In learning device-independent QM from [https://arxiv.org/abs/1303.3081], Scarani tells us that all quantum statistics involving one particle can be reproduced with local (hidden) variables. He then constructs the statistics of a qubit in such a model. Insightful as this is, it is still not clear to me why all single-particle statistics can be reproduced. Is it because the local variable can hold information about the single-particle state? Why can't it do the same in a many-particle system?