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 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 second edition of this book has been revised throughout, and updated to include new topics, such as boson sampling, coherent feedback, nonlinear control, as well as several new solved problems. 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. Features Provides the first systematic graduate-level textbook for the field of quantum continuous variables and includes 77 problems for the reader, with accompanying solutions Explores applications to entanglement theory, nonlocality, quantum technologies and quantum control Describes, in detail, a comprehensive list of experimental platforms where the formalism applies Alessio Serafini earned his PhD from the University of Salerno. He is currently a Professor at University College London. His research focuses mainly on quantum optics, quantum information with continuous variables, and the theory of quantum control.
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 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 second edition of this book has been revised throughout, and updated to include new topics, such as boson sampling, coherent feedback, nonlinear control, as well as several new solved problems. 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. Features Provides the first systematic graduate-level textbook for the field of quantum continuous variables and includes 77 problems for the reader, with accompanying solutions Explores applications to entanglement theory, nonlocality, quantum technologies and quantum control Describes, in detail, a comprehensive list of experimental platforms where the formalism applies Alessio Serafini earned his PhD from the University of Salerno. He is currently a Professor at University College London. His research focuses mainly on quantum optics, quantum information with continuous variables, and the theory of quantum control.
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.
With the exception of references made to optical elements and detection techniques when discussing Gaussian operations and POVMs, and in their applications, the theory developed in this book has been largely presented as an abstract mathematical construction. This chapter intends to liaise our theoretical pictures to the wide world of controllable, coherent quantum systems that share the continuous variable description. Emphasis will be on the emergence of the canonical commutation relations in different contexts and on the introduction of the relevant Hamiltonian descriptions with the understanding that, once the basic formal analogy is established, the approaches and results of the previous chapters will be in principle applicable to the system at hand. Note, as a caveat, that some of the Hamiltonians listed in this chapter are not quadratic, notably those involving optical nonlinearities, superconducting systems and cold atoms, so that the Gaussian theory would not apply to them. In fact, we shall take the opportunity to briefly discuss the more exhaustive control, beyond the Gaussian regime, allowed by nonlinear Hamiltonians. Upon the description of the corresponding degrees of freedom, we will also venture to sketch two well-known proposals for the realisation of hardware for quantum computation, based respectively on optical modes and trapped ions, where quantum continuous variables play a significant role. The aim of this chapter is hence not providing a thorough and exhaustive discussion of each continuous variable system, but rather to grant the reader a degree of awareness on the possibilities offered by the diverse experimental platforms to which the theory applies. Thus, the conceptual elements will typically overshadow the (often very important when applications are concerned!) technical aspects, and glaring simplifications will be adopted. Given the central role played by the optical embodiment of quantum continuous variables, an exception to this cursory treatment has been made for the quantisation of the electromagnetic field, carried out below in detail from the classical theory. This will also present the opportunity to provide a full derivation of the input-output formalism and the associated quantum Langevin equations, which to our knowledge were never previously dealt with at a pedagogical level in the time domain, and to discuss a bag of quantum optical tools, such as the rotating wave approximation, which, although featuring in a number of introductory quantum optics textbooks, find such wide application to deserve a place in any attempt at a self-contained treatment.
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.
Quantum Mechanics II: Advanced Topics offers a comprehensive exploration of the state-of-the-art in various advanced topics of current research interest. A follow-up to the authors’ introductory book Quantum Mechanics I: The Fundamentals, this book expounds basic principles, theoretical treatment, case studies, worked-out examples and applications of advanced topics including quantum technologies. A thoroughly revised and updated this unique volume presents an in-depth and up-to-date progress on the growing topics including latest achievements on quantum technology. In the second edition six new chapters are included and the other ten chapters are extensively revised. Features Covers classical and quantum field theories, path integral formalism and supersymmetric quantum mechanics. Highlights coherent and squeezed states, Berry’s phase, Aharonov—Bohm effect and Wigner function. Explores salient features of quantum entanglement and quantum cryptography. Presents basic concepts of quantum computers and the features of no-cloning theorem and quantum cloning machines. Describes the theory and techniques of quantum tomography, quantum simulation and quantum error correction. Introduces other novel topics including quantum versions of theory of gravity, cosmology, Zeno effect, teleportation, games, chaos and steering. Outlines the quantum technologies of ghost imaging, detection of weak amplitudes and displacements, lithography, metrology, teleportation of optical images, sensors, batteries and internet. Contains several worked-out problems and exercises in each chapter. Quantum Mechanics II: Advanced Topics addresses various currently emerging exciting topics of quantum mechanics. It emphasizes the fundamentals behind the latest cutting-edge developments to help explain the motivation for deeper exploration. The book is a valuable resource for graduate students in physics and engineering wishing to pursue research in quantum mechanics.
Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. As we have seen in the previous chapter, such systems always require the adoption of infinite-dimensional Hilbert spaces, even when a finite number of degrees of freedom is considered. Besides making their dynamics potentially intractable, this fact also implies that the study of properties related to the information and correlation content of continuous variable systems turns out to be particularly difficult. A widely known way to make continuous variable systems more tractable is to consider the restriction to Gaussian states. Already well trodden in older research on quantum optics, the study of Gaussian systems has more recently become a central theme of continuous variable quantum information. If open quantum systems, coupled to an external environment, are considered, the Gaussian regime applies when the system couples linearly to its environment and when system and environment are governed by Hamiltonians which are at most quadratic in the canonical operators (or, equivalently, when the overall Hamiltonian of system and environment is at most quadratic). These conditions may seem rather restrictive but, as we shall point out in the course of the book, are actually ordinarily met by a vast number of existing experimental set-ups in the areas of quantum optics, trapped ions, optomechanics, atomic ensembles and certain superconducting degrees of freedom (all of these platforms are reviewed in Chapter 9 ). What we will refer to as Gaussian measurements – POVMs that preserve the Gaussian character of initial states – are also customarily carried out with comparatively high efficiency, while the description of quantum noise can be incorporated seamlessly into the Gaussian picture. What is perhaps even more important to note is that the restriction to the Gaussian realm, while not exhaustive of the full wealth of dynamics and situations allowed in the whole Hilbert space, still lets one include most of the processes relevant to quantum technologies. Squeezing, whereby certain canonical quadratures have uncertainties below the vacuum state noise, and can hence be used in precision measurements, quantum entanglement, and cooling, where the entropy is drained out of a system in order to reach a pure quantum state and thus initialise information protocols or low noise experiments, may all be described within the Gaussian regime. Within the quantum information discourse, the Gaussian restriction has been often criticised on the grounds that no real genuinely quantum effects can ever be observed without leaving it at some point, since the statistics obtained from Gaussian states through Gaussian measurements can be reproduced by classical probability distributions. While technically correct, the reader should be reminded that such critiques often disregard the fact that the object described by a Gaussian state does entail a genuinely quantum description in a Hilbert space. Hence, for instance, the preparation of a pure Gaussian quantum state is the preparation of a pure vector of the Hilbert space. Likewise, a highly squeezed Gaussian state prepared for a metrological protocol is a state with very low noise in a certain physical observable, regardless of whether it can be mimicked by a classical distribution or not. The testing and exploitation of quantum non-locality (stronger than classical correlations that imply the presence of quantum entanglement) with Gaussian states does instead require a departure from Gaussian measurements. However, as we will see, certain non-Gaussian measurements are customarily implemented with current technology, and it is hence still relevant to study in detail the creation of such a resource for Gaussian states. This chapter will introduce Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. Our treatment will highlight the privileged role played by quadratic canonical ( i.e. , ‘symplectic’) transformations in this context, which we will take care to describe in full generality, and will hinge on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. We will then proceed to explore an equivalent characterisation in terms of statistical moments of the canonical operators, and to introduce the implications of non-commutativity on such statistics in the form of the Robertson–Schrödinger uncertainty relation.
The control of individual quantum systems promises a new technology for the 21st century - quantum technology. This book is the first comprehensive treatment of modern quantum measurement and measurement-based quantum control, which are vital elements for realizing quantum technology. Readers are introduced to key experiments and technologies through dozens of recent experiments in cavity QED, quantum optics, mesoscopic electronics, and trapped particles several of which are analysed in detail. Nearly 300 exercises help build understanding, and prepare readers for research in these exciting areas. This important book will interest graduate students and researchers in quantum information, quantum metrology, quantum control and related fields. Novel topics covered include adaptive measurement; realistic detector models; mesoscopic current detection; Markovian, state-based and optimal feedback; and applications to quantum information processing.