Tomamichel M. Quantum Information Processing with Finite Resources: Mathematical Foundations

Springer Cham, Heidelberg, New York, Dordrecht, London, 2016. – 148 p. – ISBN: 3319218905This book provides the reader with the mathematical framework required to fully explore the potential of small quantum information processing devices. As decoherence will continue to limit their size, it is essential to master the conceptual tools which make such investigation possible.
A strong emphasis is given to information measures that are essential for the study of devices of finite size, including Rényi 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 Rényi 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.
The book is aimed at graduate students in Physics and Information Theory. Mathematical fluency is necessary, but no prior knowledge of quantum theory is required.Finite Resource Information Theory
Motivating Example: Source Compression
Outline of the Book
Modeling Quantum Information
General Remarks on Notation
Linear Operators and Events
Functionals and States
Multi-partite Systems
Functions on Positive Operators
Quantum Channels
Background and Further Reading
Norms and Metrics
Norms for Operators and Quantum States
Trace Distance
Fidelity
Purified Distance
Background and Further Reading
Quantum Rényi Divergence
Classical Rényi Divergence
Classifying Quantum Rényi Divergences
Minimal Quantum Rényi Divergence
Quantum Rényi Divergence
Background and Further Reading
Conditional Rényi Entropy
Conditional Entropy from Divergence
Definitions and Properties
Duality Relations and Their Applications
Chain Rules
Background and Further Reading
Smooth Entropy Calculus
Min- and Max-Entropy
Smooth Entropies
Properties of the Smooth Entropies
Fully Quantum Asymptotic Equipartition Property
Background and Further Reading
Selected Applications
Binary Quantum Hypothesis Testing
Entropic Uncertainty Relations
Randomness Extraction
Background and Further Reading
Appendix: Some Fundamental Results in Matrix Analysis