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Measuring the coset state $|g_0+K⟩$ in the group shift operator eigenbasis samples uniformly from the annihilator of a subgroup of a finite abelian group $K^{\perp }$. Indeed,so the character-sum cancellation lemma gives probability $|K|/|G|$ on $K^{\perp }$ and zero elsewhere. The modulus of $\chi (g_0)$ is one, so the offset has no effect on the distribution.

For a finite abelian group $G$, $U(h)|g⟩=|g+h⟩$ defines the regular representation on the basis labelled by $G$. The linear characters give the common eigenbasisCharacter orthogonality proves orthonormality, and changing variables by the translation in a group proves the eigenvalue formula.