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We study the projection on classical spins starting from quantum equilibria.We show Gibbsianness or quasi-locality of the resulting classical spin systemfor a class of gapped quantum systems at low temperatures including quantumground states. A consequence of Gibbsianness is the validity of a largedeviation principle in the quantum system which is known and here recovered inregimes of high temperature or for thermal states in one dimension. On theother hand we give an example of a quantum ground state with strong nonlocalityin the classical restriction, giving rise to what we call measurement inducedentanglement, and still satisfying a large deviation principle.

Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement