Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

TL;DR

This paper reviews tensor network states, specifically matrix product states and projected entangled pair states, explaining their role in describing entanglement, symmetries, and topological order in many-body quantum systems.

Contribution

It provides a comprehensive overview of the mathematical foundations, symmetries, and applications of MPS and PEPS in understanding complex quantum entanglement patterns.

Findings

Tensor networks encode entanglement and serve as non-local order parameters.

Symmetries in tensor networks reflect global entanglement patterns.

Mathematical theorems underpin the structure and applications of MPS and PEPS.

Abstract

The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. We review how matrix product states and projected entangled pair states describe many-body wavefunctions in terms of local tensors. These tensors express how the entanglement is routed, act as a novel type of non-local order parameter, and we describe how their symmetries are reflections of the global entanglement patterns in the full system. We will discuss how tensor networks enable the construction of real-space renormalization group flows and fixed points, and examine the entanglement structure of states exhibiting topological quantum order. Finally, we provide a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications.