Local Transformations and Long-Range Entanglement

Abstract

To understand the origin of the topological phenomena discussed in the previous chapters, we need a microscopic theory for topological order. It was realized that the key microscopic feature of topologically ordered systems is the existence of long-range many-body entanglement in the ground-state wave function. Useful tools from quantum information theory to characterize many-body entanglement are local transformations, including local unitary (LU) transformations and stochastic local (SL) transformations. In this chapter, we apply these tools to the study of gapped quantum phases and phase transitions and establish the connection between topological order and long/short-range entanglement. This allows us to obtain a general theory to study topological order and symmetry-breaking order within the same framework. This leads to a basic understanding of the structure of the full quantum phase diagram.