[2309.13014] Completeness of qufinite ZXW calculus, a graphical language for finite-dimensional quantum theory Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Quantum Physics arXiv:2309.13014 (quant-ph) [Submitted on 22 Sep 2023 ( v1 ), last revised 25 Apr 2026 (this version, v3)] Title: Completeness of qufinite ZXW calculus, a graphical language for finite-dimensional quantum theory Authors: Quanlong Wang , Boldizsár Poór , Razin A. Shaikh View a PDF of the paper titled Completeness of qufinite ZXW calculus, a graphical language for finite-dimensional quantum theory, by Quanlong Wang and 1 other authors View PDF HTML (experimental) Abstract: Finite-dimensional quantum theory serves as the theoretical foundation for quantum information and computation. Mathematically, it is formalized in the category FHilb, comprising all finite-dimensional Hilbert spaces and linear maps between them. However, there has not been a graphical language for FHilb which is both universal and complete and thus incorporates a set of rules rich enough to derive any equality of the underlying formalism solely by rewriting. In this paper, we introduce the qufinite ZXW calculus - a graphical language for reasoning about finite-dimensional quantum theory. We set up a unique normal form to represent an arbitrary tensor and prove the completeness of this calculus by demonstrating that any qufinite ZXW diagram can be rewritten into its normal form. This result implies the equivalence of the qufinite ZXW calculus and the category FHilb, leading to a purely diagrammatic framework for finite-dimensional quantum theory with the same reasoning power. In addition, we identify several domains where the application of the qufinite ZXW calculus holds promise. These domains include spin networks, interacting mixed-dimensional systems in quantum chemistry, quantum programming, high-level description of quantum algorithms, and mixed-dimensional quantum computing. Our work paves the way for a comprehensive diagrammatic description of quantum physics, opening the doors of this area to the wider public. Comments: 67 pages Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2309.13014 [quant-ph] (or arXiv:2309.13014v3 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2309.13014 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Boldizsár Poór [ view email ] [v1] Fri, 22 Sep 2023 17:23:58 UTC (695 KB) [v2] Mon, 29 Jan 2024 13:20:02 UTC (442 KB) [v3] Sat, 25 Apr 2026 21:37:38 UTC (29,180 KB) Full-text links: Access Paper: View a PDF of the paper titled Completeness of qufinite ZXW calculus, a graphical language for finite-dimensional quantum theory, by Quanlong Wang and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2023-09 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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