A Group Theoretic Approach to Quantum Information

This book is the first one addressing quantum information from the viewpoint of group symmetry. Quantum systems have a group symmetrical structure. This structure enables to handle systematically quantum information processing. However, there is no other textbook focusing on group symmetry for quantum information although there exist many textbooks for group representation. After the mathematical preparation of quantum information, this book discusses quantum entanglement and its quantification by using group symmetry. Group symmetry drastically simplifies the calculation of several entanglement measures although their calculations are usually very difficult to handle. This book treats optimal information processes including quantum state estimation, quantum state cloning, estimation of group action and quantum channel etc. Usually it is very difficult to derive the optimal quantum information processes without asymptotic setting of these topics. However, group symmetry allows to derive these optimal solutions without assuming the asymptotic setting. Next, this book addresses the quantum error correcting code with the symmetric structure of Weyl-Heisenberg groups. This structure leads to understand the quantum error correcting code systematically. Finally, this book focuses on the quantum universal information protocols by using the group SU(d). This topic can be regarded as a quantum version of the Csiszar-Korner's universal coding theory with the type method. The required mathematical knowledge about group representation is summarized in the companion book, Group Representation for Quantum Theory.

這本書是第一本從群對稱的觀點來探討量子資訊的書籍。量子系統具有群對稱的結構，這種結構使得我們能夠系統地處理量子資訊。然而，儘管有許多關於群表示的教科書，卻沒有其他教科書專注於群對稱在量子資訊中的應用。在對量子資訊進行數學準備之後，本書討論了量子純化以及如何利用群對稱來量化量子純化。群對稱大大簡化了幾個純化度量的計算，儘管這些計算通常很難處理。本書還涵蓋了包括量子狀態估計、量子狀態複製、群作用估計和量子通道等最佳資訊處理。通常在這些主題中，要推導出最佳的量子資訊處理是非常困難的，除非假設了漸進設定。然而，群對稱使得我們能夠在不假設漸進設定的情況下推導出這些最佳解。接下來，本書探討了具有Weyl-Heisenberg群對稱結構的量子錯誤修正碼。這種結構有助於我們系統地理解量子錯誤修正碼。最後，本書著重於使用SU(d)群的量子通用資訊協議。這個主題可以被視為Csiszar-Korner通用編碼理論的量子版本，並使用了類型方法。有關群表示的所需數學知識已在配套書籍《量子理論的群表示》中總結。

Masahito Hayashi was born in Japan in 1971. He received the B.S. degree from the Faculty of Sciences in Kyoto University, Japan, in 1994 and the M.S. and Ph.D. degrees in Mathematics from Kyoto University, Japan, in 1996 and 1999, respectively.

He worked in Kyoto University as a Research Fellow of the Japan Society of the Promotion of Science (JSPS) from 1998 to 2000, and worked in the Laboratory for Mathematical Neuroscience, Brain Science Institute, RIKEN from 2000 to 2003, and worked in ERATO Quantum Computation and Information Project, Japan Science and Technology Agency (JST) as the Research Head from 2000 to 2006. He also worked in the Superrobust Computation Project Information Science and Technology Strategic Core (21st Century COE by MEXT) Graduate School of Information Science and Technology, The University of Tokyo as Adjunct Associate Professor from 2004 to 2007. He worked in the Graduate School of Information Sciences, Tohoku University as Associate Professor from 2007 to 2012. In 2012, he joined the Graduate School of Mathematics, Nagoya University as Professor. He also worked in Centre for Quantum Technologies, National University of Singapore as Visiting Research Associate Professor from 2009 to 2012 and as Visiting Research Professor from 2012 to now. In 2011, he received the Information Theory Society Paper Award (2011) for Information-Spectrum Approach to Second-Order Coding Rate in Channel Coding. In 2016, he received the Japan Academy Medal from the Japan Academy and the JSPS Prize from Japan Society for the Promotion of Science.

He is a member of the Editorial Board of the International Journal of Quantum Information and International Journal On Advances in Security. His research interests include classical and quantum information theory, information-theoretic security, and classical and quantum statistical inference.

林正人於1971年出生於日本。他於1994年獲得京都大學理學院的學士學位，並於1996年和1999年分別獲得京都大學數學系的碩士和博士學位。

他於1998年至2000年期間在京都大學擔任日本學術振興會（JSPS）的研究員，並於2000年至2003年期間在RIKEN腦科學研究所的數學神經科學實驗室工作。他還於2000年至2006年期間在日本科學技術振興機構（JST）的ERATO量子計算與信息項目擔任研究負責人。他還於2004年至2007年期間在東京大學資訊科學與技術研究生院的超強韌性計算項目信息科學與技術戰略核心（MEXT 21世紀COE）擔任兼職副教授。他於2007年至2012年期間在東北大學資訊科學研究生院擔任副教授。2012年，他加入名古屋大學數學研究生院擔任教授。他還於2009年至2012年期間在新加坡國立大學量子技術中心擔任訪問研究副教授，並於2012年至今擔任訪問研究教授。2011年，他因在通道編碼中的信息譜方法獲得了信息理論學會論文獎（2011年）。2016年，他獲得了日本學士院獎和日本學術振興會獎。

他是《國際量子信息期刊》和《國際安全進展期刊》的編輯委員會成員。他的研究興趣包括經典和量子信息理論、信息理論安全性以及經典和量子統計推斷。