Quantum Information Theory: Mathematical Foundation

Fabio Benatti

John Watrous

This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.

Jincheng Zhang

Quantum information theory, a foundational field within quantum computing and cryptography, relies heavily on the precise manipulation and analysis of quantum states. Traditional approaches often struggle to adequately represent and quantify the complexities inherent in these states. This paper proposes a novel framework based on hypergeometric distributions to formalize the mathematical foundations of quantum information theory, offering a rigorous and powerful tool for predicting and analyzing quantum phenomena. Hypergeometric distributions, a specialized distribution in probability theory, provide a natural and efficient way to model the correlated nature of quantum states, leading to a more robust and adaptable mathematical foundation. The core mechanism involves developing new mathematical tools to quantify fidelity and entanglement, thereby facilitating the design of quantum algorithms. This work aims to bridge the gap between theoretical concepts and practical applications, potentially unlocking new algorithmic possibilities and pushing the boundaries of quantum information processing.

Masahito Hayashi

Alexander Semenovich Holevo

Abstract Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.

Ved Prakash Gupta, Prabha Mandayam, Viakalathur Shankar Sunder