These considerations are the basis of the implementations of POVMs corresponding to platonic solids. The implementation of these POVMs is discussed in Sections \[S 4\]–\[S ico\].

Orthogonal measurement of POVMs {#Sektion ortho}
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In this section we briefly rephrase Neumark’s theorem describing the reduction of POVMs to orthogonal measurements [@Peres]. This theorem allows to implement POVMs by performing unitary transformations on the joint system consisting of the system to be measured and an ancilla register. The unitary transformations are followed by an orthogonal measurement in the computational basis.

Let $(A_j)$ with $j \in \{1, \ldots , n \}$ be a POVM with corresponding Hilbert space ${\mathbb C}^d$ where each $A_j={|\Psi_j\rangle} {\langle \Psi_j|} \in {\mathbb C}^{d \times d}$ is a positive operator of rank one. Due to the properties of POVMs we have $\sum_j A_j =I_d$ where $I_d$ denotes the identity matrix of size $d$. The choice of corresponding vectors ${|\Psi_j\rangle}$ is not unique since we can multiply each ${|\Psi_j\rangle}$ with a phase factor that is physically irrelevant. It is therefore reasonable to choose the phase factors in such a way that the implementation of the POVM is simplified. Our constructions in Sections 4–10 implicitly make use of this. For $n 