This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book develops theoretical methods in the area of quantum technologies, namely, quantum teleportation, quantum channel capacities, quantum estimation, and quantum key distribution. It focuses on the graduate students approaching the study of quantum continuous variables, or for researchers keen on gaining a first grounding in the theoretical methods to describe such systems, be it from the angle of quantum optics, of controlled quantum dynamics or, more widely, of quantum technologies. The book introduces the instruments to deal with the Hilbert space description of Gaussian systems, and analyses their quantum correlations, their information and communication capacity, their application to quantum metrology, and their use in quantum key distribution. It reports some slightly more specific notions concerning the singular value decomposition and Schur complements.
This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book develops theoretical methods in the area of quantum technologies, namely, quantum teleportation, quantum channel capacities, quantum estimation, and quantum key distribution. It focuses on the graduate students approaching the study of quantum continuous variables, or for researchers keen on gaining a first grounding in the theoretical methods to describe such systems, be it from the angle of quantum optics, of controlled quantum dynamics or, more widely, of quantum technologies. The book introduces the instruments to deal with the Hilbert space description of Gaussian systems, and analyses their quantum correlations, their information and communication capacity, their application to quantum metrology, and their use in quantum key distribution. It reports some slightly more specific notions concerning the singular value decomposition and Schur complements.
The theoretical toolbox we have developed as the book unfolded will now be put to use to describe a series of tasks, related to the transmission or extraction of information, which may be implemented with quantum continuous variable systems. We will open with quantum teleportation, the transmission of a quantum state through shared quantum entanglement and classical communication, and determine certain thresholds it must attain to be considered genuinely quantum; we shall then proceed to establish the maximum classical communication capacity of a class of Gaussian channels; next, the optimal estimation of a dynamical parameter using Gaussian states as probes will be considered; we will then show that Gaussian states may be used for the secure distribution of a cryptographic key; finally, our enquiry will close with the illustration of a sampling algorithm aimed at achieving a ‘quantum advantage’ over competing classical methods. It is worth mentioning that, in discussing teleportation thresholds and communication capacities, although the optimal schemes we shall identify will turn out to be Gaussian, we will confront full optimisations at the Hilbert space level, and will thus introduce methods that go beyond the Gaussian description. Gaussian quantum key distribution has already been commercialised, Gaussian boson sampling has already been applied to claim a quantum advantage, and all the other techniques we will touch on in this chapter are gradually approaching the status of full-blown engineering technologies, which would arguably form the most convincing empirical evidence one may have in support of the underlying physical theory.
Quantum Continuous Variables introduces the theory of continuous variable quantum systems, from its foundations based on the framework of Gaussian states to modern developments, including its applications to quantum information and forthcoming quantum technologies. This new book addresses the theory of Gaussian states, operations, and dynamics in great depth and breadth, through a novel approach that embraces both the Hilbert space and phase descriptions.
The volume includes coverage of entanglement theory and quantum information protocols, and their connection with relevant experimental set-ups. General techniques for non-Gaussian manipulations also emerge as the treatment unfolds, and are demonstrated with specific case studies.
This book will be of interest to graduate students looking to familiarise themselves with the field, in addition to experienced researchers eager to enhance their understanding of its theoretical methods. It will also appeal to experimentalists searching for a rigorous but accessible treatment of the theory in the area.
The field of quantum optics has witnessed significant theoretical and experimental developments in recent years. This book provides an in-depth and wide-ranging introduction to the subject, emphasising throughout the basic principles and their applications. The book begins by developing the basic tools of quantum optics, and goes on to show the application of these tools in a variety of quantum optical systems, including lasing without inversion, squeezed states and atom optics. The final four chapters are devoted to a discussion of quantum optical tests of the foundations of quantum mechanics, and to particular aspects of measurement theory. Assuming only a background of standard quantum mechanics and electromagnetic theory, and containing many problems and references, this book will be invaluable to graduate students of quantum optics, as well as to researchers in this field.
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.
In the last two decades there has been an enormous progress in the experimental investigation of single quantum systems. This progress covers fields such as quantum optics, quantum computation, quantum cryptography, and quantum metrology, which are sometimes summarized as `quantum technologies'. A key issue there is entanglement, which can be considered as the characteristic feature of quantum theory. As disparate as these various fields maybe, they all have to deal with a quantum mechanical treatment of the measurement process and, in particular, the control process. Quantum control is, according to the authors, `control for which the design requires knowledge of quantum mechanics'. Quantum control situations in which measurements occur at important steps are called feedback (or feedforward) control of quantum systems and play a central role here. This book presents a comprehensive and accessible treatment of the theoretical tools that are needed to cope with these situations. It also provides the reader with the necessary background information about the experimental developments. The authors are both experts in this field to which they have made significant contributions. After an introduction to quantum measurement theory and a chapter on quantum parameter estimation, the central topic of open quantum systems is treated at some length. This chapter includes a derivation of master equations, the discussion of the Lindblad form, and decoherence – the irreversible emergence of classical properties through interaction with the environment. A separate chapter is devoted to the description of open systems by the method of quantum trajectories. Two chapters then deal with the central topic of quantum feedback control, while the last chapter gives a concise introduction to one of the central applications – quantum information. All sections contain a bunch of exercises which serve as a useful tool in learning the material. Especially helpful are also various separate boxes presenting important background material on topics such as the block representation or the feedback gain-bandwidth relation. The two appendices on quantum mechanics and phase-space and on stochastic differential equations serve the same purpose. As the authors emphasize, the book is aimed at physicists as well as control engineers who are already familiar with quantum mechanics. It takes an operational approach and presents all the material that is needed to follow research on quantum technologies. On the other hand, conceptual issues such as the relevance of the measurement process for the interpretation of quantum theory are neglected. Readers interested in them may wish to consult instead a textbook such as Decoherence and the Quantum-to-Classical Transition by Maximilian Schlosshauer. Although the present book does not contain applications to gravity, part of its content might become relevant for the physics of gravitational-wave detection and quantum gravity phenomenology. In this respect it should be of interest also for the readers of this journal.
In the last decade many important advances have taken place in the field of quantum optics, with numerous potential applications. Ideal for graduate courses on quantum optics, this textbook provides an up-to-date account of the basic principles of the subject. Focusing on applications of quantum optics, the textbook covers recent developments such as engineering of quantum states, quantum optics on a chip, nano-mechanical mirrors, quantum entanglement, quantum metrology, spin squeezing, control of decoherence and many other key topics. Readers are guided through the principles of quantum optics and their uses in a wide variety of areas including quantum information science and quantum mechanics. The textbook features end-of-chapter exercises with solutions available for instructors at www.cambridge.org/9781107006409. It is invaluable to both graduate students and researchers in physics and photonics, quantum information science and quantum communications.