From Classical to Quantum Shannon Theory (1106.1445v8)

Abstract: The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and post-millenium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quantum information theory (Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution (Part III), and we develop many of the tools necessary for understanding information transmission or compression (Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory.

• The paper integrates classical information theory with quantum mechanics by presenting quantum communication protocols like teleportation and super-dense coding.
• It examines the density operator formalism and quantum channels to model uncertainty and develop noise-resilient quantum communications.
• The paper establishes quantum entropy measures and coding theorems that underpin secure quantum communication and future network paradigms.

Overview of "From Classical to Quantum Shannon Theory"

The document titled "From Classical to Quantum Shannon Theory" by Mark M. Wilde provides a thorough and structured exploration of the underpinnings and advancements from classical information theory to its quantum counterpart. In this extensive treatise, Wilde bridges foundational quantum mechanics with the principles of Shannon's information theory, offering insights into modern quantum communications and information processing.

Introduction to Quantum Shannon Theory

Wilde introduces the paper of quantum Shannon theory as an evolution of classical information theory established by Claude Shannon. Quantum Shannon theory examines the communication capabilities of quantum systems, emphasizing the unique properties of quantum mechanics such as superposition and entanglement. This theoretical framework extends Shannon's classical notions into the quantum regime, tackling the challenges posed by quantum uncertainty and the no-cloning theorem.

Fundamental Concepts and Postulates

The narrative begins by revisiting the basics of the quantum theory, focusing on the principles necessary for understanding quantum information processing. Among these are the ideas of indeterminism, interference, and entanglement, which distinguish quantum from classical mechanics. The book meticulously details the representation of quantum bits (qubits) and qudits, the operations that preserve their coherence, and the fundamental postulates governing quantum measurement.

The Density Operator Formalism

Wilde explores the density operator formalism, a pivotal concept for modeling imperfect knowledge of quantum states, a typical scenario in practical quantum systems due to noise and imperfections. The text describes how density matrices encapsulate this uncertainty and how they are utilized in predicting the outcomes of measurements. The formalism is foundational for discussing quantum channels and the algebraic structures that define transformations between quantum states.

Quantum Channels and Evolutions

A significant portion of the work addresses quantum channels, equivalent to noisy classical channels, but extended to quantum systems. Wilde presents the Choi-Kraus theorem, outlining the axiomatic development of quantum channels and their mathematical representation by completely positive trace-preserving maps. This theoretical background is instrumental in describing how quantum information is transmitted in the presence of noise.

Quantum Entropy and Information Measures

The discourse progresses to introduce quantum analogs of classical information measures, such as entropy and mutual information, adapted to account for the probabilistic nature of quantum states. Through the treatment of these measures, Wilde sets the stage for quantum coding theorems and their implications for compressing and transmitting quantum data efficiently, akin to Shannon's source and channel coding theorems.

Pivotal Protocols: Teleportation and Super-Dense Coding

Wilde rigorously explores essential protocols that epitomize quantum communication's potential. Quantum teleportation and super-dense coding are dissected, both of which leverage entanglement to achieve tasks unattainable in a classical context. These protocols serve as cornerstones for demonstrating the profound resource of entanglement in quantum information processing and prepare the reader for more complex quantum communication scenarios.

Quantum Shannon Theory and Practical Implications

The manuscript closes by examining the practical implications of quantum Shannon theory, including quantum key distribution's role in ensuring secure communications and the bounds on capacities of quantum channels under various scenarios. Wilde postulates future directions in network quantum Shannon theory, hinting at potential paradigms such as the quantum internet.

Conclusion

Mark M. Wilde's "From Classical to Quantum Shannon Theory" is an exhaustive text that provides both a historical perspective and a contemporary view of the field of quantum information. By weaving together classical principles with quantum phenomena, it highlights the transformative potential of quantum theory in information sciences. This paper serves as an essential resource for understanding the sophisticated blend of physics, mathematics, and information theory that forms the backbone of quantum communication and computation advancements. Researchers and practitioners in quantum physics and quantum information theory can draw rich insights and methodologies from Wilde's detailed exposition, fueling further progress in this rapidly evolving domain.