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.

After the preceding four introductory chapters, we are finally ready to discuss optical quantum information processing. While in principle quantum communication can be implemented with atoms or electrons, in practice the implementation of choice for long-distance quantum communication will almost certainly be optical. In this chapter, we will develop some of the key topics in quantum communication with single photons, and we will discuss continuous-variable quantum communication in Chapter 8. In some ways, quantum communication is the most technologically advanced part of quantum information processing. Most protocols in this chapter, such as teleportation and cryptography, have been convincingly demonstrated in the lab, and there are already several commercial organizations that sell cryptographic systems based on quantum mechanical principles. In particular, the promise of secure communication with quantum cryptography is one of the driving forces behind the development of single-photon sources and detectors. In this chapter, we first construct several optical representations of the qubit, including polarized photons and time-bin encoding. Since the single-photon implementation of a qubit is central to optical quantum information processing, this constitutes a large part of the chapter. In Section 5.2 we discuss quantum teleportation and entanglement swapping with single photons. We also present a method for entanglement distillation using entanglement swapping. In Section 5.3 we introduce decoherence-free subspaces, which can be used to reduce noise in the transmission of quantum information, and even establish communication channels in the absence of shared reference frames.

Solid-state systems, by their very nature, have a vast number of different possible quantum degrees of freedom. In Chapter 11, we saw that some of these degrees of freedom make good qubits. However, there are plenty more which are less suitable, since they cannot easily be localized and externally controlled. Once the qubit has been chosen, it is important to think about how it interacts with the other, uncontrolled quantum excitations in its environment. Such an interaction leads to unpredictable behaviour and can cause decoherence – the irretrievable loss of quantum information from the qubit – and this will be the topic of this chapter. The most obvious decoherence mechanism for any optical manipulation scheme is the spontaneous emission of photons. The theory behind this follows analogously from the theory we discussed in Chapter 7, with a suitable definition of a transition dipole for the relevant transitions. However, solid-state systems bring with them lattice vibrations, or phonons, which have no direct atomic analogue. We will therefore focus on phonons in this chapter, first discussing how we model them, and second how they interact with the electron-based qubit that we discussed in the last chapter. Later, we will see how this leads to a loss of coherence, and how optical methods can be used to slow the rate of coherence loss. Phonon interactions are complex and not easy to model exactly, but we will show that with certain approximations very successful theories can be developed.

We have so far spoken almost exclusively of photons and linear-optical elements, and seen just how powerful those two components can be for information processing. They provide unbreakable cryptographic tools, and allow for efficient quantum computing. However, many more possibilities become available when we allow photons to interact with atoms and solid matter in a quantum mechanical way. In particular, a quantum memory, the principal difficulty for linear-optical quantum computing, can be created. In this chapter we will take the first steps towards a full understanding of a photon's interaction with atoms. We will show how to describe the interactions within a system consisting of photons and few-level atoms and show how this interaction can be manipulated and exploited to provide quantum information processors based on both atomic and photonic qubits. We will also show that photon emission from atoms can degrade the quantum information contained within atoms, and we will present a formalism to model this effect. We begin with a general discussion of atom-photon interactions.

Atomic systems as qubits

Let us first consider an electron in an isolated atom. It is bound there by the Coulomb force due to the charge distribution of all the other electrons and the nucleus. The potential that describes this coupling is given by V(r).

In optical quantum information processing, two of the most basic elements are the sources of quantum mechanical states of light, and the devices that can detect these states. In this chapter, we narrow this down to photon sources and photodetectors. We will describe first how detectors work, starting from abstract ideal detectors, via a complete description of realistic detectors in terms of POVMS, to a brief overview of current photodetectors. Subsequently, we will define what is a single-photon source, and how we can determine experimentally whether a source produces single photons or something else. Having laid down the ground rules, we will survey some of the most popular ways photons are produced in the laboratory. Finally, we take a look at the production of entangled photon sources and quantum non-demolition measurements of photons.

A mathematical model of photodetectors

Photodetectors are devices that produce a macroscopic signal when triggered by one or more photons. In the ideal situation, every photon that hits the detector contributes to the macroscopic signal, and there are no ‘ghost’ signals, or so-called dark counts. In this situation we can define two types of detector, namely the ‘photon-number detector’, and ‘detectors without number resolution’.

First, the photon number detector is a (largely hypothetical) device that tells us how many photons there are in a given optical mode that is properly localized in space and time. This property is called ‘photon-number resolution’.

Random variables

Probability is an essential idea in both information theory and quantum mechanics. It is a highly developed mathematical and philosophical subject of its own, worthy of serious study. In this brief appendix, however, we can only sketch a few elementary concepts, tools, and theorems that we use elsewhere in the text.

In discussing the properties of a collection of sets, it is often useful to suppose that they are all subsets of an overall “universe” set U. The universe serves as a frame within which unions, intersections, complements, and other set operations can be described. In much the same way, the ideas of probability exist with a frame called a probability space Σ. For simplicity, we will consider only the discrete case. Then Σ consists of an underlying set of points and an assignment of probabilities. The set is called a sample space and its elements are events. The probability function, or probability distribution, assigns to each event e a real number P(e) between 0 and 1, such that the sum of all the probabilities is 1.

The probability P(e) is a measure of the likelihood that event e occurs. An impossible event has P(e) = 0 and a certain event would have P(e) = 1; in other cases, P(e) has some intermediate value.

The probability space itself contains all possible events that may occur, identified in complete detail, which makes it too elaborate for actual use. Suppose we flip a coin.

Beyond state vectors

We cannot always assign a definite state vector |ψ〉 to a quantum system Q. It may be that Q is part of a composite system RQ that is in an entangled state |Ψ(RQ)〉. Or it may be that our knowledge of the preparation of Q is insufficient to determine a particular state |ψ〉. Consider, for instance, a qubit sent from Alice to Bob during the BB84 key distribution protocol from Section 4.4. The state of this qubit could be |0〉, |1〉, |+〉 or |−〉, each with equal likelihood. In either case – whether Q is a subsystem of an entangled system, or the state of Q is determined by a probabilistic process – we cannot specify a quantum state vector |ψ〉 for Q.

Nevertheless, in either case we are in a position to make statistical predictions about the outcomes of measurements on Q. In this chapter we describe the mathematical machinery for doing this.

Mixtures of states

Suppose the state of Q arises by a random process, so that the state |ψα〉 is prepared with probability pα. The possible states |ψα〉 need not be orthogonal (as you can see in the BB84 example above). We call this situation a mixture of the states |ψα〉, or a mixed state for short.

One way to interpret a mixed state is to return to the idea of an ensemble of systems, which we introduced in Section 3.3.