From ground states to local Hamiltonians

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-neighbor interactions on some types 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.

Figure 1

The convex set of points given by $[x=\mathrm{Tr}\left(\rho H_1\right),y=\mathrm{Tr}\left(\rho H_2\right)]$. The convex set is the union of two half disks on the left and right and a rectangle in the middle. Points $A,B,C,D$ are by definition faces of this convex set, yet for each point there is no line that touches the convex set only at the point.

Figure 2

$H_W$ is a line that touches the convex set ${\mathcal{D}}_{\mathcal{K}}$ at the top line segment $F_W$, while $H_U$ goes through point $F_V$, but does not contain points in $F_W\setminus F_V$.