We consider the following scenario. We are given a quantum state $\rho$, which is split into $n$ different operators, and then leads us to a partial quantum state $Mb$ (where $b$ can be seen as an element of $(Z/dZ)^n$).
The different operators can be seen as projective representations of finite dimension for the group $(Z/dZ)^n$, that will be denoted $Ub$. To give a more easily understandable (and a more pratical associated experiment), we only study the case $n=2$.
We want to understand how our operators act when the quantum states $Mb$ that we observe correspond to dimension 1 projectors on the space $C^d\times C^d$, such that the operators $Ub$ permute the states $Mb$. In this particular case, one can describe explicitly the operators $Ub$, by proving that they can be described by a projective irreducible representation of $(Z/dZ)^2$, which can be easily classified.