Every Rank-Two Entangled State is Projectively Steerable
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Quantum Physics arXiv:2606.08189 (quant-ph) [Submitted on 6 Jun 2026] Title:Every Rank-Two Entangled State is Projectively Steerable Authors:Yu-Xuan Zhang, Jing-Ling Chen View a PDF of the paper titled Every Rank-Two Entangled State is Projectively Steerable, by Yu-Xuan Zhang and Jing-Ling Chen View PDF HTML (experimental) Abstract:Pure entangled states are already steerable by suitable projective measurements within their Schmidt supports, whereas rank two is the first genuinely mixed rank at which entanglement and Einstein--Podolsky--Rosen steering could bifurcate. We prove that this bifurcation does not occur even under the restricted measurement class of projective measurements: every rank-two bipartite entangled state in arbitrary finite local dimensions is projectively steerable in at least one direction, and is two-way projectively steerable when the effective local dimensions are equal. The proof is a boundary theorem rather than a steering-inequality optimization or a two-qubit reduction. A dimension--rank obstruction forces a projective outcome on the larger effective party to hit a pure boundary point of the trusted state cone. At such a contact, a nonzero support--kernel tangent block is simultaneously an NPT minor and a projective-steering certificate; if the first contact is degenerate, the rank-two constraint closes that branch and forces a second, active contact. Thus the entire first mixed-state stratum is a complete projective-steering sector, and the same support--kernel geometry gives a practical low-rank certificate whenever the relevant boundary contact is nondegenerate. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2606.08189 [quant-ph] (or arXiv:2606.08189v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2606.08189 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Jing-Ling Chen [view email] [v1] Sat, 6 Jun 2026 14:20:05 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled Every Rank-Two Entangled State is Projectively Steerable, by Yu-Xuan Zhang and Jing-Ling ChenView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-06 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?)