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Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states
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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.
Forward citations
Cited by 8 Pith papers
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Two sequentially generated locally purifiable density operators describe the same state for all system sizes exactly when their purification tensors are linked by a matrix product isometry, under step-injective or cyc...
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Trading Mathematical for Physical Simplicity: Bialgebraic Structures in Matrix Product Operator Symmetries
MPO symmetries are governed by pre-bialgebras; the anomalous Z2 symmetry of the XX model yields a non-semisimple representation category and new mixed-state fixed points.
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Spectral Gap Bounds for Quantum Markov Semigroups via Correlation Decay
A purified canonical Hamiltonian connects correlation decay of a quantum state to spectral gaps of Davies generators, yielding size-independent gap bounds for 1D chains and quantum double models.
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Strange correlator and string order parameter for non-invertible symmetry protected topological phases in 1+1d
Strange correlators and string order parameters for non-invertible SPT phases in 1+1d are systematically constructed from the interface algebra, giving local detectors of these phases.
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Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry
Anomaly-free fusion category symmetries have a canonical trivial phase, and the three Rep†(D8) symmetry-protected topological phases are explicitly realized by Q-system lattice models connected by an S3 duality.
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Les Houches Lecture Notes on Tensor Networks
A well-organized five-lecture review of tensor networks (MPS/PEPS/MPO) covering algorithms, phase classification, string-nets, strange correlators, and dualities; it contains no new research results.
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