Entanglement is a property of composite physical systems distinct from any classical description and therefore has fundamental significance in quantum theory. Although a great deal of progress has been made in understanding entanglement, there is still much unknown about this phenomenon and interest in the nature of entanglement alone continues to motivate research in this area. Moreover, a better understanding of entanglement inevitably leads to a deeper knowledge of physical processes in quantum systems. This thesis explores continuous-variable entanglement in many-body nonlinear bosonic systems with the overall aim of broadening our knowledge of entanglement, and in turn, the quantum optical systems under consideration. The secondary aim of this thesis is to further the development of entanglement as a resource for use in quantum information, communication or computing. After introducing some of the terminology and basics of entanglement theory (Chapter 1), we then provide a brief review of the main theoretical tools employed throughout this thesis (Chapter 2). We begin our investigations of entanglement by considering an evanescently coupled pair of nonlinear crystals contained in a pumped optical cavity (Chapter 3). This two-mode system is known as the intracavity Kerr nonlinear coupler and it is the output beams emerging from the cavity that provide a source of continuous-variable entangled states. Within this framework, we examine the concept of steering, a term introduced by Schr{\o}dinger to generalize the effect described by the Einstein-Podolsky-Rosen paradox. In particular, we investigate bipartite asymmetric steering. This is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories, depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. We predict that asymmetric steering should arise in experiments using the intracavity nonlinear coupler, at least in the case where Alice and Bob can only make Gaussian measurements. We continue our investigations of nonlinear quantum optical systems and entanglement by studying an experimentally feasible intra-cavity coupled down-conversion scheme that makes use of concurrent nonlinearities (Chapter 4). From a theoretical perspective, we demonstrate that this scheme gives rise to a set of continuous-variable multipartite entangled output beams. Specifically, we verify that genuine quadripartite entanglement is present in this system by calculating the output fluctuation spectra and then considering violations of multipartite entanglement inequalities. The entanglement characteristics both above and below the oscillation threshold of the cavity are considered. Finally, we build on these studies of quadripartite entanglement and examine a modified intra-cavity concurrent down-conversion scheme (Chapter 5). We analyze the feasibility and potential of this scheme for generating continuous-variable cluster states. Cluster states have been proposed as a resource state for continuous-variable one-way quantum computing. We demonstrate genuine quadripartite entanglement and investigate the degree of entanglement present. We find that above the oscillation threshold, the basic cluster state geometry under consideration suffers from phase diffusion. We alleviate this problem by incorporating a small injected signal into our analysis. We also investigate squeezed joint operators, which must approach zero in the limit of large squeezing if the cluster state defining relation is to be satisfied. While the squeezed joint operators approach zero in the undepleted regime, we find that this is not the case when we consider the full interaction Hamiltonian and the presence of a cavity. In fact, we find that the decay of these operators is minimal in a cavity and even depletion alone inhibits cluster state formation.
Entanglement is a property of composite physical systems distinct from any classical description and therefore has fundamental significance in quantum theory. Although a great deal of progress has been made in understanding entanglement, there is still much unknown about this phenomenon and interest in the nature of entanglement alone continues to motivate research in this area. Moreover, a better understanding of entanglement inevitably leads to a deeper knowledge of physical processes in quantum systems. This thesis explores continuous-variable entanglement in many-body nonlinear bosonic systems with the overall aim of broadening our knowledge of entanglement, and in turn, the quantum optical systems under consideration. The secondary aim of this thesis is to further the development of entanglement as a resource for use in quantum information, communication or computing. After introducing some of the terminology and basics of entanglement theory (Chapter 1), we then provide a brief review of the main theoretical tools employed throughout this thesis (Chapter 2). We begin our investigations of entanglement by considering an evanescently coupled pair of nonlinear crystals