The quantum no-broadcasting theorem has an analogue in modal quantum theory (MQT), a toy model based on finite fields. The failure of broadcasting in MQT is related to the failure of distributivity of the lattice of subspaces of the state space.
Cite
@article{arxiv.2304.13561,
title = {A no-broadcasting theorem for modal quantum theory},
author = {Phillip Diamond and Benjamin Schumacher and Michael D. Westmoreland},
journal= {arXiv preprint arXiv:2304.13561},
year = {2023}
}