Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. 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. 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 introduces Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. The treatment highlights the privileged role played by quadratic canonical i.e '., symplectic' transformations in this context, and hinges on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. It also discusses another equivalent parametrisation of a generic Gaussian state, deriving a more direct rendition of the spectrum of a Gaussian state in terms of observable parameters.
Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. 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. 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 introduces Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. The treatment highlights the privileged role played by quadratic canonical i.e '., symplectic' transformations in this context, and hinges on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. It also discusses another equivalent parametrisation of a generic Gaussian state, deriving a more direct rendition of the spectrum of a Gaussian state in terms of observable parameters.
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 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.
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.
The characterisation of quantum correlations, also known as entanglement, is central to quantum information theory. This chapter devotes to techniques and criteria based on second moments to detect the entanglement of Gaussian states. It also devotes 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. The chapter addresses the issues of entanglement distillation and quantum non-locality tests, in whose contexts we shall confront other non-Gaussian operations and states, such as photon subtracted states. It introduces a general criteria based on higher order statistical moments, which will be put to use to detect the quantum correlations of an entangled cat-like state and addresses the issues of entanglement distillation and quantum non-locality tests.
Theoretical chapter devoted to the detailed description of continuous variable (CV) systems by consideringthe "phase space," that is spanned by position and momentum for massive particles, quadratures for a quantum electromagnetic field, and phase and charge for electrical circuits. It introduces tools like the Glauber, Husimi, or Dirac phase–space functions, and in more details the Wigner function, that are convenient to describe CV quantum states and their time evolution using the Moyal equation. The chapter gives examples of Wigner functions and their time evolution in the presence of dissipation. It then defines symplectic quantum maps that are simple and important cases of Hamiltonian evolution and are simply related to the covariance matrix containingvariances and correlations. It details the characterization of the quantum processes using the Williamson reduction and Bloch–Messiah decomposition. It discusses Gaussian and non-Gaussian states and the specific measurement procedures for CV states, such as homodyne and double homodyne detection. It introduces the EPR entangled state and, finally, describes how to characterize entanglement and unconditionally teleport Gaussian quantum states.
In recent years the paradigm based on entanglement as the unique measure of quantum correlations has been challenged by the rise of new correlation concepts, such as quantum discord, able to reveal quantum correlations that are present in separable states. It is in general difficult to compute quantum discord, because it involves a minimization over all possible local measurements in a bipartition. In the realm of continuous-variable (CV) systems, a Gaussian version of quantum discord has been put forward upon restricting to Gaussian measurements. It is natural to ask whether non-Gaussian measurements can lead to a stronger minimization than Gaussian ones. Here we focus on two relevant classes of two-mode Gaussian states: squeezed thermal states and mixed thermal states, and allow for a range of experimentally feasible non-Gaussian measurements, comparing the results with the case of Gaussian measurements. We provide evidence that Gaussian measurements are optimal for Gaussian states.