Sector length distributions of recursively definable graph states through analytic combinatorics
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Quantum Physics arXiv:2604.09766 (quant-ph) [Submitted on 10 Apr 2026] Title:Sector length distributions of recursively definable graph states through analytic combinatorics Authors:Eloïc Vallée, Kenneth Goodenough, Paul E. Gunnells, Tim Coopmans, Jordi Tura View a PDF of the paper titled Sector length distributions of recursively definable graph states through analytic combinatorics, by Elo\"ic Vall\'ee and 4 other authors View PDF HTML (experimental) Abstract:The sector length distribution or Shor-Laflamme distribution (SLD) of quantum states is governed by the $k$-body correlations amongst the different systems, and has been used to study entanglement and error correction. A succinct description of a quantum state's SLD can be obtained by representing it through the coefficients of an appropriate weight enumerator polynomial, yielding bounds on fidelity under depolarizing noise and on multipartite entanglement. However, such expressions quickly grow out of hand and are generally difficult to achieve analytically, reflecting the computational hardness of the SLD. We sidestep this problem and, instead of a single state's SLDs, encode a family of quantum state's SLD as a generating function. We then find closed-form expressions for a large class of graph states which we call `recursively definable' and which include many common graphs such as path graphs, cycle graphs, star graphs, grid graphs, and more. As direct corollary, we obtain analytical expressions for such graph states' concentratable entanglement, bounds on their depolarizing fidelity, and a multipartite entanglement criterion. Our work opens up the use of generating functions and more generally analytic combinatorics to solve problems in quantum information theory. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.09766 [quant-ph] (or arXiv:2604.09766v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.09766 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Eloïc Vallée [view email] [v1] Fri, 10 Apr 2026 18:00:01 UTC (659 KB) Full-text links: Access Paper: View a PDF of the paper titled Sector length distributions of recursively definable graph states through analytic combinatorics, by Elo\"ic Vall\'ee and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)