[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83426-en":3,"doc-seo-83426-105":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},83426,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Hockey Stick f-divergences","The paper develops a systematic, unified framework for quantum f-divergences derived from quantum “hockey stick” divergences. It extends the theory to non-normalized states and to more general hockey stick divergence definitions, including an integral representation with an additional real parameter. The results are further generalized to von Neumann algebras, with main theorems giving Neyman–Pearson representations via error probabilities and connecting regularized hockey stick Rényi divergences to Petz- and sandwiched Rényi divergences over different parameter ranges.","arXiv :2607 .08760v1 [ quant-ph] 9 Jul 2026  \nHockey stick f-divergences  \nFumio Hiai, 1, 2, ∗ Mil´an Mosonyi,2,† and Marco Tomamichel3, 4,‡  \n1 Graduate School of Information Sciences, Tohoku University,  \nAoba-ku, Sendai 980-8579, Japan  \n2 Department of Analysis and Operations Research, Institute of Mathematics,  \nBudapest University of Technology and Economics,  \nM˝uegyetem rkp. 3., H-1111 Budapest, Hungary  \n3 Center for Quantum Technologies, National University of Singapore Singapore 117543  \n4 Department of Electrical and Computer Engineering,  \nNational University of Singapore, Singapore 117583  \nIn this paper we give a systematic and unified treatment and extensions of various results on anew notion of quantum f-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in [7, 23, 30] . In particular, we consider non-normalized states and hockey stick f-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from [23, 30] to this setting. Our main results here are the representation of the hockey stick f-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in [30], an extension of Jenˇcov´a’s result [26] on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in [23] showing that the regularized hockey stick R´enyi α-divergences coincide with the Petz-type R´enyi divergences for α ∈ (0 , 1) and with the sandwiched R´enyi divergences for α > 1. Moreover, we give some partial results on the characterization of when different notions of quantum f-divergences give the same value on a pair of quantum states.  \nContents  \nI. Introduction 2  \nII. Preliminaries 3  \nIII. Classical f-divergences 3  \nA. The perspective function 3  \nB. Classical f-divergences 7  \nC. Classical f-divergences from hockey stick divergences 9  \nD. Classical f-divergences from Neyman-Pearson error probabilities 11  \nE. Differentiability 17  \nIV. f-divergences from quantum hockey stick divergences: finite dimension 17  \nA. Quantum f-divergences 17  \nB. Hockey stick divergences 18  \nC. Measured hockey stick divergences and the Neyman-Pearson error probablities 21  \nD. Neyman-Pearson tests and error probabilities at t ! +∞ 22  \nE. Hockey stick f-divergences 24  \nF. Basic properties and order 27  \nG. Symmetry 28  \nH. Hockey stick f-divergences and Neyman-Pearson error probabilities 30  \nV. Hockey stick f-divergences in von Neumann algebras 33  \nA. Hockey stick divergences in von Neumann algebras 33  \nB. Differentiability of measured hockey stick divergences 37  \nC. Representation from Neyman-Pearson tests 39  \nD. Joint lower semicontinuity 43  \nE. Martingale convergence 45  \n∗ Electronic address: [hiai.fumio@gmail.com](hiai.fumio@gmail.com)  \n†Electronic [address: milan.mosonyi@gmail.com](address: milan.mosonyi@gmail.com)  \n‡Electronic [address: marco.tomamichel@nus.edu.sg](address: marco.tomamichel@nus.edu.sg)  \n2  \nF. Sufficiency via measured hockey stick f-divergences 47  \nVI. Regularized version of hockey stick R´enyi α-divergences 48  \nA. Haagerup’s reduction theorem 49  \nB. Regularized measured R´enyi α-divergences 49  \nCD .. DDmαmαeaseas ,,hshs((ϱϱσσ))  αα((ϱϱσσ)) forfor αα  ((10,, ∞) 5502  \nVII. Equality cases between Dmeasf(ϱ∥σ), Dmeasf ,hs (ϱ∥σ), Dmaxf ,hs (ϱ∥σ) and Dmaxf(ϱ∥σ) 57  \nA. Case Dmeasf(ϱ∥σ) = Dmeasf ,hs (ϱ∥σ) 57  \nB. Case Dmeasf ,hs (ϱ∥σ) = Dmaxf(ϱ∥σ) 59  \nC. Case Dmeasf ,hs (ϱ∥σ) = Dmaxf ,hs (ϱ∥σ)? 60  \nAcknowledgments 63  \nA. Radon-Nikodym derivatives 63  \nB. Differentiability 64  \nC. Joint lower semicontinuity of the Petz-type f-divergences in the finite dimensional case 66  \nReferences 66  \nI. I","cbCait1uxsY9WDIv","https://ap.wps.com/l/cbCait1uxsY9WDIv","pdf",755020,1,68,"English","en",105,"# Introduction\n# Preliminaries\n## Classical f-divergences\n## f-divergences from quantum hockey stick divergences: finite dimension\n## Hockey stick f-divergences in von Neumann algebras\n## Regularized version of hockey stick Rényi α-divergences\n## Equality cases between various divergences","[{\"question\":\"What is the main contribution of the paper about quantum hockey stick f-divergences?\",\"answer\":\"It provides a unified treatment that defines quantum f-divergences from quantum hockey stick divergences and extends the theory to more general settings, including non-normalized states and von Neumann algebras.\"},{\"question\":\"How do the paper’s main results connect hockey stick f-divergences to Neyman–Pearson theory?\",\"answer\":\"The core results represent hockey stick f-divergences using Neyman–Pearson error probabilities, extending known finite-dimensional representations to broader frameworks.\"},{\"question\":\"What relationship is shown between regularized hockey stick Rényi α-divergences and other Rényi divergences?\",\"answer\":\"The paper extends prior results by showing that regularized hockey stick Rényi α-divergences coincide with Petz-type Rényi divergences for α in (0,1) and with sandwiched Rényi divergences for α\\u003e1.\"}]",1784187522,171,{"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},"hockey-stick-f-divergences","",{"@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/hockey-stick-f-divergences/83426/",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 main contribution of the paper about quantum hockey stick f-divergences?","Question",{"text":75,"@type":76},"It provides a unified treatment that defines quantum f-divergences from quantum hockey stick divergences and extends the theory to more general settings, including non-normalized states and von Neumann algebras.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the paper’s main results connect hockey stick f-divergences to Neyman–Pearson theory?",{"text":80,"@type":76},"The core results represent hockey stick f-divergences using Neyman–Pearson error probabilities, extending known finite-dimensional representations to broader frameworks.",{"name":82,"@type":73,"acceptedAnswer":83},"What relationship is shown between regularized hockey stick Rényi α-divergences and other Rényi divergences?",{"text":84,"@type":76},"The paper extends prior results by showing that regularized hockey stick Rényi α-divergences coincide with Petz-type Rényi divergences for α in (0,1) 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