Qubits are not the only information carriers that can be used for quantum information processing. In this chapter, we will focus on quantum communication with ‘continuous quantum variables’, or continuous variables for short. In the context of quantum information processing we will also call continuous variables ‘qunats’. We have seen in Chapter 2 that a natural representation of a continuous variable is given by the position of a particle. The conjugate continuous variable is then the momentum of the particle. Unfortunately, the eigenstates of the position and momentum operators are not physical, and we have to construct suitable approximations to these states that can be created in the laboratory. Any practical information processing device must then take into account the deviation of the actual states from the ideal position and momentum eigenstates. Rather than the position and momentum of a particle, we will consider here the two position and momentum quadratures of an electromagnetic field mode. These operators obey the same commutation relations as the canonical position and momentum operators, but they are not the physical position and momentum of field excitations. Approximate eigenstates of the quadrature operators can be constructed in the form of squeezed coherent states. We define a quantum mechanical phase space for quadrature operators, similar to a classical phase space for position and momentum. Probability distributions in the classical phase space then become quasi-probability distributions over the quadrature phase space.We will develop one of these distributions, namely the Wigner function, and identify certain phase-space transformations of the Wigner function with linearoptical and squeezing operations.

The field of quantum information processing has reached a level of maturity, and spans such a wide variety of topics, that it merits further specialization. In this book, we consider quantum information processing with optical systems, including quantum communication, quantum computation, and quantum metrology. Optical systems are the obvious choice for quantum communication, since photons are excellent carriers of quantum information due to their relatively slow decoherence. Indeed, many aspects of quantum communication have been demonstrated to the extent that commercial products are now available. The importance of optical systems for quantum communication leads us to ask whether we can construct integrated systems for communication and computation in which all processing takes place in optical systems. Recent developments indicate that while full-scale quantum computing is still extremely challenging, optical systems are one of the most promising approaches to a fully functional quantum computer.

This book is aimed at beginning graduate students who are starting their research career in optical quantum information processing, and it can be used as a textbook for an advanced master's course. The reader is assumed to have a background knowledge in classical electrodynamics and quantum mechanics at the level of an undergraduate physics course. The nature of the topic requires familiarity with quantized fields, and since this is not always a core topic in undergraduate physics, we derive the quantum mechanical formulation of the free electromagnetic field from first principles. Similarly, we aim to present the topics in quantum information theory in a self-contained manner.

In the previous chapter we have discussed some important quantum information processing techniques based on continuous variables, or ‘qunats’. In this chapter, we describe methods for extending these techniques to quantum computation with optical qunats. We will first show in Section 9.1 how the optical qunat states are initialized using single-mode squeezing, and we give optical implementations of some of the most important single-qunat gates, such as the Heisenberg-Weyl operators and the Fourier transform. In Section 9.2, we study the theory of two-mode Gaussian operations, and we construct a general optical circuit that can implement any two-mode Gaussian operation. In Section 9.3, we introduce the stabilizer formalism for qunats, and use it to derive a Gottesman-Knill theorem. This leads to the general requirement of nonlinear gates for qunats, and their optical implementation will be discussed in Section 9.4. Having thus established the optical implementation of universal quantum computing with continuous variables, we extend this to the one-way model in Section 9.5, where we introduce cluster states for continuous variables. Finally, in Section 9.6, we discuss various quantum error-correction techniques for qunat quantum computing.

Single-mode optical qunat gates

We have already seen in Chapter 2 that the single qunat gates are fundamentally different from qubit gates. In the context of optical implementations, this difference arises due to the complementary nature of qubits and qunats: qubits that are represented by single photons can be considered particles, whereas qunats are encoded in the wave properties of the optical field.

In Chapter 7 we discussed the interaction between atoms and photons. We found that the coupling between a single atom and a single photon is rather weak, unless the atom and the photon are contained within a small cavity. The only way to significantly affect a free atom with light is to use a state of light with many photons, and in particular we showed how to use classical laser fields to prepare particular atomic states. However, there is another way of enhancing the interaction in a useful way. We can use many atoms rather than a single one. In this chapter, we will show that the interaction between a single photon and an ensemble of atoms can be greatly increased through the phenomenon of collective enhancement. We will show that this gives an atomic ensemble a relatively large susceptibility, and that multilevel atoms can be used to generate large optical nonlinearities. These offer the possibility of slowing, and even stopping, a beam of light; a phenomenon that goes hand in hand with electromagnetically induced transparency. We will discuss how a quantum state of light can be stored in and retrieved from an ensemble of atoms, and how the states of different atomic ensembles can be entangled with one another. We will show that collective states of atoms can even be used as a single qubit or several qubits, and finally, we will show that atomic ensembles can be use to mediate a photon–photon interaction.

In general, all forms of information processing can be considered in a quantum mechanical context, including for example communication, channel capacities, and the quantum limits to information extraction. In this chapter we introduce the basic features of quantum information processing. We first present qubits as abstractions of two-level addressable quantum systems, before generalizing this concept to higher-dimensional qudit systems, and continuous variables. We introduce the stabilizer formalism towards the beginning of our discussion as a convenient way to track quantum information. In Section 2.2 we derive the no-cloning theorem, which leads us to cryptography, teleportation, and quantum repeaters. Section 2.3 describes quantum computing with qubits. We start with a brief description of the circuit model, and use it to define cluster states and the one-way model. We study the properties of cluster states and discuss error correction. In the last section we introduce quantum computing over continuous variables.

Quantum information

Classically, information is carried by ‘bits’, which are physical systems that have two (macroscopic) states denoted by 0 and 1. These bits are described classically, which immediately raises the question of how we should extend this information-theoretic description to the quantum mechanical case. This will lead to the quantum unit of information, the quantum bit or ‘qubit’.

