Generalized Stein's lemma and asymptotic equipartition property for subalgebra entropies
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Wuhan Institute of Quantum Technology（武汉量子技术研究院）；Univ Lyon, Inria, ENS Lyon, UCBL, LIP（里昂大学、法国国家信息与自动化研究所、巴黎高等师范学校里昂分校、里昂第一大学、计算机科学实验室）；Wallenberg Centre for Quantum Technology, Chalmers University of Technology（查尔姆斯理工大学瓦伦堡量子技术中心）
Li Gao, Mizanur Rahaman
备注：
Updated version. Many changes are made and typos are fixed. Major addition is the second-order analysis for the hypothesis-testing relative entropy in the sublagebra case. Sec. 3.3 contains this new feature. Some changes are made in section 4
The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras.
After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.
Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Department of Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Cambridge（剑桥大学）；Inria, ENS de Lyon, UCBL, LIP（法国国家信息与自动化研究所，里昂高等师范学校，里昂第一大学，计算机科学实验室）
Quantum Channel Stein's Lemma with an Exponential Strong Converse
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
Additivity of quantum relative entropies as a single-copy criterion
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Electrical and Computer Engineering, College of Design and Engineering, National University of Singapore（新加坡国立大学设计与工程学院电气与计算机工程系）
Sample complexity of quantum resource testing via one-shot quantum blurring
QuSoft；Institute for Logic, Language and Computation, University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Scuola Normale Superiore（比萨高等师范学院）
