This book provides readers with a concise introduction to current studies on operator-algebras
and their generalizations operator spaces and operator systems with a special focus on their
application in quantum information science. This basic framework for the mathematical
formulation of quantum information can be traced back to the mathematical work of John von
Neumann one of the pioneers of operator algebras which forms the underpinning of most current
mathematical treatments of the quantum theory besides being one of the most dynamic areas of
twentieth century functional analysis. Today von Neumann's foresight finds expression in the
rapidly growing field of quantum information theory. These notes gather the content of lectures
given by a very distinguished group of mathematicians and quantum information theorists held
at the IMSc in Chennai some years ago and great care has been taken to present the material as
a primer on the subject matter. Starting from the basic definitions of operator spaces and
operator systems this text proceeds to discuss several important theorems including
Stinespring's dilation theorem for completely positive maps and Kirchberg's theorem on tensor
products of C*-algebras. It also takes a closer look at the abstract characterization of
operator systems and motivated by the requirements of different tensor products in quantum
information theory the theory of tensor products in operator systems is discussed in detail.
On the quantum information side the book offers a rigorous treatment of quantifying
entanglement in bipartite quantum systems and moves on to review four different areas in which
ideas from the theory of operator systems and operator algebras play a natural role: the issue
of zero-error communication over quantum channels the strong subadditivity property of quantum
entropy the different norms on quantum states and the corresponding induced norms on quantum
channels and lastly the applications of matrix-valued random variables in the quantum
information setting.
