This thesis deals with the study of quantum communication protocols with Continuous Variable (CV) systems. Continuous Variable systems are those described by canonical conjugated coordinates x and p endowed with infinite dimensional Hilbert spaces, thus involving a complex mathematical structure. A special class of CV states, are the so-called Gaussian states. With them, it has been possible to implement certain quantum tasks as quantum teleportation, quantum cryptography and quantum computation with fantastic experimental success. The importance of Gaussian states is two-fold; firstly, its structural mathematical description makes them much more amenable than any other CV system. Secondly, its production, manipulation and detection with current optical technology can be done with a very high degree of accuracy and control. Nevertheless, it is known that in spite of their exceptional role within the space of all Continuous Variable states, in fact, Gaussian states are not always the best candidates to perform quantum information tasks. Thus non-Gaussian states emerge as potentially good candidates for communication and computation purposes.
This thesis deals with the study of quantum communication protocols with Continuous Variable (CV) systems. Continuous Variable systems are those described by canonical conjugated coordinates x and p endowed with infinite dimensional Hilbert spaces, thus involving a complex mathematical structure. A special class of CV states, are the so-called Gaussian states. With them, it has been possible to implement certain quantum tasks as quantum teleportation, quantum cryptography and quantum computation with fantastic experimental success. The importance of Gaussian states is two-fold; firstly, its structural mathematical description makes them much more amenable than any other CV system. Secondly, its production, manipulation and detection with current optical technology can be done with a very high degree of accuracy and control. Nevertheless, it is known that in spite of their exceptional role within the space of all Continuous Variable states, in fact, Gaussian states are not always the best candidates to perform quantum information tasks. Thus non-Gaussian states emerge as potentially good candidates for communication and computation purposes.
The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.
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 idea to assimilate classical information theory with quantum mechanics resulted in the creation of a new field in physics known as quantum information. One of the first papers in this new field occurred in the early 1970's when Stephen Wiesner wrote the seminal manuscript titled: Conjugate Coding. However, its importance wasn't imme- diately recognized and wasn't published until 1983. The 1980's and 1990's saw a number of important papers published in quantum information leading to the subfields of quantum cryptography, quantum teleportation, quantum entanglement, distinguishability of quantum states, and quantum cloning. It was also during the 1980's, that a new model of computing, known as quantum computation, was beginning to emerge. It offered the possibility of solving certain problems faster than a classical computer by exploiting various properties of quantum mechanics. Research in this field was undoubtedly stimulated by a well known talk given by Richard Feynman in 1981 at MIT on quantum simulations. Both quantum information and quantum computation were initially developed with quantum discrete variables in mind. However, over the course of the last decade, there has been a significant increase in using quantum continuous variables. This thesis will focus on the topic of quantum information and quantum computation using continuous variables. Specifically, we will theoretically consider the cloning of continuous-variable entanglement, the distinguishability of Gaussian states, new continuous-variable quantum cryptography protocols and finally, the universality of quantum computation using continuous-variable cluster states.
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.
In this thesis we consider quantum communication as realised in the continuous variable quantum optical regime. The work presented includes results concerning quantum communication applications, such as entanglement distillation and quantum key distributions, as well as fundamental questions regarding measurement uncertainty and the boundary between classical and quantum correlations. In the first part of the thesis we investigate various protocols in which one engages in a trade-off between determinism and the quality of quantum correlations. In discrete variable quantum optics non-determinism is ubiquitous, with photon sources, detectors and various gates commonly operating probabilistically. In the continuous variable regime one has access to many quantum states and operations, including exotic effects such as entanglement, in a fully deterministic fashion and one might regard giving up this feature as highly undesirable. However it has been discovered that certain vital tasks such as entanglement distillation, indispensable for large scale quantum information networks, cannot be realised deterministically. We address precisely these limitations, firstly by conducting the most general analysis to date of a non-deterministic but heralded quantum amplifier that can circumvent the standard quantum limits, and distill Gaussian entanglement. We then analyse the security of long standing technique in continuous variable quantum key distribution, known as post-selection, which again allows the distillation of higher quality correlations. We provide a complete security proof in the asymptotic limit and show that there is in fact an relationship between this method and the noiseless amplifier. Furthermore, the protocol allows secure key to be generated for channel parameters where all previous CVQKD protocols fail. Thirdly we consider this equivalence between post-selection and heralded operations more generally, and conduct a detailed analysis of the post-selection based emulation of the noiseless amplifier. Turning to more fundamental issues, we consider links between several recently proposed entropic measures of quantum correlations and their potential operational links to continuous variable quantum key distribution. Finally we consider entropic uncertainty relations for continuous variables on infinite dimensional Hilbert spaces and apply these to one-sided device independent continuous variable quantum key distribution protocols.n
The theoretical toolbox has developed to describe a series of tasks, related to the transmission or extraction of information, which may be implemented with quantum continuous variable systems. This chapter opens with quantum teleportation, the transmission of a quantum state through shared quantum entanglement and classical communication, and determines certain thresholds it must attain to be considered genuinely quantum. It establishes the maximum classical communication capacity of a class of Gaussian channels and considers the optimal estimation of a dynamical parameter by using Gaussian states as probes. The chapter illustrates the fact that Gaussian states may be used for the secure distribution of a cryptographic key. Gaussian quantum key distribution has already been commercialised, and all the other techniques 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.