Spectral geometric mean and trace characterizations
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Quantum Physics arXiv:2605.18888 (quant-ph) [Submitted on 16 May 2026] Title:Spectral geometric mean and trace characterizations Authors:Airat Bikchentaev, Trung Hoa Dinh, Anh Vu Le, Mohammad Sal Moslehian View a PDF of the paper titled Spectral geometric mean and trace characterizations, by Airat Bikchentaev and 3 other authors View PDF HTML (experimental) Abstract:We use nearly parallel pure states to characterize positive linear functionals $\phi$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $\phi(A \natural B) \leq \sqrt{\phi(A) \phi(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $\phi(A \natural B) \leq \phi((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace. Subjects: Quantum Physics (quant-ph); Functional Analysis (math.FA); Operator Algebras (math.OA) Cite as: arXiv:2605.18888 [quant-ph] (or arXiv:2605.18888v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.18888 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Trung Hoa Dinh [view email] [v1] Sat, 16 May 2026 21:33:16 UTC (13 KB) Full-text links: Access Paper: View a PDF of the paper titled Spectral geometric mean and trace characterizations, by Airat Bikchentaev and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: math math.FA math.OA 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?)