2.1.1 Classical and quantum bits

We can define the qubit as a two-dimensional quantum system, which means that the state space of the system is a two-dimensional complex Hilbert space ℋ.

Classically, light is an electromagnetic phenomenon, described by Maxwell's equations. However, under certain conditions, such as low intensity or in the presence of certain nonlinear optical materials, light starts to behave differently, and we have to construct a ‘quantum theory of light’. We can exploit this quantum behaviour of light for quantum information processing, which is the subject of this book. In this chapter, we develop the quantum theory of the free electromagnetic quantum field. This means that we do not yet consider the interaction between light and matter; we postpone that to Chapter 7. We start from first principles, using the canonical quantization procedure in the Coulomb gauge: we derive the field equations of motion from the classical Lagrangian density for the vector potential, and promote the field and its canonical momentum to operators and impose the canonical commutation relations. This will lead to the well-known creation and annihilation operators, and ultimately to the concept of the photon. After quantization of the free electromagnetic field we consider transformations of the mode functions of the field. We will demonstrate the intimate relation between these linear mode transformations and the effect of beam splitters, phase shifters, and polarization rotations, and show how they naturally give rise to the concept of squeezing. Finally, we introduce coherent and squeezed states.

Before we begin our detailed discussion of optical quantum information processing, we need to introduce certain figures of merit that quantify how well our information processor is performing. For a quantum computer, the time and resources it takes to complete a task are good measures, but we are faced with the problem that we do not know what the final design for a quantum computer will be. We therefore need additional figures of merit that are applicable in a wide range of situations. We first introduce the concept of the ‘density operator’, which will be used to describe quantum states about which we have incomplete knowledge. We then define the ‘fidelity’, which is used for assessing how close we are to a particular desired quantum state, and we discuss different measures of entanglement. The later part of the chapter will focus on figures of merit that are particularly relevant for assessing optical states, namely the first-order correlation functions, and the visibility of interference phenomena.

Density operators and superoperators

Classical physics often confronts us with situations where we can say only a limited amount about the state of a system: we can measure certain ‘bulk’ variables such as the temperature and pressure, but we do not know all of the details of the microscopic make-up of the state. For example, we typically have very little knowledge of the positions and velocities of all the atoms that constitute the system.

In this chapter we consider the physical limits to information extraction. This is an important aspect of optical quantum information processing in that many high-precision experiments (such as gravitational wave detection) are implemented in optical systems, i.e., interferometers. It is not surprising that just as in computation and communication, the use of quintessentially quantum mechanical properties allows us to improve the sensitivity in interferometry. We start this chapter with a derivation of the Fisher information and the Cramér-Rao bound, which tell us how much information we can extract about a parameter in a set of measurements. In Section 13.2 we introduce the statistical distance between two probability distributions. This can in turn be used to determine how many times the system needs to be queried before we can determine which probability distribution governs the system. In addition, we make a connection between the statistical distance and the angle between states in Hilbert space. In Sections 13.3 and 13.4 we derive bounds on how fast quantum states evolve to orthogonal states, and how entangled states can be used to improve parameter estimation. Finally, in Section 13.5 we present a number of approaches for implementing quantum metrology in optical systems, most importantly in optical interferometers.

Parameter estimation and Fisher information

In the theory of computation, discrete variables have the benefit that a practically perfect readout is often possible.

In this chapter we will discuss solid-state quantum computing, concentrating on systems where qubit manipulation, initialization or readout is performed optically. We will begin with a discussion of crystals with a periodic lattice and derive Bloch's theorem, which sets constraints on the form of electronic wave functions in crystals. We will then introduce semiconductor heterostructures and show that these have a discrete energy-level structure with transitions corresponding to the optical region of the electromagnetic spectrum. The discrete levels can be used as several different kinds of qubit, and we will discuss two that can be manipulated optically, namely an electron spin and an exciton. We will touch upon crystal defects and their importance in optical quantum computing. The emphasis will be on the NV− centre in diamond, which has produced some of the most important experimental results in recent years. Towards the end of the chapter, we will discuss specific implementations of single- and two-qubit gates in solid-state structures, before concluding with some methods for scaling up a solid-state device to a full-scale quantum computer.

Basic concepts of solid-state systems

In order to understand the optical characteristics of semiconductors, we must first review some basic concepts from solid-state physics. In particular, we will need the form and properties of the electronic wave functions in a periodic crystal structure. Unfortunately, the calculation of electronic states in a solid is impossible to do exactly.

In the previous chapter we developed some of the basic aspects of quantum information processing with single photons as qubits. Apart from noting the obvious benefit of using light for quantum communication, we identified some difficulties in manipulating quantum information that is encoded in photons. In particular, it is difficult to construct two-qubit gates for photonic qubits. In this chapter we will have to face this difficulty head-on in our discussion of quantum computation with single photons and linear-optical elements. The possibility of a quantum computer based on single photons, linear optics, and photon counting was demonstrated in a landmark paper by Knill, Laflamme, and Milburn in 2001. Their protocol is commonly referred to as the ‘klm protocol’. In subsequent years, the klm protocol has been dramatically slimmed down in terms of complexity and the necessary resources to create the universal set of quantum gates. For pedagogical reasons we give the streamlined version here, and we briefly describe the original klm protocol in Appendix 2. We will make extensive use of the results in Section 1.4 about mode transformations, and Section 2.3 about cluster-state quantum computing. We start this chapter, however, with a description of linear-optical networks, and how they fail as deterministic quantum computers. In Section 6.2 we discuss the principle of post-selection, and how this can be used to create probabilistic gates with deterministic feed-forward control.