[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84509-en":3,"doc-seo-84509-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84509,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Ricci Flow for the Bures–Helstrom Qubit Metric","The Bures–Helstrom metric is the minimal monotone Riemannian metric on the qubit state space, identified (under the quantum Fisher normalization) with a geodesic hemisphere of the unit round three–sphere. The document gives an explicit Ricci-flow description in rotational symmetry, where a Hamilton–DeTurck gauge yields a single linear forced heat equation for the squared warping function Ψ=Φ2. As the metric is Einstein, the flow is a homothetic shrinker g(t)=(1−4t)gBH, with extinction time T=1/4, curvature 6/(1−4t), and a volume-normalized fixed point whose stability is determined by the Laplacian spectrum and reflection-even boundary-compatible modes.","arXiv :2606 . 19493v 3 [ cs .IT] 10 Jul 2026  \nRicci flow for the Bures–Helstrom qubit metric  \nAndrew Lesniewski  \nDepartment of Mathematics  \nBaruch College  \nOne Bernard Baruch Way  \nNew York, NY 10010  \nUSA  \nIn memory of Mary Beth Ruskai (1944–2023), friend and collaborator  \nAbstract  \nThe Bures–Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three–sphere. We describe its Ricci flow explicitly. Ina general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton–DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function Ψ = Φ2 satisfies the linear forced heat equation  \nDt Ψ = Ψττ − 2 ,  \nwhile the fixed-lapse coordinate form contains the associated transport term. Since the Bures– Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker  \ng (t) = (1 − 4t)gBH ,  \nwith scalar curvature 6/(1 − 4t) and extinction time T = 1/4 . Thus the metric remains inside the monotone cone for all t \u003C T and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume–normalized flow, for which the Bures–Helstrom metric is a fixed point. Its linearization is the shifted round–sphere Laplacian ∆S3 + 3, with zonal spectrum  \nσℓ = −(ℓ − 1)(ℓ + 3) .  \nThe perturbations compatible with the totally geodesic pure–state boundary are the reflection– even modes ℓ = 0 , 2 , 4 , . . . ; after removing the scaling mode ℓ = 0 the fixed point is stable, with spectral gap 5 .  \n1 Introduction  \nThe faithful states of a qubit form the open Bloch ball  \nB = nρ = 12 (I + r · σ) : |r| \u003C 1o,  \nwhere σ = {σ1 ,σ2 ,σ3 } are the Pauli matrices. The Riemannian metrics gρ on B that contract under every completely positive trace–preserving map [13] are the monotone metrics (or Petz metrics) . They are classified by operator monotone functions [2] and give the infinitesimal form of the contractivity of quantum relative entropy and quantum Fisher information [15, 16 , 12] . Their Riemannian geometry, in particular their curvature, has been studied in detail [5, 8] . Among them, the Bures–Helstrom metric, equivalently the symmetric logarithmic derivative quantum Fisher metric, is the minimal monotone metric and the canonical statistical metric on the qubit state space.  \nThe Bures–Helstrom metric measures the local distinguishability of nearby quantum states: if  \nρϵ = ρ + ϵX,  \nthen, to second order, the squared statistical distance between ρ and ρ ϵ is proportional to gρ(X, X) . Thus the metric is the local quadratic form governing quantum Fisher information, or equivalently the Hessian of the corresponding relative-entropy landscape.  \nThe physical motivation comes from the renormalization group. The contraction of statistical distinguishability under irreversible, completely positive dynamics is a form of informational coarse-graining: a quantum channel or Lindblad semigroup dissipates distinguishability much asa renormalization-group step dissipates short-distance detail, and so induces a flow of the metric on the space of states. This is the informational analogue of Friedan’s result [6, 7] that therenormalization-group flow of the target-space metric of a two-dimensional nonlinear sigma model is, to leading order, the Ricci flow introduced by Hamilton [9],  \n∂tg = −2Ric(g) + LVg,  \nwhere LV is a reparametrization (gauge) term. In this dictionary the coarse-graining parameter isan informational scale, and the loss of distinguishability is the analogue of a beta function.  \nAgainst this backdrop, this paper studies the intrinsic Ricci flow of the Bures–Helstrom metric. Monotone metrics measure distinguishability of quantum states, while Ricci flow evolves a metric by its own curvature; it therefore f","cbCaihhTYSaOCOpj","https://ap.wps.com/l/cbCaihhTYSaOCOpj","pdf",318958,1,14,"English","en",105,"# Introduction\n## Background: monotone (Petz) metrics and quantum Fisher geometry\n## Bures–Helstrom metric and its geometric meaning\n## Motivating dictionary: renormalization-group flow and Ricci flow\n# Ricci flow formulation in rotational symmetry\n## Rotationally symmetric metric ansatz and warping function equation\n## DeTurck frame vs fixed-lapse coordinate form","[{\"question\":\"What is the Bures–Helstrom metric in the qubit state space?\",\"answer\":\"It is the minimal monotone Riemannian metric, equivalently the symmetric logarithmic derivative quantum Fisher metric, on the Bloch ball of faithful qubit states.\"},{\"question\":\"How is the Ricci flow described for the Bures–Helstrom metric?\",\"answer\":\"Because the metric is Einstein, the geometric flow is a homothetic shrinker g(t)=(1−4t)gBH, reaching extinction at time T=1/4.\"},{\"question\":\"What equation governs the warping function in the DeTurck frame?\",\"answer\":\"In the moving DeTurck frame, the squared warping function Ψ=Φ2 satisfies a linear forced heat equation DtΨ=Ψττ−2.\"}]",1784196210,35,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"ricci-flow-for-the-bureshelstrom-qubit-metric","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/ricci-flow-for-the-bureshelstrom-qubit-metric/84509/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the Bures–Helstrom metric in the qubit state space?","Question",{"text":75,"@type":76},"It is the minimal monotone Riemannian metric, equivalently the symmetric logarithmic derivative quantum Fisher metric, on the Bloch ball of faithful qubit states.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the Ricci flow described for the Bures–Helstrom metric?",{"text":80,"@type":76},"Because the metric is Einstein, the geometric flow is a homothetic shrinker g(t)=(1−4t)gBH, reaching extinction at time T=1/4.",{"name":82,"@type":73,"acceptedAnswer":83},"What equation governs the warping function in the DeTurck frame?",{"text":84,"@type":76},"In the moving DeTurck frame, the squared warping function Ψ=Φ2 satisfies a linear forced heat equation DtΨ=Ψττ−2.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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