauli matrices in that order, soA pure state has $|\mathbf{r}|=1$: examples are $|1⟩$ at the north pole and $(|1⟩+|0⟩)/\sqrt{2}$ at $(1,0,0)$. An incoherent mixed statehas $\mathbf{r}=(0,0,2p-1)$ 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.