Quantum Information Theory

Certainly to go down as one of the standard references in quantum information theory, this book is accessible to anyone who has a good background in linear algebra, elementary quantum physics, and elementary information theory. To avoid complicated techniques from functional analysis and operator theory, and the attending use of infinite-dimensional Hilbert spaces, the author has chosen to stay within the confines of a finite number of states. This means of course that readers interested in continuous variable quantum communication will have to look elsewhere.Here are just a few of the highlights of the book:- The review of classical Shannon theory. The physicist reader actually may not be familiar with Shannon’s results as they are not typically a part of the physics curriculum, either undergraduate or graduate.- The role of the trace distance in the analysis of the quantum protocols. Readers will see the importance of the trace distance in the discussions of the many protocols and an “operational definition” in terms of ordinary hypothesis testing, but may still be unsure as to why the trace distance is used to distinguish the efficacy of the different protocols. A perusal of the research literature reveals a plethora of metrics to measure the distance between quantum states. This raises the obvious question as to whether the results in this book, and quantum information theory in general, are dependent on the metric chosen to distinguish two states.- Detailed proofs are given for the major results, but the author assigns exercises throughout the book to test the reader’s understanding. The presence of these exercises makes the book a textbook rather than just a monograph.- The author’s honesty about the technological feasibility of the protocols that are discussed in the book.- The insightful discussions on entanglement and how it can be viewed as a communications “resource”. Many “resource inequalities” that compare classical and quantum capabilities for information transfer occur throughout the book, and they serve as good compact summaries for the reader. The author is thankfully not hesitant to give somewhat hand-waiving explanations of entanglement, which is done quite often in physics and improves the reader’s intuition on this bizarre phenomenon. These explanations are not excessively philosophical though, so readers who want to indulge themselves in philosophical musings about entanglement will have to look elsewhere.- The discussion on purification and how noisy quantum theory can unexpectedly be viewed as a subset of noiseless quantum theory. The author shows, interestingly, that one can view noise as resulting from the entanglement of a system with another system. Some readers may naturally wonder if this viewpoint can be extended to the case of infinite dimensions and if it is compatible with some of the work in quantum stochastic calculus and quantum noise.It is difficult to ascertain whether the protocols discussed in the book can actually be represented in the physical world as quantum evolution of states via the Schrodinger equation. In this regard, very early in the book, the author points out that this equation will not appear, with the focus being on the representation of quantum theory using quantum information. Quantum evolution is viewed more as a phenomenon that must satisfy a given set of axioms that is manifested in the Choi-Kraus theorem, and represented by the now ubiquitous “quantum channel”. The physicist reader may therefore be somewhat skeptical as to the actual physical realizability of the protocols. This skepticism may be aggravated by the lack of discussion on experimental results in this book. In this regard, it might be fair to rename quantum information theory as Hilbert space information theory in spite of the eye brows this may raise in discussions of the subject.

"Quantum Information Theory" by Mark M. Wilde is a comprehensive textbook on the mathematical foundations of quantum information processing.This book is important for several reasons:Quantum Computing: Quantum information theory is the foundation of quantum computing, which has the potential to revolutionize computing, cryptography, and many other fields. Understanding the mathematical principles behind quantum information processing is crucial for the development of quantum algorithms, quantum error correction, and quantum communication protocols.Interdisciplinary connections: Quantum information theory bridges concepts from physics (quantum mechanics), computer science (information theory), and mathematics (linear algebra, operator theory). This book provides a rigorous treatment of these connections, making it a valuable resource for researchers and students from diverse backgrounds.Comprehensive treatment: Wilde's book covers a wide range of topics, including quantum entanglement, quantum channels, quantum error correction, and quantum information-theoretic protocols. Its comprehensive nature makes it an excellent resource for both beginners and experts in the field.Mathematical rigor: The book provides a mathematically rigorous treatment of quantum information theory, which is essential for understanding the underlying principles and developing new results in the field.Research and applications: Quantum information theory has numerous applications in quantum computing, quantum cryptography, and quantum communication. This book provides a solid foundation for researchers working on these applications and for exploring new areas, such as quantum machine learning and quantum simulation.Overall, "Quantum Information Theory" by Mark M. Wilde is an important resource for anyone interested in understanding the mathematical foundations of quantum information processing and its applications.

Everything I know about Quantum Shannon Theory, I learned from Mark via his book. It is self-contained and wonderfully-explained.

Could simplify many derivations. Very slowreading. However, an excellent text overall.Recommended for upper level graduate students and researchers in the field.

Great book, that goes into detail without being overwhelming on the math part. Mr Wilde seems to know which parts are hard to understand for students, so that it is easy to follow his book.He also covered a wide amount of topics, starting from the basics and going into more complex parts.My only complaint is the shipping by Amazon, the book was sent inside a package without any cushion.