Gaussian operations, as one usually refers to CP-maps that send Gaussian states to Gaussian states, are apt to model a wide range of situations of practical interest and may often be controlled and applied on demand in the laboratory, in a variety of set-ups (that will be the subject matter of Chapter 9 ). Furthermore, they can be classified in a hierarchy of compact, powerful and far-reaching theoretical descriptions, which will be introduced by degrees and discussed in the current chapter. Note that here we are dealing with discrete quantum operations, rather than dynamical equations, which will be dealt with in the next chapter. We shall start from Gaussian unitary operations, corresponding to the symplectic group already introduced in Chapter 3 , and will also clarify how other basic manipulations such as tensoring and partial tracing carry over to the Gaussian picture. Then, we shall broaden our view to include all deterministic (trace-preserving) maps resulting from Gaussian unitary dilations, which are often known as the set of ‘bosonic Gaussian channels’ and encompass open dynamics subject to noise and ‘decoherence’ (the loss of quantum coherence due to the interaction with an environment). Next, we will move on to consider POVMs corresponding to the well-known homodyne and heterodyne detection schemes which, when acting on a subsystem, preserve the Gaussian character of the initial state both when the system is filtered by recording the measurement outcomes, and when acting unconditionally (that is, when the outcome of the measurement is not recorded). Our description will be further enlarged to the wider class of ‘general-dyne’ detections, and we will also introduce methods to account for imperfect detections. The ability to accommodate noise in the analytical description, often daunting with discrete variable and, more generally, outside the Gaussian regime, is probably one of the greatest strengths of the formalism under discussion. Finally, we shall introduce the Gaussian version of the Choi–Jamiolkowski isomorphism, and attain a unified description of all the Gaussian maps previously defined. Throughout our exposition, we will refer to quantum optical systems to exemplify the practical relevance of the operations encountered.
Gaussian operations, as one usually refers to CP-maps that send Gaussian states to Gaussian states, are apt to model a wide range of situations of practical interest and may often be controlled and applied on demand in the laboratory, in a variety of set-ups (that will be the subject matter of Chapter 9 ). Furthermore, they can be classified in a hierarchy of compact, powerful and far-reaching theoretical descriptions, which will be introduced by degrees and discussed in the current chapter. Note that here we are dealing with discrete quantum operations, rather than dynamical equations, which will be dealt with in the next chapter. We shall start from Gaussian unitary operations, corresponding to the symplectic group already introduced in Chapter 3 , and will also clarify how other basic manipulations such as tensoring and partial tracing carry over to the Gaussian picture. Then, we shall broaden our view to include all deterministic (trace-preserving) maps resulting from Gaussian unitary dilations, which are often known as the set of ‘bosonic Gaussian channels’ and encompass open dynamics subject to noise and ‘decoherence’ (the loss of quantum coherence due to the interaction with an environment). Next, we will move on to consider POVMs corresponding to the well-known homodyne and heterodyne detection schemes which, when acting on a subsystem, preserve the Gaussian character of the initial state both when the system is filtered by recording the measurement outcomes, and when acting unconditionally (that is, when the outcome of the measurement is not recorded). Our description will be further enlarged to the wider class of ‘general-dyne’ detections, and we will also introduce methods to account for imperfect detections. The ability to accommodate noise in the analytical description, often daunting with discrete variable and, more generally, outside the Gaussian regime, is probably one of the greatest strengths of the formalism under discussion. Finally, we shall introduce the Gaussian version of the Choi–Jamiolkowski isomorphism, and attain a unified description of all the Gaussian maps previously defined. Throughout our exposition, we will refer to quantum optical systems to exemplify the practical relevance of the operations encountered.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
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.
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.
Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
Gaussian states, operations, and measurements are central building blocks for continuous-variable quantum information processing which paves the way for abundant applications, especially including network-based quantum computation and communication. To make the most use of the Gaussian processes, it is required to understand and utilize suitable mathematical tools such as the symplectic space, symplectic algebra, and Wigner representation. Applying these mathematical tools to practical quantum scenarios, we developed various schemes for quantum transduction, interference-based bosonic mode permutation and bosonic sensing. We demonstrated that generic coupler characterized by Gaussian unitary process can be transformed into a high-fidelity transducer, assuming the access to infinite squeezing and adaptive feedforward with homodyne measurements. To address the practical limitation of finite squeezing, we explored the interference-based protocols. These protocols let us freely permute bosonic modes only assuming the access to single-mode Gaussian operations and multiple uses of a given multi-mode Gaussian process. Thus, such a scheme not only enables universal decoupling for bosonic systems, which is useful for suppressing undesired coupling between the system and the environment, but also faithful bidirectional single-mode quantum transduction. Moreover, noticing that the Gaussian processes are appropriate theoretical models for optical sensors, we studied the quantum noise theory for optical parameter sensing and its potential in providing great measurement precision enhancement. We also extended the Gaussian theories to discrete variable systems, with several examples such as quantum (gate) teleportation. All the analyses originated from the fundamental quantum commutation relations, and therefore are widely applicable.