Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems
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Quantum Physics arXiv:2601.00266 (quant-ph) [Submitted on 1 Jan 2026] Title:Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems Authors:Wai-Keong Mok, Tobias Haug, Wen Wei Ho, John Preskill View a PDF of the paper titled Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems, by Wai-Keong Mok and 3 other authors View PDF HTML (experimental) Abstract:Recent advances in quantum simulators allow direct experimental access to the ensemble of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics, a phenomenon known as deep thermalization. While infinite-temperature systems generate Haar-random ensembles, realistic physical constraints such as finite temperature or conservation laws require a more general framework. It has been proposed that deep thermalization is governed in general by the emergence of Scrooge ensembles, maximally entropic distributions of pure states consistent with the underlying constraints. Here we provide rigorous arguments supporting this proposal. To characterize this universal behavior, we invoke Scrooge $k$-designs, which approximate Scrooge ensembles, and identify three physically distinct mechanisms for their emergence. First, global Scrooge designs can arise from long-time chaotic unitary dynamics alone, without the need for measurements. Second, if the global state is highly scrambled, a local Scrooge design is induced when the complementary subsystem is measured. Third, a local Scrooge ensemble arises from an arbitrary entangled state when the complementary system is measured in a highly scrambled basis. Numerical simulations across a range of many-body systems identify coherence, entanglement, non-stabilizerness, and information scrambling as essential resources for the emergence of Scrooge-like behavior. Taken together, our results establish a unified theoretical framework for the emergence of maximally entropic, information-stingy randomness in quantum many-body systems. Comments: Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph) Cite as: arXiv:2601.00266 [quant-ph] (or arXiv:2601.00266v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.00266 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Wai-Keong Mok [view email] [v1] Thu, 1 Jan 2026 09:01:38 UTC (7,602 KB) Full-text links: Access Paper: View a PDF of the paper titled Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems, by Wai-Keong Mok and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 Change to browse by: cond-mat cond-mat.stat-mech 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?)