 Solution

For an unpolarized ensemble ${\rho }_i=I/2$. Every unitary gate satisfies $U(I/2)U^{†}=I/2$, soThe purity remains $1/2$. Equal populations here come with zero quantum coherence, unlike the state in (b). There is no Ramsey interferometry fringe: the two orthogonal components of any equal-weight ensemble decomposition give cancelling population responses. Quantum state preparation that reduces the ensemble entropy is needed for the usual polarized clock interrogation.

The original plot compares the three excited-state populations. The constant $1/2$ lines coincide despite the very different purities in (b) and (c).