From Ground States to Local Hamiltonians

TL;DR

This paper investigates the conditions under which a subspace can be the ground-state space of a local Hamiltonian with specific interaction patterns, linking geometric properties of reduced states to Hamiltonian construction methods.

Contribution

It provides a geometric framework for understanding when a subspace can be realized as a ground state of local Hamiltonians, and discusses construction methods from physical perspectives.

Findings

Necessary condition for ground-state space based on reduced states and maximum entropy.

Link between convex geometry of reduced states and existence of local Hamiltonians.

Methods for constructing local Hamiltonians including perturbation and frustration-free approaches.

Abstract

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.