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.
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.
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.
We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. We transform the validity of a Gaussian witness to a Bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of Bosonic Gaussian channels. Then we have necessary and sufficient criteria of separability not only for Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from Gaussian states. The criterion can be further explicitly expressed with covariance matrix of the Gaussian state or covariance matrix of Gaussian kernel of the non-Gaussian state. This opens a way for precise detection of non-Gaussian entanglement.
Entanglement is central to quantum physics, yet detecting and exploiting it in non-Gaussian systems remains a major challenge. In continuous variable platforms, standard inseparability criteria based on Gaussian statistics-such as the Duan-Simon criterion-fail when quantum correlations are encoded in higher moments of the field quadratures. Here we introduce a framework for detecting non-Gaussian entanglement using higher-order quadrature cumulants. In the Gaussian limit, the lowest order condition reduces to the Duan criterion, while higher-order violations reveal entanglement inaccessible to second-order methods. We experimentally demonstrate a tomography-free certification of inseparability in photon-subtracted squeezed states of light, where Gaussian witnesses fail despite the presence of entanglement. We further show that such higher-order inseparability enables enhanced quantum teleportation of Wigner negativity as compared to Gaussian protocols. These results identify higher-order cumulants as natural observables for non-Gaussian entanglement and open new routes to harnessing non-Gaussian resources in continuous variable quantum technologies.
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.
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.
The dimensionality of entanglement in high-dimensional quantum systems is quantified by the Schmidt number, whose large value signals potential advantages for quantum communication and quantum computation. A commonly used approach to practical Schmidt-number detection in finite-dimensional systems is based on the estimation of fidelities to pure states with large entanglement dimensionality or lower bounds to these fidelities. However, for mode-entangled continuous-variable systems, the corresponding measurements are typically difficult to realize. Here, we introduce and assess fidelity bounds for Schmidt-number detection based on practically more accessible measurements of the first and second statistical moments of arbitrary continuous-variable quantum states. We provide analytical formulas that allow evaluating these bounds and test them for families of Gaussian and non-Gaussian states, including two-mode squeezed thermal states and photon-added displaced two-mode squeezed vacuum states. We compare our bounds to other methods for the certification of high-dimensional entanglement based on the same measurements.