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.