contained in a pumped optical cavity (Chapter 3). This two-mode system is known as the intracavity Kerr nonlinear coupler and it is the output beams emerging from the cavity that provide a source of continuous-variable entangled states. Within this framework, we examine the concept of steering, a term introduced by Schr{\o}dinger to generalize the effect described by the Einstein-Podolsky-Rosen paradox. In particular, we investigate bipartite asymmetric steering. This is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories, depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. We predict that asymmetric steering should arise in experiments using the intracavity nonlinear coupler, at least in the case where Alice and Bob can only make Gaussian measurements. We continue our investigations of nonlinear quantum optical systems and entanglement by studying an experimentally feasible intra-cavity coupled down-conversion scheme that makes use of concurrent nonlinearities (Chapter 4). From a theoretical perspective, we demonstrate that this scheme gives rise to a set of continuous-variable multipartite entangled output beams. Specifically, we verify that genuine quadripartite entanglement is present in this system by calculating the output fluctuation spectra and then considering violations of multipartite entanglement inequalities. The entanglement characteristics both above and below the oscillation threshold of the cavity are considered. Finally, we build on these studies of quadripartite entanglement and examine a modified intra-cavity concurrent down-conversion scheme (Chapter 5). We analyze the feasibility and potential of this scheme for generating continuous-variable cluster states. Cluster states have been proposed as a resource state for continuous-variable one-way quantum computing. We demonstrate genuine quadripartite entanglement and investigate the degree of entanglement present. We find that above the oscillation threshold, the basic cluster state geometry under consideration suffers from phase diffusion. We alleviate this problem by incorporating a small injected signal into our analysis. We also investigate squeezed joint operators, which must approach zero in the limit of large squeezing if the cluster state defining relation is to be satisfied. While the squeezed joint operators approach zero in the undepleted regime, we find that this is not the case when we consider the full interaction Hamiltonian and the presence of a cavity. In fact, we find that the decay of these operators is minimal in a cavity and even depletion alone inhibits cluster state formation.
Asymmetric steering is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. This phenomenon is possible because of an asymmetry in the definition of the original Einstein-Podolsky-Rosen paradox and in this article we show theoretically that it may be demonstrated, at least in the case where Alice and Bob can only make Gaussian measurements, using the intracavity nonlinear coupler.
This thesis focuses on fundamental resources in quantum information theory with continuous variables and their applications to quantum optics and quantum thermodynamics. Our analysis mainly considers Gaussian states of continuous variable systems formed by two, three, or more modes, and adopts a variety of techniques to study their static and dynamical properties. Our original results include three main case studies: (i) quantifying genuine multipartite Bell nonlocality in permutationally invariant Gaussian states; (ii) characterising bipartite Einstein-Podolsky-Rosen (EPR) steering with non-Gaussian measurements; and (iii) benchmarking the reaction-coordinate mapping in non-equilibrium quantum harmonic chains. In project (i), we investigate genuine multipartite nonlocality of multimode Gaussian states, as detected by the violation of Svetlichny inequality. We consider both displaced parity and pseudospin measurements and identify the largest violation of the inequality on permutationally invariant states. In the pseudospin case, we show that the maximum violation of Svetlichny inequality allowed by quantum mechanics can be approached for asymptotically large squeezing. In project (ii) we study the detection of EPR steering (an asymmetric quantum correlation intermediate between entanglement and Bell nonlocality) in bipartite Gaussian states by pseudospin measurements. Our analysis reveals previously unexplored regimes where non-Gaussian measurements are more effective than Gaussian ones to witness steering of Gaussian states in the presence of local noise. Finally, in project (iii) we analyze the validity of the reaction-coordinate mapping to characterise heat transport in an exactly solvable model: a two-node linear "quantum wire" connecting two baths at different temperatures. We find surprisingly that the stationary state of the original problem can be reproduced accurately by a weak-coupling treatment even in the regime of very strong residual dissipation.
