  Frustration of classical many-body systems can be used to distinguish
ferromagnetic interactions from anti-ferromagnetic ones via the Toulouse
conditions. A quantum version of the Toulouse conditions provides a similar
classification based on the local ground states. We compute the global ground
states for a family of models with Heisenberg-like interactions and analyse
their behaviour with respect to frustration, entanglement and degeneracy. For
that we develop analytical and numerical analysing tools capable to quantify
the interplay between those three quantities. We find that the quantum Toulouse
conditions provide a proper classification, however, refinements can be found.
Our results show how the different local ground states affect the interplay and
pave the way for further generalisation and possible applications to other
quantum many-body systems.
