{\rm
and} \; \; \left( \begin{array}{cc}0&1\\1&0\end{array}\right)$$ where $\omega:={\rm exp}(-2\pi i/m)$ is a $m$th complex root of unity. We can define a projective representation of the group $D_m$ by mapping the element $r \in D_m$ to the first matrix and the element $s \in D_m$ to the second matrix.

We consider the orbit of a vector under the action of the dihedral group $D_m$. Let $(\alpha, \beta)^T \in {\mathbb C}^2$ with $|\alpha|^2 + |\beta|^2 =1$. Since a global phase factor of a vector is physically irrelevant we assume $\alpha \in {\mathbb R}$ without loss of generality. Under the action of the dihedral group the orbit contains the vectors $(\alpha,\beta \omega^j)^T$ and $(\beta, \alpha
\omega^{j})^T$ with $j \in \{0, \ldots , m-1\}$. An example of the orbit is shown in Figure \[Figur diederpunkte\].

We have at most $n=2m$ vectors. In the following, we assume that the orbit contains $2m$ elements. If the orbit of $D_m$ contains less than $2m$ points we have either the case that all points are on the $xy$-plane (and the POVM consists of a single orbit under the group $C_m$) or we have only the two points $(1,0)^T$ and $(0,1)^T$ defining an orthogonal measurement. Since $$\sum_j \left(
\begin{array}{c} 