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，卻沒有其他專注於量子資訊的群對稱教科書。在數學準備部分介紹了量子資訊後，本書討論了使用群對稱來量化量子紛結以及其量化方法。群對稱大大簡化了幾個紛結度量的計算，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，本書介紹了具有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，本書著重介紹了使用SU(d)群的量子通用資訊協議。這個主題可以被視為Csiszar-Korner的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經在配套書籍《量子理論的群表示》中總結了起來。