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

TL;DR

<3-5 sentence high-level summary>Matrix Product States and Projected Entangled Pair States provide a unifying tensor-network framework in which ground states of local Hamiltonians are efficiently describable when entanglement satisfies area laws. The review develops the formal theory, including fundamental theorems, bulk-boundary correspondences, and symmetry classifications (SPT/SET) in 1D and 2D, with extensions to fermionic, continuous, and infinite tensor networks. It highlights how transfer matrices encode correlations and entanglement spectra, how RG fixed points and MPO-injectivity characterize topological order, and how bulk physics is reflected at boundaries via entanglement Hamiltonians. The work connects these tensor-network structures to RG flows, excitations, and holographic ideas, underlining their central role in understanding phases of matter and facilitating efficient simulations of strongly correlated systems.

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.

Figures (14)

• Figure 1: Construction of projected entangled pair states on a 2D lattice.
• Figure 2: Tensor network description of an MPS (a), a PEPS on a square lattice (b), and their corresponding marginals.
• Figure 3: Definition of (a) Matrix Product Operators (MPO) and (b) Projected Entangled Pair Operators (PEPO)
• Figure 4: Sufficient condition for a translationally invariant MPO $\rho$ to be positive.
• Figure 5: (a) Tensors generating an MPU after blocking. (b) the MPU can be described as a quantum circuit, with alternating layers of unitary operators $u$ and $v$ acting on nearest neighbors withe even-odd indices or odd-even indices, respectively.

Theorems & Definitions (21)

• Definition 4.1
• Definition 4.2
• Proposition 4.3
• Theorem 4.4: Fundamental Theorem for proportional MPVs
• Corollary 4.5: Fundamental Theorem for equal MPVs
• Theorem 4.6
• Definition 4.7
• Definition 4.8
• Theorem 4.9: Fundamental Theorem for normal PEPS
• Corollary 4.10