Matrix Product Operator Algebras I: Representations of Weak Hopf Algebras and Projected Entangled Pair States
6 affiliations
- 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
- 2Dpto. de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040 Madrid, Spain
- 3Instituto de Ciencias Matemáticas, Nicolás Cabrera 13-15, 28049 Madrid, Spain
- 4Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
- 5Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
- 6Munich Center for Quantum Science and Technology, Schellingstr. 4, 80799 München, Germany
Abstract
Matrix Product Operators (MPOs) are tensor networks representing operators acting on 1D systems. They model a wide variety of situations, including communication channels with memory effects, quantum cellular automata, mixed states in 1D quantum systems, or holographic boundary models associated to 2D quantum systems. A scenario where MPOs have proven particularly useful is to represent algebras of non-trivial symmetries. Concretely, the boundary of both symmetry protected and topologically ordered phases in 2D quantum systems exhibit symmetries in the form of MPOs. In this paper, we develop a theory of MPOs as representations of algebraic structures. We establish a dictionary between algebra and MPO properties which allows to transfer results between both setups, covering the cases of pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion of pulling-through algebras, which abstracts the minimal requirements needed to define topologically ordered 2D tensor networks from MPO algebras. We show, as one of our main results, that any semisimple pivotal weak Hopf algebra is a pulling-through algebra. We demonstrate the power of this framework by showing that they can be used to construct Kitaev's quantum double models for Hopf algebras solely from an MPO representation of the Hopf algebra, in the exact same way as MPO symmetries obtained from fusion categories can be used to construct Levin-Wen string-net models, and to explain all their topological features; it thus allows to describe both Kitaev and string-net models on the same formal footing.
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BibTeX
@article{molnar2026matrix,
title = {{Matrix Product Operator Algebras I: Representations of Weak Hopf Algebras and Projected Entangled Pair States}},
author = {András Molnár and Alberto Ruiz-de-Alarcón and José Garre-Rubio and Norbert Schuch and J. I. Cirac and David Pérez-Garcı́a},
journal = {Communications in Mathematical Physics},
year = {2026},
month = aug,
doi = {10.1007/s00220-026-05670-w},
url = {https://doi.org/10.1007/s00220-026-05670-w},
note = {CC BY},
}
RIS
TY - JOUR
TI - Matrix Product Operator Algebras I: Representations of Weak Hopf Algebras and Projected Entangled Pair States
AU - András Molnár
AU - Alberto Ruiz-de-Alarcón
AU - José Garre-Rubio
AU - Norbert Schuch
AU - J. I. Cirac
AU - David Pérez-Garcı́a
PY - 2026
DA - 2026/08/28
JO - Communications in Mathematical Physics
DO - 10.1007/s00220-026-05670-w
UR - https://doi.org/10.1007/s00220-026-05670-w
AB - Matrix Product Operators (MPOs) are tensor networks representing operators acting on 1D systems. They model a wide variety of situations, including communication channels with memory effects, quantum cellular automata, mixed states in 1D quantum systems, or holographic boundary models associated to 2D quantum systems. A scenario where MPOs have proven particularly useful is to represent algebras of non-trivial symmetries. Concretely, the boundary of both symmetry protected and topologically ordered phases in 2D quantum systems exhibit symmetries in the form of MPOs. In this paper, we develop a theory of MPOs as representations of algebraic structures. We establish a dictionary between algebra and MPO properties which allows to transfer results between both setups, covering the cases of pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion of pulling-through algebras, which abstracts the minimal requirements needed to define topologically ordered 2D tensor networks from MPO algebras. We show, as one of our main results, that any semisimple pivotal weak Hopf algebra is a pulling-through algebra. We demonstrate the power of this framework by showing that they can be used to construct Kitaev's quantum double models for Hopf algebras solely from an MPO representation of the Hopf algebra, in the exact same way as MPO symmetries obtained from fusion categories can be used to construct Levin-Wen string-net models, and to explain all their topological features; it thus allows to describe both Kitaev and string-net models on the same formal footing.
SN - 0010-3616
SN - 1432-0916
ER -