Limits of Rank Recovery in Bilinear Observation Problems

Quantum Physics arXiv:2601.09754 (quant-ph) [Submitted on 13 Jan 2026] Title:Limits of Rank Recovery in Bilinear Observation Problems Authors:Seungbeom Choi View a PDF of the paper titled Limits of Rank Recovery in Bilinear Observation Problems, by Seungbeom Choi View PDF HTML (experimental) Abstract:Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure. Comments: Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT) Cite as: arXiv:2601.09754 [quant-ph] (or arXiv:2601.09754v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.09754 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Seungbeom Choi [view email] [v1] Tue, 13 Jan 2026 06:11:25 UTC (68 KB) Full-text links: Access Paper: View a PDF of the paper titled Limits of Rank Recovery in Bilinear Observation Problems, by Seungbeom ChoiView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 Change to browse by: cs cs.IT math math.IT 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)