Entanglement of the valence-bond-solid state on an arbitrary graph

Abstract

Abstract. The Affleck-Kennedy-Lieb-Tasaki (AKLT) spin interacting model can be defined on an arbitrary graph. We explain the construction of the AKLT Hamiltonian. Given certain conditions, the ground state is unique and known as the Valence-BondSolid (VBS) state. It can be used in measurement-based quantum computation as a resource state instead of the cluster state. We study the VBS ground state on an arbitrary connected graph. The graph is cut into two disconnected parts: the block and the environment. We study the entanglement between these two parts and prove that many eigenvalues of the density matrix of the block are zero. We describe a subspace of eigenvectors of the density matrix corresponding to non-zero eigenvalues. The subspace is the degenerate ground states of some Hamiltonian which we call the block Hamiltonian.

Entanglement in valence-bond-solid states on symmetric graphs

“…The ground state is unique and known as the VBS state. Although there are several possible representations for the VBS state (see [90,100]), the most convenient one for our purposes is the Schwinger boson representation [42,49,50]. In this representation, we introduce a pair of bosonic creation and annihilation operators at each vertex to realize SU(2) Lie algebra.…”

“…For the construction of the generalized AKLT Hamiltonian and the condition of the uniqueness of the ground state, see [49,50,90,100]. In this paper, we shall focus on the basic model, i.e., M k,l = 1 for any k, l .…”

“…This gives an explicit proof of the area law in this system (see [99] for a review of area laws and the entanglement entropy). General entanglement properties of the VBS state on an arbitrary graph [37,49,50] have been studied in [100]. The authors showed that the eigenspace of the reduced density matrix of the block is spanned by the degenerate ground states of the block Hamiltonian.…”