> d$ the vectors ${|\Psi_j\rangle}$ cannot be mutually orthogonal. As a simple example we consider a system with Hilbert space ${\mathbb C}^2$ and the following vectors: $${|\Psi_1\rangle} = \sqrt{\frac{1}{3}}
\left(\begin{array}{c}1\\1\end{array}\right), \;\; {|\Psi_2\rangle} =
\sqrt{\frac{1}{3}}\left(\begin{array}{c}1\\ \omega
\end{array}\right) \;{\rm and} \;\; {|\Psi_3\rangle}=\sqrt{\frac{1}{3}}
\left(\begin{array}{c}1 \\ \omega^2\end{array} \right).$$ Here is $\omega:= {\rm exp}(-2\pi i /3)$ a third root of unity. We therefore have $$A_1 = \frac{1}{3}\left( \begin{array}{cc}1 & 1 \\ 1 & 1 
\end{array} \right), \; \; A_2=\frac{1}{3} \left( \begin{array}{cc}1 & 
\omega^2 \\ \omega & 1 \end{array} \right)\; {\rm and} \;\; A_3= 
\frac{1}{3}\left( 
\begin{array}{cc}1 & \omega \\ \omega^2 & 1 \end{array} \right)$$ as POVM operators. In Section \[Section 2.1\] we consider a generalization of this POVM.

Assuming orthogonal measurements as basic measurements, we have to extend the system by at least $n-d$ dimensions to make a measurement with $n$ different measurement outcomes possible. In order to simplify notation, we consider the given system with $d$ dimensions as a subsystem of a system with $n$ dimensions. Since we are interested in quantum circuits we have to embed the system into a qubit register. This can be done by assuming that the POVM consists of $n=2^l$ operators. Note that this is no loss of generality since 