The characterisation of quantum correlations, also known as entanglement, is central to quantum information theory. Operationally, entanglement gives rise to the notion of quantum non-locality, which challenges the ‘reality’ of quantum mechanics (adopting the terminology of Einstein, Podolski and Rosen) and arguably represents its most distinctive feature, in the sense that it calls for a departure from certain foundational tenets of classical physics. It should thus not come as a surprise that entanglement turns out to be a key resource for the implementation of a number of quantum communication protocols and also necessary to achieve computational speed-ups with quantum hardware, in the sense that accessing regions of the Hilbert space with entangled states allows one to shorten the computational depth of certain algorithms. Not surprisingly, generating and maintaining entanglement is generally difficult in practice, as it requires strong interactions between selected components of a system that must at the same time be isolated from the environment. Thanks to the techniques developed in quantum optical set-ups to achieve squeezed light, entanglement is, however, comparatively accessible with optical quantum continuous variables. As we shall see, in fact, continuous variable entanglement is closely related to the notion of squeezing. It is therefore of the utmost importance to be able to qualify and quantify the entanglement of quantum continuous variables, which is the subject matter of the present chapter. The first part of the chapter is devoted to techniques and criteria based on second moments to detect the entanglement of Gaussian states and also to the description of possible ways to quantify such quantum correlations, bearing in mind that the determination of entanglement monotones of clear operational significance is still an open problem, even in the restricted arena of Gaussian states. Although the criteria based on second moments are sufficient, but not necessary, for the detection of non-Gaussian entanglement too, we will push our analysis further by introducing more general criteria based on higher order statistical moments, which will be put to use to detect, on paper, the quantum correlations of an entangled cat-like state. We will also address the issues of entanglement distillation and quantum non-locality tests, in whose contexts we shall encounter other non-Gaussian operations and states, such as photon-subtracted states.
A large number of inequalities have been proposed for the detection of bipartite continuous variable entanglement and Einstein-Podolsky-Rosen steering. Many of these are based on either measured or inferred variances and are relatively easily measured via homodyne detection of optical outputs. In this work we examine a number of bipartite inequalities in the process of cascaded downconversion and sum-frequency generation inside an optical cavity. This has previously been predicted to be a potential source of continuous-variable tripartite entanglement, therefore we know that there are no separable bipartitions. In this work we investigate the performance of the chosen inequalities in the stable, below threshold regime. We show that detection of the existing entanglement is sensitive to the actual inequalities chosen, with some criteria missing it completely. Our results are explained by the fact that violation of the chosen inequalities is sufficient but not necessary to demonstrate the non-classical relationships between the modes, and highlights the outstanding problem of developing a reliable continuous-variable entanglement measure.
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.
This PhD Dissertation collects results of my own work on the topic of continuous variable (CV) quantum teleportation, which is one of the most important applications of quantum entanglement, as well as on the understanding, quantification, detection, and applications of a type of quantum correlations known as Einstein-Podolsky-Rosen (EPR) steering, for both bipartite and multipartite systems and with a main focus on CV systems. For the first results, we examine and compare two fundamentally different teleportation schemes; the well-known continuous variable scheme of Vaidman, Braunstein and Kimble, and a recently proposed hybrid scheme by Andersen and Ralph. We analyse the teleportation of ensembles of arbitrary pure single-mode Gaussian states using these schemes and compare their performance against classical strategies that utilize no entanglement (benchmarks). Our analysis brings into question any advantage due to non-Gaussianity for quantum teleportation of Gaussian states. For the second part of the results, we study bipartite EPR-steering. We propose a novel powerful method to detect steering in quantum systems of any dimension in a systematic and hierarchical way. Our method includes previous results of the literature as special cases on one hand, and goes beyond them on the other. We proceed to the quantification of steering-type correlations, and introduce a measure of steering for arbitrary bipartite Gaussian states, prove many useful properties, and provide with an operational interpretation of the proposed measure in terms of the key rate in one-sided device independent quantum key distribution. Finally, we show how the Gaussian steering measure gives a lower bound to a more general quantifier of which Gaussian states are proven to be extremal. We proceed to the study of multipartite steering, and derive laws for the distribution of Gaussian steering among different parties in multipartite Gaussian states. We define an indicator of collective steering-type correlations, which is interpreted operationally in terms of the guaranteed secret key rate in the multi-party cryptographic task of quantum secret sharing. The final results look at the cryptographical task of quantum secret sharing, whose security has remained unproven almost two decades after its original conception. By utilizing intuition and ideas from steering, we manage to establish for the first time an unconditional security proof for CV entanglement-based quantum secret sharing schemes, and demonstrate their practical feasibility. Our results establish quantum secret sharing as a viable and practically relevant primitive for quantum communication technologies.