Classification of quantum states through conserved operators

Quantum Physics arXiv:2610.20988 (quant-ph) [Submitted on 14 Oct 2026] Title:Classification of quantum states through conserved operators Authors:Federico Petrovich, R. Rossignoli, M. Cerezo View a PDF of the paper titled Classification of quantum states through conserved operators, by Federico Petrovich and 2 other authors View PDF HTML (experimental) Abstract:Determining whether quantum states can be interconverted by a prescribed set of transformations is a fundamental problem in quantum information. Here we develop a general classification framework based on conserved operators (COs), i.e., not necessarily hermitian matrices having the state of interest as an eigenstate. We first show that COs belonging to the algebra generating the allowed transformations can be obtained from the kernel of a quantum covariance matrix. Their number determines the dimension of the accessible orbit, while their algebraic structure and eigenvalues provide further constraints on state convertibility. We establish general bounds on the number of COs and prove that highest-weight states, like separable states in the case of $n$ qudits, attain the maximum. We also show how these operators generate parent Hamiltonians having the characterized state as an exact ground state. We then also consider positive zero-eigenvalue COs, which encode forbidden directions of reduced density matrices and incorporate rank-based constraints into the same framework. Applying this approach to multiqubit SLOCC, we obtain complementary classifications from algebraic and rank COs. While for $n\leq 3$ qubits both lead to the usual classes, for $n=4$ qubits they distinguish 17 and 20 types separately, and 29 when combined. The combined classification yields 287 fixed types and $4$ continuous families for $n=5$ qubits, and 4926 fixed types and 147 continuous families for $n=6$ qubits, providing, to our knowledge, the most refined exhaustive classification of SLOCC-convertibility classes based on such data. Comments: Subjects: Quantum Physics (quant-ph) Report number: LA-UR-26-27624 Cite as: arXiv:2610.20988 [quant-ph] (or arXiv:2610.20988v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2610.20988 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Federico Petrovich [view email] [v1] Wed, 14 Oct 2026 22:03:28 UTC (1,536 KB) Full-text links: Access Paper: View a PDF of the paper titled Classification of quantum states through conserved operators, by Federico Petrovich and 2 other authorsView PDFHTML (experimental)TeX Source view license Ancillary-file links: Ancillary files (details): supplementary_data.pdf Current browse context: quant-ph new | recent | 2026-10 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?)