Abstract This book deals with the basic concept of information, its importance, and the similarities and differences between classical and quantum information. First, classical information theory and its application to communication are discussed. Quantum mechanics and communication based on the laws of quantum mechanics are then considered. Quantum entanglement, a key resource in quantum communication, is analysed both from the fundamental perspective and from the perspective of its information-processing capability. This book also introduces quantum computation (in the context of the search and factorisation algorithms) and the basics of quantum error correction. Throughout this book, the connections between information theory, thermodynamics, and (quantum) physics are emphasised and discussed. Other topics covered in the book are data compression, related measures of information such as relative entropy, joint entropy, conditional entropy, and mutual information, along with the capacity of a noisy channel.
Abstract This book deals with the basic concept of information, its importance, and the similarities and differences between classical and quantum information. First, classical information theory and its application to communication are discussed. Quantum mechanics and communication based on the laws of quantum mechanics are then considered. Quantum entanglement, a key resource in quantum communication, is analysed both from the fundamental perspective and from the perspective of its information-processing capability. This book also introduces quantum computation (in the context of the search and factorisation algorithms) and the basics of quantum error correction. Throughout this book, the connections between information theory, thermodynamics, and (quantum) physics are emphasised and discussed. Other topics covered in the book are data compression, related measures of information such as relative entropy, joint entropy, conditional entropy, and mutual information, along with the capacity of a noisy channel.
Information theory lies at the heart of modern technology, underpinning all communications, networking, and data storage systems. This book sets out, for the first time, a complete overview of both classical and quantum information theory. Throughout, the reader is introduced to key results without becoming lost in mathematical details. Opening chapters present the basic concepts and various applications of Shannon's entropy, moving on to the core features of quantum information and quantum computing. Topics such as coding, compression, error-correction, cryptography and channel capacity are covered from classical and quantum viewpoints. Employing an informal yet scientifically accurate approach, Desurvire provides the reader with the knowledge to understand quantum gates and circuits. Highly illustrated, with numerous practical examples and end-of-chapter exercises, this text is ideal for graduate students and researchers in electrical engineering and computer science, and practitioners in the telecommunications industry. Further resources and instructor-only solutions are available at www.cambridge.org/9780521881715.
Abstract This chapter discusses the principles of quantum information. The quantum mechanical equivalent of the Shannon noisy-channel communication theorem is obtained, along with some profound statements about the behavior of quantum information during generalised quantum measurements. The latter are very important in studies of quantum entanglement, but also in exploring the connections between thermodynamics, information theory, and quantum physics. This chapter also discusses equalities and inequalities related to entropy, the Holevo bound, capacity of a bosonic channel, information gained through measurements, relative entropy and thermodynamics, entropy increase due to erasure, and Landauer's erasure and data compression.
Information theory, introduced by Claude Shannon in 1948, provides a mathematical model, predominantly probabilistic, for addressing the challenges associated with measuring the amount of information in a message and efficiently and reliably storing the message (compression and protection). This theory has always been intricately connected to communication theory. Information theory not only relies on communication theory but also draws from fields such as statistics, statistical physics, and inference theory. In this chapter, a concise overview is presented, covering the mathematical foundations of classical and quantum information, their manipulation and associated qualifications. The fundamental concepts of classical information theory, including Shannon entropy, joint entropy, conditional entropy, and mutual information, are introduced. Additionally, basic concepts related to quantum information and quantum correlation measurements are presented.
This book is intended as an introductory textbook at graduate or advanced undergraduate level. Students with diverse backgrounds, in mathematics, physics and computer science, can use it to become acquainted with the quantum theory of information. Its scope is deliberately chosen to be not too broad, while being self-consistent and usable for a semester-long course for beginners. The subject is presented using a balanced mix of intuitive arguments and rigorous formalism. The emphasis is not on the physical interpretation, but rather on the formal and mathematical features of quantum information theory, in order to be accessible from different backgrounds, the only requirement being a basic knowledge of linear algebra. In particular, a background in quantum mechanics is not needed, since quantum theory is used as a mere mathematical framework without entering into physical interpretations. The book introduces all necessary concepts from quantum theory and classical information theory, to cover the elements of quantum Shannon theory: quantum states and channels, data compression and entropy, error correcting codes and channel capacities, as well as the basics of entanglement theory and quantum cryptography. Examples are provided to help the reader in understanding the different concepts, together with exercises at the end of each chapter.
Abstract This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.
This chapter provides a comprehensive introduction to the foundational concepts and advanced developments of mathematical perspectives on quantum information theory. Beginning with the distinctions between classical and quantum information, we explore key measures of quantum correlations such as entanglement entropy, mutual information, and entanglement of purification. Various entanglement quantifiers – including concurrence and negativity – are discussed in the context of bipartite and mixed states. The chapter also introduces quantum channels, noise models, and their physical realizations in systems ranging from superconducting qubits to black holes. We present key quantum algorithms, including Shor’s factoring algorithm and Grover’s search, along with their underlying principles such as the quantum Fourier transform. Bridging theory and practice, this chapter sets the stage for further exploration into quantum error correction, quantum thermodynamics, and the role of information in fundamental physics. The material is designed to be accessible to graduate students, researchers, and professionals across the disciplines of physics, computer science, and engineering.