Quantum Information Processing with Finite Resources -- Mathematical Foundations

TL;DR

This book develops the mathematical foundations for quantum information processing with limited resources, focusing on finite quantum systems and information measures like Rényi and smooth entropies.

Contribution

It introduces a rigorous mathematical framework for analyzing quantum information tasks with small quantum devices, emphasizing finite-resource measures and their properties.

Findings

Provides a self-contained presentation of quantum norms and metrics.

Develops quantum generalizations of Rényi divergence and conditional entropy.

Applies the framework to quantum cryptography and statistical tasks.

Abstract

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.