  The structure of the ground spaces of quantum systems consisting of local
interactions is of fundamental importance to different areas of physics. In
this Letter, we present a necessary and sufficient condition for a subspace to
be the ground space of a k-local Hamiltonian. Our analysis are motivated by the
concept of irreducible correlations studied by [Linden et al., PRL 89, 277906]
and [Zhou, PRL 101, 180505], which is in turn based on the principle of maximum
entropy. It establishes a better understanding of the ground spaces of local
Hamiltonians and builds an intimate link of ground spaces to the correlations
of quantum states.
