{"paper":{"title":"Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Fusion category symmetries realizable on tensor-product Hilbert spaces must be weakly integral, with indices fixed by categorical data under defect assumptions.","cross_cats":["hep-th","math.CT","quant-ph"],"primary_cat":"cond-mat.str-el","authors_text":"Kansei Inamura, Rui Wen, Sakura Schafer-Nameki","submitted_at":"2026-05-14T17:59:45Z","abstract_excerpt":"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 cons"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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. We construct a lattice model that provides a QCA-refined realization for any weakly integral fusion category symmetry on a tensor product Hilbert space.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The physical assumptions on defects that allow the indices to be determined solely by categorical data; without these the index determination may not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fusion category symmetries realizable on tensor-product Hilbert spaces must be weakly integral, with indices fixed by categorical data under defect assumptions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"820a7cee3c4ecc7e096240fa7a58149acce91495ade84b59116565e162d33160"},"source":{"id":"2605.15194","kind":"arxiv","version":1},"verdict":{"id":"52d20cef-4d5e-4231-930c-b74d4ee5611d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:49:00.331342Z","strongest_claim":"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. We construct a lattice model that provides a QCA-refined realization for any weakly integral fusion category symmetry on a tensor product Hilbert space.","one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The physical assumptions on defects that allow the indices to be determined solely by categorical data; without these the index determination may not hold.","pith_extraction_headline":"Fusion category symmetries realizable on tensor-product Hilbert spaces must be weakly integral, with indices fixed by categorical data under defect assumptions."},"references":{"count":68,"sample":[{"doi":"","year":2007,"title":"Dua lity and defects in rational conformal field theory","work_id":"acb8c81e-c521-4745-9920-2f86994ddcad","ref_index":1,"cited_arxiv_id":"hep-th/0607247","is_internal_anchor":true},{"doi":"","year":2018,"title":"On finite symmetries and their gauging in two dimensions,","work_id":"db1acc3b-e257-4442-8c91-9ee1ac627dc1","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"ICTP Lectures on (Non-)Invertible Generalized Symmetries","work_id":"63ae42b5-8b0a-40e3-9f9d-50a57be4043c","ref_index":3,"cited_arxiv_id":"2305.18296","is_internal_anchor":true},{"doi":"","year":2023,"title":"What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Sym- metries","work_id":"3e4726a4-120f-43de-89d7-81589b94d413","ref_index":4,"cited_arxiv_id":"2308.00747","is_internal_anchor":false},{"doi":"","year":2024,"title":"Lectures on Generalized Symmetries","work_id":"aac2c85e-c2a1-4c51-a019-7fa153696908","ref_index":5,"cited_arxiv_id":"2307.07547","is_internal_anchor":true}],"resolved_work":68,"snapshot_sha256":"3a44b03f4e4445dcdd9b400cc73dba9dd668a0a5a713896df12d21a190d6a388","internal_anchors":16},"formal_canon":{"evidence_count":2,"snapshot_sha256":"a6cf2244aa09ea5a114ec3a416431ea5c75c2127e3b1b95f360fa93a71073c76"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}