Traditional quantum physics solves ground states for a given Hamiltonian,

while quantum information science asks for the existence and construction of

certain Hamiltonians for given ground states. In practical situations, one

would be mainly interested in local Hamiltonians with certain interaction

patterns, such as nearest neighbour interactions on some type of lattices. A

necessary condition for a space $V$ to be the ground-state space of some local

Hamiltonian with a given interaction pattern, is that the maximally mixed state

supported on $V$ is uniquely determined by its reduced density matrices

associated with the given pattern, based on the principle of maximum entropy.

However, it is unclear whether this condition is in general also sufficient. We

examine the situations for the existence of such a local Hamiltonian to have

$V$ satisfying the necessary condition mentioned above as its ground-state

space, by linking to faces of the convex body of the local reduced states. We

further discuss some methods for constructing the corresponding local

Hamiltonians with given interaction patterns, mainly from physical points of

view, including constructions related to perturbation methods, local

frustration-free Hamiltonians, as well as thermodynamical ensembles.