Matrix multiplication

Quantum mechanics

Theoretical physics

Product (mathematics)

Pure mathematics

Cluster state

Squashed entanglement

Physics

Quantum entanglement

Mathematics

Separable state

Homogeneous space

Tensor product

Geometry

Quantum

Combinatorics

Quantum discord

Topology (electrical circuits)

Reviews of Modern Physics

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.

Matrix product states and projected entangled pair states: Concepts, symmetries, theorems