Uniform matrix product states with a boundary
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Quantum Physics arXiv:2512.11968 (quant-ph) [Submitted on 12 Dec 2025] Title:Uniform matrix product states with a boundary Authors:Marta Florido-Llinàs, Álvaro M. Alhambra, David Pérez-García, J.
Ignacio Cirac View a PDF of the paper titled Uniform matrix product states with a boundary, by Marta Florido-Llin\`as and 3 other authors View PDF Abstract:Canonical forms are central to the analytical understanding of tensor network states, underpinning key results such as the complete classification of one-dimensional symmetry-protected topological phases within the matrix product state (MPS) framework. Yet, the established theory applies only to uniform MPS with periodic boundary conditions, leaving many physically relevant states beyond its reach. Here we introduce a generalized canonical form for uniform MPS with a boundary matrix, thus extending the analytical MPS framework to a more general setting of wider physical significance. This canonical form reveals that any such MPS can be represented as a block-invertible matrix product operator acting on a structured class of algebraic regular language states that capture its essential long-range and scale-invariant features. Our construction builds on new algebraic results of independent interest that characterize the span and algebra generated by non-semisimple sets of matrices, including a generalized quantum Wielandt's inequality that gives an explicit upper bound on the blocking length at which the fixed-length span stabilizes to an algebra. Together, these results establish a unified theoretical foundation for uniform MPS with boundaries, bridging the gap between periodic and arbitrary-boundary settings, and providing the basis for extending key analytical and classification results of matrix product states to a much broader class of states and operators. Comments: Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph) Cite as: arXiv:2512.11968 [quant-ph] (or arXiv:2512.11968v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.11968 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Marta Florido-Llinàs [view email] [v1] Fri, 12 Dec 2025 19:00:06 UTC (112 KB) Full-text links: Access Paper: View a PDF of the paper titled Uniform matrix product states with a boundary, by Marta Florido-Llin\`as and 3 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: cond-mat cond-mat.str-el math math-ph math.MP 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?)