Masahito Hayashi Knihy

This book uniquely addresses quantum information through the lens of group symmetry, a perspective not covered by existing textbooks despite the abundance of resources on group representation. It begins with the mathematical foundations of quantum information and explores quantum entanglement, utilizing group symmetry to simplify the calculation of various entanglement measures that are typically complex. The text delves into optimal information processes, such as quantum state estimation, cloning, group action estimation, and quantum channels, demonstrating how group symmetry facilitates the derivation of optimal solutions without relying on asymptotic settings. Additionally, the book examines quantum error-correcting codes through the symmetric structure of Weyl-Heisenberg groups, providing a systematic understanding of these codes. It culminates in a discussion of quantum universal information protocols using the group SU(d), which can be seen as a quantum adaptation of Csiszar-Korner's universal coding theory employing the type method. For readers seeking the necessary mathematical background, a companion book summarizes the relevant concepts of group representation for quantum theory.

This graduate-level textbook provides a unified viewpoint of quantum information theory that merges key topics from both the information-theoretic and quantum- mechanical viewpoints. The text provides a unified viewpoint of quantum information theory and lucid explanations of those basic results, so that the reader fundamentally grasps advances and challenges. This unified approach makes accessible such advanced topics in quantum communication as quantum teleportation, superdense coding, quantum state transmission (quantum error-correction), and quantum encryption.

The book offers a comprehensive exploration of quantum information theory, integrating essential concepts from both information theory and quantum mechanics. It presents clear explanations of fundamental results, making it suitable for graduate-level readers seeking a cohesive understanding of the subject.

The book delves into group representation theory as it applies to quantum theory, highlighting its significance in various quantum phenomena. It explores topics such as angular momentum, hydrogen-type Hamiltonians, spin-orbit interactions, and the quark model. Additionally, it addresses applications in quantum optics and quantum information processing, including techniques for quantum error correction. This comprehensive approach provides a robust framework for understanding the interplay between group theory and quantum mechanics.

This graduate textbook offers a comprehensive view of quantum information theory, clearly explaining the necessary mathematical foundations while merging key topics from both information-theoretic and quantum-mechanical perspectives. It provides accessible explanations of advanced topics in quantum communication, including quantum teleportation, superdense coding, quantum state transmission (quantum error-correction), and quantum encryption. Since the previous publication, significant advancements have been made in areas such as quantum state discrimination, quantum channel capacity, bipartite and multipartite entanglement, security analysis of quantum communication, the reverse Shannon theorem, and the uncertainty relation. The book employs an improved method for evaluating leaked information in quantum security, revealing a notable connection between quantum security and quantum coherence. These enhancements facilitate a better understanding of quantum state transmission. Additionally, various newly discovered uncertainty relations are discussed. With a wealth of new developments, the book introduces readers to the latest challenges and advancements in quantum information. Each chapter is supplemented with exercises and solutions to aid comprehension.

This book presents the basics of quantum information, e.g., foundation of quantum theory, quantum algorithms, quantum entanglement, quantum entropies, quantum coding, quantum error correction and quantum cryptography. As important quantum protocols, this book contains quantum teleportation, quantum dense coding, quantum data compression.