Entanglement in Valence-Bond-Solid States

TL;DR

This review explores quantum entanglement in Valence-Bond-Solid states, focusing on their mathematical structure, entanglement measures, and implications for condensed matter physics and quantum computation.

Contribution

It provides a comprehensive, pedagogical overview of entanglement in VBS states, including explicit calculations of entanglement entropy and the density matrix spectrum.

Findings

Density matrix is proportional to a projector.

Entanglement entropy approaches finite limits for large blocks.

Diagonalization of density matrices yields insights into entanglement structure.

Abstract

This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Renyi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.