T HE FUNDAMENTAL KNOWLEDGE BEHIND MODERN INformation technology, the most visible side of science which permeates almost every aspect of our daily lives, rests on two pillars: information science and quantum mechanics. Knowing how to efficiently communicate and process information and understanding the physical phenomena and material properties that are essential to building functional devices have provided us with unprecedented convenience and mastery over nature, possibly with the price of becoming the slaves of our computers and gadgets. Nevertheless, information science and quantum mechanics enter the backbone of current technologies as distinct and independent fields. Therefore, one is naturally tempted to ask several questions. Is there land between these two realms? If yes, what can grow out of that land? In particular, can information theory tell us something relevant about the foundations of quantum mechanics? On the other hand, can one formulate a quantum information science and in what sense would this be different from the classical version? In this chapter we introduce several basic concepts that will allow us to better understand the implications of these questions and grasp the essential ideas behind the answers formulated so far. Discussing these fundamental concepts will also shed light on the role of topological quantum matter in the economy of quantum computation. Here we focus on the key ideas that were launched into the arena by classical information theory and classical computation science. The metamorphosis of these ideas after the encounter with quantum mechanics will be discussed in the final three chapters of this book.
T HE FUNDAMENTAL KNOWLEDGE BEHIND MODERN INformation technology, the most visible side of science which permeates almost every aspect of our daily lives, rests on two pillars: information science and quantum mechanics. Knowing how to efficiently communicate and process information and understanding the physical phenomena and material properties that are essential to building functional devices have provided us with unprecedented convenience and mastery over nature, possibly with the price of becoming the slaves of our computers and gadgets. Nevertheless, information science and quantum mechanics enter the backbone of current technologies as distinct and independent fields. Therefore, one is naturally tempted to ask several questions. Is there land between these two realms? If yes, what can grow out of that land? In particular, can information theory tell us something relevant about the foundations of quantum mechanics? On the other hand, can one formulate a quantum information science and in what sense would this be different from the classical version? In this chapter we introduce several basic concepts that will allow us to better understand the implications of these questions and grasp the essential ideas behind the answers formulated so far. Discussing these fundamental concepts will also shed light on the role of topological quantum matter in the economy of quantum computation. Here we focus on the key ideas that were launched into the arena by classical information theory and classical computation science. The metamorphosis of these ideas after the encounter with quantum mechanics will be discussed in the final three chapters of this book.
We may be on the brink of a quantum transfor -mation driven by our increasing ability to realize, detect, control, and manipulate high-purity quantum states. For decades the quantum world has crept into our lives through modern information technology, the most visible side of science that permeates almost every aspect of our daily conduct. Yet quantumness in its full glory remains a promise for the future, as existing mainstream technologies are essentially based on properties of mixed quantum states realized in, e.g., metal–oxide–semiconductor heterostructures. The scientific backbone of current technologies combines quantum theory with information theory and the theory of computation, everything being sprinkled with a healthy dose of mathematics and brought into reality by remarkable engineering achievements. This multidisciplinary approach has enabled us to understand key physical phenomena and material properties, to communicate and process information efficiently, and, ultimately, to build functional devices that provide us with unprecedented convenience and mastery over nature, possibly with the price of being in danger to become the slaves of our computers and gadgets. However, our technological success is built on a combination of essentially distinct and largely independent scientific fields. Harnessing high-purity quantum states, on the other hand, sets these previously separated “continents of knowledge” on a collision path. The main goal of the last three chapters of this book is to provide a bird's eye view of the “collision zone” and prepare the reader to explore, enjoy, and perhaps exploit the newly raising mountains. We start with a brief introduction to basic concepts in quantum information, which also includes a summary of the operational formulation of quantum mechanics (the “natural” language of quantum information theory) and a synopsis of classical information theory ( Chapter 10 ). For a thorough treatment of concepts, problems, and techniques in quantum information theory the reader is referred to the books by Nielsen and Chuang [ 389 ] and Wilde [ 548 ]. The book by Watrous [ 530 ] focuses on the mathematical theory of quantum information. An undergraduate level introduction to this subject can be found, for example, in Schumacher and Westmoreland [ 458 ]. Some of the philosophical implications of the relationship between quantum mechanics and information theory are discussed in Timpson [ 505 ]. Our survey continues with Chapter 11 , where we discuss a few key aspects pertaining to quantum computation, which can be viewed as the quintessential quantum technology involving the controlled realization and manipulation of high-purity quantum states within a large Hilbert space. Finally, Chapter 12 focuses on topological quantum computation, a promising approach to overcoming the fundamental challenge associated with quantum decoherence based on harnessing the underlying robustness of topological quantum states. This also marks our final return to the realm of topological quantum matter.
