Quantum Information is the science that aims to use the unusual behavior of the microscopic world, governed by the laws of Quantum Mechanics, in order to improve the way in which we compute or communicate information. Though the first ideas in this direction come from the early 80’s, it is in the last decade when Quantum Information has suffered an spectacular development. It is impossible to resume in a paper like this one the importance and complexity of the field. Therefore, I will limit to briefly explain some of the initial ideas (considered classical by now), and to briefly suggest some of the modern lines of research. By the nature of this exposition, I have decided to avoid rigor and to concentrate more in ideas and intuitions. Anyhow, I have tried to provide with enough references, in such a way that an interested reader could find there proper theorems and proofs.
Quantum Information is the science that aims to use the unusual behavior of the microscopic world, governed by the laws of Quantum Mechanics, in order to improve the way in which we compute or communicate information. Though the first ideas in this direction come from the early 80’s, it is in the last decade when Quantum Information has suffered an spectacular development. It is impossible to resume in a paper like this one the importance and complexity of the field. Therefore, I will limit to briefly explain some of the initial ideas (considered classical by now), and to briefly suggest some of the modern lines of research. By the nature of this exposition, I have decided to avoid rigor and to concentrate more in ideas and intuitions. Anyhow, I have tried to provide with enough references, in such a way that an interested reader could find there proper theorems and proofs.
What are the information processing capabilities of physical systems?As recently as the first half of the 20 th century this question did not even have a definite meaning.What is information, and how would one process it?It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate.Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms?Rolf Landauer had the essential insight -"Information is physical" -that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics.This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave.Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences".Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]?One might begin to suspect foul play in the universe's operating principles.Interesting insights into the physics of information accumulated through the 1970s and 1980s -most sensationally in the proposal for a "quantum computer".If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer.But the applications of information in physics -and vice versa -have been far more widespread than this popular discovery.These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature.In this thesis I present a short review of ideas in quantum information theory.The first chapter contains introductory material, sketching the central ideas of probability and information theory.Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems.In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems.The final chapter presents a new resource: quantum information.This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.ii This thesis is dedicated to my mother.iii This thesis was begun as an intrepid adventure.It caught
ENGLISH ABSTRACT: What are the information processing capabilities of physical systems? As recently as the first half of the 20th century this question did not even have a definite meaning. What is information, and how would one process it? It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate. Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms? Rolf Landauer had the essential insight - "Information is physical" - that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics. This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave. Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences". Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]? One might begin to suspect foul play in the universe's operating principles. Interesting insights into the physics of information accumulated through the 1970s and 1980s - most sensationally in the proposal for a "quantum computer". If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer. But the applications of information in physics - and vice versa - have been far more widespread than this popular discovery. These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature. In this thesis I present a short review of ideas in quantum information theory. The first chapter contains introductory material, sketching the central ideas of probability and information theory. Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems. In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems. The final chapter presents a new resource: quantum information. This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.
Quantum Information Science is a cross-disciplinary subject that has arisen in the last twenty years. It concerns itself with the consequences of our most complete description of the physical world (that is, quantum mechanics) for the reliable, secure, private, and rapid processing of information, both in communication and computation. While its invention is often ascribed to the famous theoretical physicist Richard Feynman in the 1980s, his contributions were only one of many that initiated the field around that time. While he perceived that new types of physical devices, in which the quantum laws of superposition and entanglement function at the logical level, could give new power in the simulation of quantum physics, it was others (Bennett, Brassard) who showed that the uncertainty principle led to fundamentally more secure ways of communicating secret messages. It was yet others (Deutsch, Vazirani) who understood that quantum theory defined a new kind of Turing machine, and that not only quantum physics simulations, but potentially many other computational problems, are sped up in this machine. And it was yet another (Shor) who found a simple, very fast algorithm for prime factorization. The word "qubit" was coined only in 1995. Its introduction is indicative of a new mindset that has developed in recent years for studying and using quantum systems. "Qubit" now stands for a durable paradigm that spans a very wide variety of fields. It thus has many sub-meanings: first, it is the basic abstract unit of information for workers ranging from optical communication engineers to NMR spectroscopists to black-hole theorists. Second, it is the name we now routinely give to physical two level systems, as they are realized by photons, atomic and ionic eigenlevels, electronic and nuclear spin, structural defects in solids, and circulating-current states of superconducting devices. Physics has long dealt with some of these two-level systems, but calling them qubits implies a whole suite of interrelated ideas and capabilities that we ascribe to these systems: the ability to precisely set, and to precisely measure, the quantum state of one unique specimen, to avoid its coupling with the environment while at the same time strongly and controllably coupling the qubits together, to exploit the resulting quantum entanglement as a resource for metrological, cryptographic, and computational tasks. Quantum Information is a very diverse subject pursued today in many different directions, by many hundreds of researchers internationally: In theoretical physics, it has enlivened and sharpened the understanding of efficient representations of entangled many-particle wavefunctions, and has provided the prospect of applications for new concepts such as anyons and majorana fermions. Information theorists has benefitted from having a rigorous extension of the basis of their field, in which classical information theory is subsumed into a greatly broader subject. Theoretical computer scientists continue the search for new quantum algorithms, and have used quantum concepts to prove new results about the classification of computational complexity. Coding theorists have had the new and subtle problem of quantum error correction to analyse and conquer. Most strikingly, the program of experimental physics has been influenced in many directions by Quantum Information Science. State-of-the-art optics experiments transmit quantum states over long distances and perform precision manipulations in single quanta (atomic ions, impurity spins, quantum dots) in the quest to have working quantum cyptography and quantum computing. Quantum Hall systems, and topological insulators, are being assiduously examined for new elementary excitations for use as qubits. In the course of ten years, superconducting devices have improved by over four orders of magnitude in their quantum coherence, a metric made precise by the ideas of quantum computing. [...] DiVincenzo, David
Abstract This comprehensive and self-consistent textbook presents quantum information science, a rapidly evolving interdisciplinary scientific field that will change modern computer science. The book is divided into four parts. The first part starts with a gentle introduction to some important mathematical concepts, followed by a discussion of computational models, linear algebra, and quantum mechanics. The second part explains more advanced topics of quantum mechanics, which are necessary to describe the decoherence of quantum states. The third part includes a presentation of entanglement, early quantum algorithms and quantum simulation, focussing on computational problems that quantum computers can solve faster than classical machines. The last part presents superconducting devices, a promising architecture for the implementation of quantum computers. With more than 200 figures and 100 exercises (with solutions), this book is ideal for a master’s course on quantum information processing. Readers starting a master’s program or a graduate course in physics, computer science, mathematics or quantum engineering will develop a good understanding of the fundamental concepts of the field.
