Dear Nikolay,
thank you for your comment. I’m not familiar with this particular representation, but if you’re referring to works such as https://arxiv.org/abs/2006.13727, then it seems to me that this is fundamentally different. Our aim is to discuss Leggett-Garg inequalities, and thus we need to be able to distinguish physical observables (observed and disturbed). Our probability vector representation allows for such a distinction. On the other hand, the MIC-POVM measurement cannot distinguish the observables since the POVM measurement provides all of the probabilities at once by measuring quantum devices with larger dimensions. Consequently, the MIC-POVM representation cannot be used to investigate assumptions underlying the formulation of Leggett-Garg inequalities such as noninvasive measurability.
Lastly, please note that the dimension of probability vectors associated with generic N-level qudit states in our construction is N(N^2-1). For more details, please refer to Sec. II of https://arxiv.org/abs/2312.16281.
All the best,
Sebastian