T OPOLOGICAL MATTER IS FUNDAMENTALLY QUANTUM mechanical, with no correspondent in the classical world. Therefore, it would be quite natural to begin our journey with a review of topics that are particularly relevant to understanding the nature of topological quantum states, such as Berry phases and anyon physics. Yet, even before discussing those topics, pausing to think about some fundamental aspects of quantum theory would be extremely worthwhile. For several decades after the triumph of quantum mechanics in the 1920s and early 1930s, thinking about the foundations of the theory and challenging its standard interpretation were viewed by a majority of physicists as a waste of time, if not plain heresy. The judgment of the physics community on the value of such an endeavor has drastically changed after the groundbreaking work of John Bell and the subsequent experimental tests during the 1970s and 1980s. Today, studying the foundations of quantum mechanics has turned into a flourishing field intimately linked with the explosive rise of quantum information theory. Yet, the subject still has a rather marginal place in standard quantum mechanics textbooks, which focus on the formalism and on the recipes necessary for doing calculations. This chapter is intended for the reader who is less familiar with the problems brought to the fore by quantum information theory and the work on the foundations of quantum mechanics as a brief introduction to these fields and a summary of the basic terminology. But why should one bother about such things when discussing the physics of topological quantum matter? Basically, because the two areas are intimately connected: on the one hand, quantum computation is a prominent item on the list of possible applications for topological quantum states, on the other hand, key concepts in quantum theory, such as quantum entanglement, play a central role in understanding topological matter. Hence, starting the journey with a thought about the foundations of quantum mechanics should enable us to grasp the delicate root that makes the quantum world utterly strange and beautiful, then follow it as it grows out of the sphere of few particle physics into the realm of many-body systems. 1.1 THE QUANTUM MEASUREMENT PROBLEM There is a certain uneasiness about quantum mechanics that every student of this field has probably experienced in some measure. And it is not just the insecurity that results from losing the firm ground of classical intuition; special relativity also deals with this type of problem and does not generate similar feelings. Nor is it at the center of any controversy similar in scale and depth with, for example, the debate [455] between Einstein and Bohr on the foundations of quantum theory. To better understand the source of the problem, let us recall the “standard” way of presenting quantum theory, i.e., starting from certain principles or postulates [126, 440] — a typical approach that can be found in most of the commonly used textbooks. More specifically, let us focus on the key postulates that define the state vectors and their evolution, as well as the observables and the measurements.
Topology is woven into the subtle fabric of quantum theory. Topological concepts and ideas developed starting in the 1980s by the high energy physics and quantum gravity communities, first viewed by many as nothing but aesthetically appealing abstract possibilities, are central components of modern condensed matter physics and play a key role in the field of quantum information and quantum computation. The discovery of integer and fractional quantum Hall effects and, more recently, of topological insulators and superconductors has brought topological quantum matter in the laboratory, with the perspective of realizing functional platforms for topological quantum computation. Rooted in a solid mathematical formalism and spanning its relevance from fundamental questions in quantum gravity and quantum information to practical aspects regarding new materials and platforms for quantum computation, quantum topology is positioned at a fertile confluence of mathematics, physics, and computer and information science. This book focuses on the aspects of quantum topology that are most relevant from the perspective of condensed matter physics and its relation to quantum computation. We start our journey by sketching the relation between topology and quantum physics and summarizing a few basic mathematical concepts that will be used throughout this work. Beyond dry formal definitions, our main goal is to grasp the delicate root that makes the topological quantum world utterly strange and beautiful and follow it into the realm of condensed matter many-body systems.
Once upon a time there was a curious man. He knew nothing about the subject called “Information Technology.” One day he visited a library and suddenly saw a book entitled “Information Technology: The Art of Managing Information.” First, he thought: “This title is not for me. Let me ignore it and look at the next title.” But then the curious man started thinking: “What is it? What is information technology? What is information? Why do I need to know how to manage it?” Since a curious man lives in all of us, it would be tempting to follow the sequence of his thoughts and try to answer these questions. Let us start with a simple question: What is information technology? This question is very important as far as this book or any other text related to information theory is concerned. The simplest answer to this question is already provided in the title of that book as: Information technology is the art of managing information. As soon as we accept this particular definition, the other two questions that appeared in his mind become extremely relevant. In this chapter, we will try to answer those two questions and develop a quantitative perception of classical and quantum information. Once a basic perception is built in the first part of this chapter, we will describe a short history of quantum computation and quantum communication at the end of this chapter.
The marriage of quantum physics and information technology has the potential to generate radically new information processing devices. Examples are quantum cryptosystems, which provide guaranteed secure communication, and quantum computers, which manipulate data quantum mechanically and could thus solve some problems currently intractable to conventional (classical) computation. This introductory chapter serves two purposes. Firstly, I discuss some of the basic aspects of quantum physics which underpin quantum information technology (QIT). These will be used (and in somes cases further expanded upon) in subsequent chapters. Secondly, and as a lead into the whole book, I outline some of the ideas of QIT and its possible uses. To appear as Chapter 1 of Introduction to Quantum Computation and Information, eds. H.-K. Lo, S. Popescu and T. P. Spiller, (World Scientific Press 1998), http://www.wspc.com.sg/. 1 1 Introduction Information technology (IT) can feed off quantum physic...
Quantum information theory represents a rich subject of discussion for those interested in the philosphical and foundational issues surrounding quantum mechanics for a simple reason: one can cast its central concerns in terms of a long-familiar question: How does the quantum world differ from the classical one? Moreover, deployment of the concepts of information and computation in novel contexts hints at new (or better) means of understanding quantum mechanics, and perhaps even invites re-assessment of traditional material conceptions of the basic nature of the physical world. In this paper I review some of these philosophical aspects of quantum information theory, begining with an elementary survey of the theory, seeking to highlight some of the principles and heuristics involved. We move on to a discussion of the nature and definition of quantum information and deploy the findings in discussing the puzzles surrounding teleportation. The final two sections discuss, respectively, what one might learn from the development of quantum computation (both about the nature of quantum systems and about the nature of computation) and consider the impact of quantum information theory on the traditional foundational questions of quantum mechanics (treating of the views of Zeilinger, Bub and Fuchs, amongst others).