The development of quantum mechanics and quantum physics in general is one of the greatest intellectual achievements of the 20th century. Initially, quantum physicists focused on mathematically describing and understanding the laws of a micro-world, comprised of atoms, molecules and photons. Nuclei and sub-atomic particles were discovered, particle physics evolved, and complex quantum systems in chemistry and solid-state physics are now described in terms of quantum physics. Within a century, quantum physics became the framework, which allows us to describe and predict, albeit with probabilities, the behavior of the microscopic world as we know it. The powerful quantum framework led to revolutionary technical advances for a century, such as semiconductor technology and the entire electronics industry based on it, enabling computers and the information age. The development of lasers found applications in many areas of our current technologies; we use them for printers, storing and retrieving information using CDs, industrial welding, medical surgery, and for range finding and communication purposes. Medical imaging using MRI is now a routine instrument, as are precision measurements using tunneling and atomic force microscopes. Superconducting technologies are used for high magnetic fields and many more current technologies are based on and enabled by the laws of quantum physics. While the framework of quantum physics has thus proven to be extremely successful and powerful, throughout the last century there were many conceptual, and to some extent philosophical, discussions on our understanding of quantum physics. As early as the 1930s, Einstein and Schrödinger realized that quantum physics allows for some peculiar states, which Schrödinger called “entangled”, which have a non-local character and thus do find an easy interpretation in our classical world. It took a few more decades until John Bell conceived a way to quantify such states, which thus became amenable to measurements. Only in the eighties and nineties of the last century did the generation of entangled states, their characterization and their subsequent application become feasible and accessible in laboratories. Early on, Bell states were proposed for applications in quantum communication, employing protocols based on teleportation. Soon, entanglement was recognized to be useful also for sensing, for precision measurements and eventually for quantum information processing. Starting with the early concept of a trapped-ion quantum computer introduced by Ignacio Cirac and Peter Zoller in 1995, many other platforms were proposed for quantum information processing, based on superconducting quantum bits, photonic processes, engineered quantum dots and many more. The first decade of the 21st century saw an explosive development of quantum information physics, which is by now a mature field and increasingly focuses on applications. Worldwide efforts are ongoing to realize quantum networks for secure communication and to implement quantum computers. Enhanced sensing has been demonstrated using advanced quantum states, and quantum simulators have been developed, already tackling problems that are hard to compute with classical computers. While architectures for full-fledged quantum computers are already available, the realization of such computers still requires the routine implementation of error correction, which is not yet available, for full scalability. Currently, noisy intermediate-scale quantum (NISQ) processors are already used for computations and are even accessible via cloud services. Since about 2000, quantum technologies have been developed for many platforms with the aim to surpass the capablities of their classical counterparts. Many such devices now make use of entangled states, which are seen as the new resource for advanced quantum technologies. For this, describing, understanding and generally mastering the generation and manipulation of entanglement is a key element. Moreover, both hardware and software technologies have to be improved in order to better control quantum devices. Mastering quantum phenomena for communication, metrology, simulation and computation purposes is currently one of the hottest topics in science since the wide applicability of quantum technologies will revolutionize these fields. The development of advanced quantum technologies is thus seen worldwide as a strategic area for generating technical knowledge and industrial evolution. This special issue intends to showcase the breadth and impact of the ongoing research in advancing quantum technologies. We have invited a selection of renowned scientists to present their work, and we are confident that this issue will find a broad readership and attract much interest within the scientific community.
Quantum information is an emerging field which has attracted a lot of attention in the last fifteen years or so. It is a broad subject which covers from the most applied questions (e.g., how to build quantum computers or secure cryptographic systems), to the most theoretical problems concerning the formalism of quantum mechanics, its complexity, and its potential to go beyond classical physics. I have written these notes as an introduction to the formalism of quantum information, trying to show how it is a natural progression from an undergraduate course on quantum mechanics (just as I felt it was for me). Hence, these are mainly notes about the language of quantum information and the meaning of its terms, not about its applications. They do not intend to be an exhaustive review about the field, but just a logical sequence of ideas, concepts and definitions, building up towards the continuous variables formalism, which actually takes over most of the notes. In this way, most of the results are stated without proof, and I haven't been careful with references; this is a work in progress, and I intend to add such things in the future. In any case, most of the topics that I introduce have been thoroughly covered in previous books and reviews, and with this notes I only intend to give my point of view, as well as to introduce the different topics and objects in exactly the way I work with and think about them, hoping it will be useful, especially for the people who are starting in the field. These notes originated in 2010 as a way of studying for my close collaboration with Raúl García-Patrón and Nicolas J. Cerf, to whom I am deeply grateful for guidance and incredibly clarifying discussions. They benefited greatly as well from the weeks I spent in Doha with Hyunchul Nha and his group in 2012, especially from discussions with him and Ho-Joon Kim.