Proposition: Destructive coarse-graining

Suppose mutually orthogonal projectors $P_k$ are refined into orthonormal rank-one projectors $P_{k\ell }$. Destructive fine readout followed by forgetting $\ell$ has residual block

It is independent of the chosen orthonormal basis within $\mathrm{Ran}{P}_k$.

Proof. Expand $P_{k}Z{P}_k$ in any orthonormal basis of its clock subspace. Partial trace removes all off-diagonal clock matrix units and sums the diagonal residual blocks.