\alpha \\ \beta \omega^j \end{array} \right) (\alpha,
{\overline \beta} \omega ^{-j}) + \sum_j \left(
\begin{array}{c} \beta \\ \alpha \omega^j \end{array} \right)
({\overline \beta}, \alpha \omega^{-j}) = m I_2,$$ we rescale $\alpha$ and $\beta$ with the factor $\sqrt{1/m}$ to obtain a POVM.

We now consider the implementation of the dihedral POVM. In order to analyze the structure, we do not consider the embedding of the constructed system into a qubit register in the first place. The orbit under the action of $D_m$ breaks into two orbits under the action of the subgroup $C_m$. The two orbits can be obtained by the action of $C_m$ on the vectors $(\alpha, \beta)^T$ and $(\beta,
\alpha)^T$. Therefore, we expect to obtain implementations of the dihedral POVMs which are similar to the implementations in the previous section. With an appropriate order of the vectors we have the matrix $$M= \left(
\begin{array}{cccc|cccc} \alpha & \alpha & \ldots & \alpha
& \beta & \beta & \ldots & \beta \\ \beta & \beta \omega & \ldots &
\beta \omega^{m-1} & \alpha & \alpha \omega & \ldots & \alpha
\omega^{m-1} \end{array} \right) \in {\mathbb C}^{2 \times 2m}.$$ For even $m$, this matrix can be extended to the unitary matrix $${\tilde 