A note on the quantum Wielandt inequality

[2504.21638] A note on the quantum Wielandt inequality Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Quantum Physics arXiv:2504.21638 (quant-ph) [Submitted on 30 Apr 2025 ( v1 ), last revised 11 Dec 2025 (this version, v3)] Title: A note on the quantum Wielandt inequality Authors: Owen Ekblad View a PDF of the paper titled A note on the quantum Wielandt inequality, by Owen Ekblad View PDF HTML (experimental) Abstract: In this note, we prove that the index of primitivity of any primitive unital Schwarz map is at most $2(D-1)^2$, where $D$ is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. As we show, the assumption of unitality is basically innocuous, but in general not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the method of Rahaman to unital primitive Schwarz maps that don't preserve trace. As a corollary of this theorem, we show that the index of primitivity of any primitive 2-positive map is at most $2(D-1)^2$, so in particular this bound holds for arbitrary primitive completely positive maps. We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac. Comments: 7 pages; the author was supported in part by US National Science Foundation Grant No. DMS-2153946 Subjects: Quantum Physics (quant-ph) ; Information Theory (cs.IT); Operator Algebras (math.OA) Cite as: arXiv:2504.21638 [quant-ph] (or arXiv:2504.21638v3 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2504.21638 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.laa.2025.11.014 Focus to learn more DOI(s) linking to related resources Submission history From: Owen Ekblad [ view email ] [v1] Wed, 30 Apr 2025 13:40:53 UTC (65 KB) [v2] Fri, 2 May 2025 20:36:20 UTC (65 KB) [v3] Thu, 11 Dec 2025 03:49:06 UTC (42 KB) Full-text links: Access Paper: View a PDF of the paper titled A note on the quantum Wielandt inequality, by Owen Ekblad View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2025-04 Change to browse by: cs cs.IT math math.IT math.OA References & Citations INSPIRE HEP NASA ADS Google 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? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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