According to quantum mechanics, bosons can occupy the same quantum state in the same system but fermions cannot because the wave function for the interchange of two identical fermions would be asymmetric, according to the Pauli exclusion principle.
I was wondering what is the definition of such a system and its reach, e.g. I can have a free fermion at LHC and another one at LANL and both of them can be at the same state (because there is no quantum entanglement?), but in a neutron star, fermion degeneracy, i.e. that all these neutrons cannot occupy the same state, is what is giving the neutron star its particular characteristics.
So the question is: what is the formal definition of a system? When are these fermions entangled and cannot occupy the same state and when aren't they?