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As a key demonstration, it formalizes the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators in finite-dimensional quantum systems, builds basis-independent operator-theoretic tools compatible with Mathlib, and supports entropic components and trace inequalities, enabling strong subadditivity and completing the Lean formalization path toward the generalized quantum Stein’s lemma.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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the paper focus on?","Question",{"text":63,"@type":64},"It centers on the sandwiched Rényi relative entropy and the data processing inequality (DPI) under quantum channels, a cornerstone monotonicity result.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What does the presented Lean library provide?",{"text":68,"@type":64},"It provides a reusable Lean 4 formal infrastructure for quantum information, including basis-independent operators, interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, and representations such as Choi, Kraus, and Stinespring.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the DPI formalized in the library?",{"text":72,"@type":64},"The work formalizes DPI for the sandwiched Rényi relative entropy for positive semidefinite operators in finite-dimensional systems, using supporting infrastructure for trace inequalities and entropy-specific ingredients like variational formulas and Haar-measure-based unitary averaging.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},328057,1790376757,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & 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Quantum Information  \nKazumi Kasaura, 1, 2, ∗ Kei Tsukamoto,3 Kento Mori,4,† Risa Mizuno,3 Takahiro Namatame,5 Yuta Oriike,6 Masaya Taniguchi,2 Sho Sonoda,2, 6,‡ and Hayata Yamasaki3, §  \n1 OMRON SINIC X Corporation  \n2 RIKEN  \n3 The University of Tokyo  \n4 MOLSIS Inc.  \n5 Kyoto University  \n6 CyberAgent, Inc.  \nQuantum information theory is built on entropic quantities; among them, the sandwiched Rényi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert–Schmidt operator spaces, Jensen’s operator inequality, generalized perspectives, operator power means, and Lieb–Ando trace inequalities. Ontop of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein’s lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.  \nThe Lean library is available as Ref. [ 1] .  \nCONTENTS  \nI. Introduction 2  \nA. Background 2  \nB. Main Results and Technical Contributions 3  \nC. Design Choices and Formalization Methodology 4  \nD. Related Work 5  \nE. Applications 5  \nF. Organization of the Paper 6  \nII. Basis-Independent Lean Formalization of Quantum Mechanics 6  \nA. Quantum Systems and States 6  \nB. Quantum Channels and Channel Representations 7  \nIII. Hierarchy of Trace Inequalities 13  \nA. Operator Monotonicity and Löwner–Heinz Theorem 14  \nB. Block-Operator and Matrix-Positivity Formalism 16  \nC. Hilbert–Schmidt Operator Space 17  \nD. Jensen’s Operator Inequality 19  \nE. Generalized Perspective Functions 20  \n∗ [kazumi.kasaura@sinicx.com](kazumi.kasaura@sinicx.com)[ ](kazumi.kasaura@sinicx.com)† [mori.kento@molsis.co.jp](mori.kento@molsis.co.jp)[ ](mori.kento@molsis.co.jp)‡ [sho.sonoda@riken.jp](sho.sonoda@riken.jp)  \n§ [hayata.yamasaki@gmail.com](hayata.yamasaki@gmail.com)  \n2  \nF. Operator Power Means 22  \nG. Lieb–Ando Trace Inequalities 23  \nIV. Quantum Entropic Quantities 24  \nA. Variational Formulas via Young and Reverse-Young Inequalities 25  \nB. Continuous Functional Calculus on Tensor Products 27  \nC. Haar Measure and Unitary Averaging 28  \nD. Data Processing Inequality for the Sandwiched Rényi Relative Entropy 30  \nV. Conclusion 32  \nAcknowledgments 33  \nReferences 33  \nI. INTRODUCTION  \nA. Background  \nQuantum information theory provides a mathematical framework for characterizing the optimal performance and fundamental limitations of quantum information processing [2] . Entropic quantities play a central role in this framework: they quantify distinguishability, uncertainty, correl","cbCaicIknYnm50ZO","https://ap.wps.com/l/cbCaicIknYnm50ZO","pdf",566615,34,"English","# Introduction\n## Background\n## Main Results and Technical Contributions\n## Design Choices and Formalization Methodology\n## Related Work\n## Applications\n## Organization of the Paper\n# Basis-Independent Lean Formalization of Quantum Mechanics\n## Quantum Systems and States\n## Quantum Channels and Channel Representations\n# Hierarchy of Trace Inequalities\n## Operator Monotonicity and Löwner–Heinz Theorem\n## Block-Operator and Matrix-Positivity Formalism\n## Hilbert–Schmidt Operator Space\n## Jensen’s Operator Inequality\n## Generalized Perspective Functions\n## Operator Power Means\n## Lieb–Ando Trace Inequalities\n# Quantum Entropic Quantities\n## Variational Formulas via Young and Reverse-Young Inequalities\n## Continuous Functional Calculus on Tensor Products\n## Haar Measure and Unitary Averaging\n## Data Processing Inequality for the Sandwiched Rényi Relative Entropy\n# Conclusion","[{\"question\":\"What central quantum concept does the paper focus on?\",\"answer\":\"It centers on the sandwiched Rényi relative entropy and the data processing inequality (DPI) under quantum channels, a cornerstone monotonicity result.\"},{\"question\":\"What does the presented Lean library provide?\",\"answer\":\"It provides a reusable Lean 4 formal infrastructure for quantum information, including basis-independent operators, interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, and representations such as Choi, Kraus, and Stinespring.\"},{\"question\":\"How is the DPI formalized in the library?\",\"answer\":\"The work formalizes DPI for the sandwiched Rényi relative entropy for positive semidefinite operators in finite-dimensional systems, using supporting infrastructure for trace inequalities and entropy-specific ingredients like variational formulas and Haar-measure-based unitary averaging.\"}]","Lean-Quantum - Toward AI-Assisted Formalization of Quantum Information | PDF",1789978581,86]