Sector length distributions of recursively definable graph states through analytic combinatorics

Quantum Physics [Submitted on 10 Apr 2026] Sector length distributions of recursively definable graph states through analytic combinatorics Eloïc Vallée, Kenneth Goodenough, Paul E. Gunnells, Tim Coopmans, Jordi Tura 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: 17 pages, 7 figures 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 Submission history From: Eloïc Vallée [view email] [v1] Fri, 10 Apr 2026 18:00:01 UTC (659 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)