Entanglement of Valence-Bond-Solid on an Arbitrary Graph

TL;DR

This paper investigates the entanglement properties of Valence-Bond-Solid states derived from AKLT models on arbitrary graphs, revealing zero eigenvalues in the reduced density matrix and characterizing the non-zero eigenvalue subspace.

Contribution

It extends the understanding of AKLT VBS states to arbitrary graphs and characterizes their entanglement structure and eigenvalue spectrum.

Findings

Many eigenvalues of the density matrix are zero.

The non-zero eigenvalues correspond to the ground states of a block Hamiltonian.

The study applies to VBS states used in measurement-based quantum computation.

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-Bond-Solid (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.