Greenberger-Horne-Zeilinger states and few-body Hamiltonians

TL;DR

This paper establishes conditions under which GHZ states can be eigenstates of Hamiltonians, showing degeneracy issues with low-body interactions and providing explicit examples of Hamiltonians with GHZ eigenstates.

Contribution

It derives general conditions for GHZ states as eigenstates and constructs specific low-body Hamiltonians with GHZ states as nondegenerate eigenstates.

Findings

Degeneracy unavoidable with 2-body interactions for more than 4 qubits

Explicit 4-qubit 2-body Hamiltonian with GHZ eigenstate

Explicit 5-qubit 3-body Hamiltonian with GHZ eigenstate

Abstract

The generation of Greenberger-Horne-Zeilinger (GHZ) states is a crucial problem in quantum information. We derive general conditions for obtaining GHZ states as eigenstates of a Hamiltonian. In general, degeneracy cannot be avoided if the Hamiltonian contains m-body interaction terms with m<=2 and a number of qubits strictly larger than 4. As an application, we explicitly construct a two-body 4-qubit Hamiltonian and a three-body 5-qubit Hamiltonian that exhibit a GHZ as a nondegenerate eigenstate.