The second law of thermodynamics is the cornerstone of physics, characterizing the convertibility between thermodynamic states through a single function, entropy. Given the universal applicability of thermodynamics, a fundamental question in quantum information theory is whether an analogous second law can be formulated to characterize the convertibility of resources for quantum information processing by a single function. In 2008, a promising formulation was proposed, linking resource convertibility to the optimal performance of a variant of the quantum version of hypothesis testing. Central to this formulation was the generalized quantum Stein's lemma, which aimed to characterize this optimal performance by a measure of quantum resources, the regularized relative entropy of resource. If proven valid, the generalized quantum Stein's lemma would lead to the second law for quantum resources, with the regularized relative entropy of resource taking the role of entropy in thermodynamics. However, in 2023, a logical gap was found in the original proof of this lemma, casting doubt on the possibility of such a formulation of the second law. In this work, we address this problem by developing alternative techniques to successfully prove the generalized quantum Stein's lemma under a smaller set of assumptions than the original analysis. Based on our proof, we reestablish and extend the second law of quantum resource theories, applicable to both static resources of quantum states and a fundamental class of dynamical resources represented by classical-quantum (CQ) channels. These results resolve the fundamental problem of bridging the analogy between thermodynamics and quantum information theory.
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）
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）
Generalised quantum Stein's lemma more robust than ever
Scuola Normale Superiore（比萨高等师范学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；International Quantum Academy (SIQA)（国际量子学院）；Nagoya University（名古屋大学）
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）
Universal quantum resource distillation via composite generalised quantum Stein's lemma
Scuola Normale Superiore（高等师范学校）；Mathematical Quantum Information RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（理研量子信息萩博研究团队、理研先驱研究所（PRI）和理研量子计算中心（RQC））；Department of Basic Science, The University of Tokyo（东京大学基础科学系）
Ke Li's lemma for quantum hypothesis testing in general von Neumann algebras
Ke Li在[arXiv：1208.1400]中陈述的引理已经在例如[arXiv：1510.04682，arXiv：1706.04590，arXiv：1612.01464，arXiv：1308.6503，arXiv：1602.08898]中用于量子假设检验、具有量子边信息的数据压缩或量子密钥分发中的各种任务。这个引理最初是在有限维中证明的，并直接推广到I型冯诺依曼代数。在这里，我们表明，使用模块化理论可以给更透明的意义，构造的引理的对象，并证明它的一般冯诺依曼代数。这就给出了量子Stein引理的一个新的证明，并给出了它的二阶渐近性的直接推广，例如[arXiv：1510.04682]和[arXiv：1208.1400]中的主要结果.
Mathematical Institute, Slovak Academy of Sciences（斯洛伐克科学院数学研究所）；Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, CentraleSupélec, Laboratoire Méthodes Formelles（巴黎萨克雷大学、法国国家科学研究中心、巴黎高等师范学院、法国国家信息与自动化研究所、中央理工-高等电力学院、形式方法实验室）
