Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata

• URL: http://arxiv.org/abs/2605.15194v1
• Date: Thu, 14 May 2026 17:59:45 GMT
• Title: Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata
• Abstract summary: We investigate realizations of (1+1)-dimensional fusion category symmetries on tensor-product Hilbert spaces, allowing for mixing with quantum cellular automata (QCAs)<n>First, we show that, under certain physical assumptions on defects, any QCA-refined realization has QCA and symmetry-operator indices determined by the categorical data, up to the freedom of redefining the symmetry operators.<n>Second, we construct a lattice model that provides a QCA-refined realization for any weakly integral fusion category symmetry on a tensor product Hilbert space.
• Score: 1.7154631869844845
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We investigate realizations of (1+1)-dimensional fusion category symmetries on tensor-product Hilbert spaces, allowing for mixing with quantum cellular automata (QCAs). It was argued recently that any such realizable symmetry must be weakly integral. We develop a systematic analysis of QCA-refined realizations of fusion categories and prove two statements. First, we show that, under certain physical assumptions on defects, any QCA-refined realization has QCA and symmetry-operator indices determined by the categorical data, up to the freedom of redefining the symmetry operators. Second, we construct a lattice model that provides a QCA-refined realization for any weakly integral fusion category symmetry on a tensor product Hilbert space. We also compute indices of the QCAs in our lattice model and show agreement with the first result. As an application of the general construction, we give an explicit QCA-refined realization of general Tambara-Yamagami categorical symmetries.

Related papers

• Typical entanglement in anyon chains: Page curves beyond Lie group symmetries [41.99844472131922]
  We study bipartite entanglement statistics in one-dimensional anyon chains constrained by fusion rules of unitary pre-modular categories.<n>Surprisingly, despite the constrained Hilbert space structure, the large $L$ expansion has no universal $O(sqrtL)$ or $O(1)$ symmetry-type corrections.<n> Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions.
  arXiv  Detail & Related papers  (2026-03-26T18:00:03Z)
• Remarks on non-invertible symmetries on a tensor product Hilbert space in 1+1 dimensions [0.0]
  We propose an index of non-invertible symmetry operators on the tensor product of finite dimensional on-site Hilbert spaces on the lattice.<n>Our index generalizes the Gross-Nesme-Vogts-Werner index of invertible symmetry operators represented by quantum cellular automata.
  arXiv  Detail & Related papers  (2026-02-12T15:17:58Z)
• Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities [0.0]
  Wigner's theorem requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities.<n>We propose that instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in $mathcalC$ act as isometries between distinct Hilbert spaces constructed from twisted sectors.
  arXiv  Detail & Related papers  (2026-02-06T19:00:00Z)
• Algebraic Fusion in a (2+1)-dimensional Lattice Model with Generalized Symmetries [0.04077787659104315]
  We develop a framework for deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries.<n>We explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.
  arXiv  Detail & Related papers  (2025-12-24T22:01:15Z)
• Anomalies on the Lattice, Homotopy of Quantum Cellular Automata, and a Spectrum of Invertible States [41.99844472131922]
  We develop a rigorous theory of anomalies on the lattice, which are obstructions to gauging global symmetries and the existence of trivial symmetric states.<n>We also construct $$-spectra of a class of invertible states and quantum cellular automata, which allows us to classify both anomalies and symmetry protected topological phases up to blend equivalence.
  arXiv  Detail & Related papers  (2025-12-01T19:00:01Z)
• Trading Mathematical for Physical Simplicity: Bialgebraic Structures in Matrix Product Operator Symmetries [20.76275069383104]
  We show that simple quantum spin chains of physical interest are not included in the rigid framework of fusion categories and weak Hopf algebras.<n>We show that our work provides a bridge between well-understood topological defect symmetries and those that arise in more realistic models.
  arXiv  Detail & Related papers  (2025-09-03T18:01:22Z)
• Predicting symmetries of quantum dynamics with optimal samples [41.42817348756889]
  Identifying symmetries in quantum dynamics is a crucial challenge with profound implications for quantum technologies.<n>We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency.<n>We prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols.
  arXiv  Detail & Related papers  (2025-02-03T15:57:50Z)
• Hilbert space geometry and quantum chaos [39.58317527488534]
  We consider the symmetric part of the QGT for various multi-parametric random matrix Hamiltonians. We find for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect.
  arXiv  Detail & Related papers  (2024-11-18T19:00:17Z)
• Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
  We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
  arXiv  Detail & Related papers  (2024-02-26T06:23:39Z)
• Electric-magnetic duality and $\mathbb{Z}_2$ symmetry enriched Abelian lattice gauge theory [2.206623168926072]
  Kitaev's quantum double model is a lattice gauge theoretic realization of Dijkgraaf-Witten topological quantum field theory (TQFT) Topologically protected ground state space has broad applications for topological quantum computation and topological quantum memory.
  arXiv  Detail & Related papers  (2022-01-28T14:13:38Z)
• Generalized string-nets for unitary fusion categories without tetrahedral symmetry [77.34726150561087]
  We present a general construction of the Levin-Wen model for arbitrary multiplicity-free unitary fusion categories. We explicitly calculate the matrix elements of the Hamiltonian and, furthermore, show that it has the same properties as the original one.
  arXiv  Detail & Related papers  (2020-04-15T12:21:28Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.