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

By I. Cirac, David Pérez-García, Norbert Schuch, Frank Verstraete. Reviews of Modern Physics, 2020

848 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/bdabd4b0b90b05fdd1be4ae32648ea58e93eec62

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Citations in the corpus

Measurement as a Shortcut to Long-Range Entangled Quantum Matter (2022, background): “Ref.[15] for a review) and serves as a practical, modular approach for preparing many topologically ordered states in constant depth.”
Generalized Symmetries in Condensed Matter (2022, neutral): “9A related perspective appears via the ‘pulling-though’ operators in the tensor network description of topological orders reviewed in (94).”
Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases (2021, background): “In particular, one can ‘push’ physical symmetries through to the ‘virtual’ level 12 [3, 22, 25, 92, 124]: There exists an operator V g such that In other words, we see that the physical operator U g is equivalent to acting with V g and V † g at the virtual level.”