  One of the predominant challenges when engineering future quantum information
processors is that large quantum systems are notoriously hard to maintain and
control accurately. It is therefore of immediate practical relevance to
investigate quantum information processing with limited physical resources, for
example to ask: How well can we perform information processing tasks if we only
have access to a small quantum device? Can we beat fundamental limits imposed
on information processing with classical resources? This book will introduce
the reader to the mathematical framework required to answer such questions. A
strong emphasis is given to information measures that are essential for the
study of devices of finite size, including R\'enyi entropies and smooth
entropies. The presentation is self-contained and includes rigorous and concise
proofs of the most important properties of these measures. The first chapters
will introduce the formalism of quantum mechanics, with particular emphasis on
norms and metrics for quantum states. This is necessary to explore quantum
generalizations of R\'enyi divergence and conditional entropy, information
measures that lie at the core of information theory. The smooth entropy
framework is discussed next and provides a natural means to lift many arguments
from information theory to the quantum setting. Finally selected applications
of the theory to statistics and cryptography are discussed.
