Convergent renormalization trajectories of Hopf-algebra boundary MPDOs under on-site noise are classified by finite *-quantum hypergroups via a new quantum Goursat lemma.
Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states
4 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
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-trough 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.
representative citing papers
Any weakly integral fusion category admits a QCA-refined realization on tensor-product Hilbert spaces with QCA and symmetry indices fixed by the categorical data under defect assumptions.
The paper shows phase equivalence for two 1D mixed-state phases connected by a phase transition by constructing low-depth channel circuits from parent Lindbladians, generalizing the analysis to intrinsically nontrivial cases beyond fixed points.
Proposes continuous matrix product operators for QFT with closed-form matrix-function expressions from continuum limits of MPOs that preserve area-law entanglement and enable new continuous unitaries beyond quantum cellular automata.
citing papers explorer
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Renormalization flows for 1D mixed states and a quantum Goursat lemma
Convergent renormalization trajectories of Hopf-algebra boundary MPDOs under on-site noise are classified by finite *-quantum hypergroups via a new quantum Goursat lemma.
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Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata
Any weakly integral fusion category admits a QCA-refined realization on tensor-product Hilbert spaces with QCA and symmetry indices fixed by the categorical data under defect assumptions.
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Establishing Mixed-State Phase Equivalence beyond Renormalization Fixed Points
The paper shows phase equivalence for two 1D mixed-state phases connected by a phase transition by constructing low-depth channel circuits from parent Lindbladians, generalizing the analysis to intrinsically nontrivial cases beyond fixed points.
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Continuous matrix product operators for quantum fields
Proposes continuous matrix product operators for QFT with closed-form matrix-function expressions from continuum limits of MPOs that preserve area-law entanglement and enable new continuous unitaries beyond quantum cellular automata.