Some QKD protocols, as I will detail in Chapter 11, produce Gaussian key elements. The reconciliation methods of the previous chapter are, as such, not adapted to this case. In this chapter, we build upon the previous techniques to treat the case of continuous-variable key elements or, more generally, the case of non-binary key elements.

In the first two sections, I describe two techniques to process non-binary key elements, namely sliced error correction and multistage soft decoding. Then, I conclude the chapter by giving more specific details on the reconciliation of Gaussian key elements.

Sliced error correction

Sliced error correction (SEC) is a generic reconciliation protocol that corrects strings of non-binary elements using binary reconciliation protocols as primitives [173]. The purpose of sliced error correction is to start from a list of correlated values and to give, with high probability, equal binary strings to Claude and Dominique. The underlying idea is to convert Claude's and Dominique's values into strings of bits, to apply a binary correction protocol (BCP) on each of them and to take advantage of all available information to minimize the number of exchanged reconciliation messages. It enables Claude and Dominique to reconcile a wide variety of correlated variables X and Y while relying on BCPs that are optimized to correct errors modeled by a binary symmetric channel (BSC).

An important application of sliced error correction is to correct correlated Gaussian random variables, namely X ∼ N(0, Σ) and Y = X + ∈ with ∈ ∼ N(0, σ). This important particular case is needed for QKD protocols that use a Gaussian modulation of Gaussian states, as described in Chapter 11.

In this last chapter I shall attempt a very brief assessment of the Bohr–Einstein debate. Who won it in the view of the audience? Who actually had the better case? I will also ask two rather more specific questions. The first is ‘Have the contributions of John Bell favoured the position of either of the protagonists?’ The second is ‘Has the dramatic arrival and progress of quantum information theory anything to tell us about the Bohr–Einstein debate?’

To help to answer these questions, let me first re-assemble a few points, putting together in turn an argument for each of the protagonists against a rather hostile critic. First Bohr, and I take as his critic John Bell himself. Bell was prepared, in fact, to give Bohr great credit for what Bell called the pragmatic approach to quantum theory, which I discussed earlier. This approach says that the world must be divided into ‘classical’ and ‘quantum’ parts, with an arbitrarily placed cut between them; that the macroscopic measuring device must be described in classical terms, but that we should not expect to picture the ‘quantum’ system in physical terms, and should be content just to possess rules of calculation that work well. But Bell regarded Bohr's philosophy of what lay behind pragmatism, complementarity, as ill-defined, unsatisfactory and bizarre.

The polarization of light

Our first example of a qubit will be the polarization of a photon. First we briefly review the subject of light polarization. The polarization of light was demonstrated for the first time by the Chevalier Malus in 1809. He observed the light of the setting sun reflected by the glass of a window in the Luxembourg Palace in Paris through a crystal of Iceland spar. He showed that when the crystal was rotated, one of the two images of the sun disappeared. Iceland spar is a birefringent crystal which, as we shall see below, decomposes a light ray into two rays polarized in perpendicular directions, while the ray reflected from the glass is (partially) polarized. When the crystal is suitably oriented one then observes the disappearance (or strong attenuation) of one of the two rays. The phenomenon of polarization displays the vector nature of light waves, a property which is shared by shear sound waves: in an isotropic crystal, a sound wave can correspond either to a vibration transverse to the direction of propagation, i.e., a shear wave, or to a longitudinal vibration, i.e., a compression wave. In the case of light the vibration is only transverse: the electric field of a light wave is orthogonal to the propagation direction.

Quantum physics is well known for being counter-intuitive, or even bizarre. Quantum correlations have no equivalence in classical physics. This was all well known for years. But several discoveries in the 1990's changed the world. First, in 1991 Artur Ekert, from Oxford University, discovered that quantum correlations could be used to distribute cryptographic keys. Suddenly physicists realized that quantum correlations and its associated bizarre non-locality, could be exploited to achieve a useful task that would be impossible without quantum physics. What a revolution! And this was not the end. Three years later, Peter Shor, from the AT&T Laboratories, discovered an algorithm that breaks the most used public key crypto-systems. Shor's algorithm requires a quantum computer, yet another bizarre quantum device, a kind of computer that heavily exploits the quantum superposition principle. The following year, in 1995, a collaboration between six physicists and computer scientists from three continents led to the discovery of quantum teleportation, a process with a science-fiction flavour.

These discoveries and others led to the emergence of a new science, marrying quantum physics and theoretical computer science, called quantum information science. Today, a steadily growing community of physicists, mathematicians and computer scientists develop the tools of this new science. This leads to new experiments and new insights into physics and information theory. It is a fascinating time.

Quantum information is still a very young science. In particular, the technology required to build a quantum computer is still unknown. Only quantum key distribution has reached a certain level of maturity, with a few start-ups already offering complete systems. Nevertheless, quantum information has been widely recognised as a source for truly innovative ideas and disruptive technologies.

Quantum information is concerned with using the special features of quantum physics for the processing and transmission of information. It should, however, be clearly understood that any physical object when analyzed at a deep enough level is a quantum object; as Rolf Landauer has succinctly stated, “A screwdriver is a quantum object.” In fact, the conduction properties of the metal blade of a screwdriver are ultimately due to the quantum properties of electron propagation in a crystalline medium, while the handle is an electrical insulator because the electrons in it are trapped in a disordered medium. It is again quantum mechanics which permits explanation of the fact that the blade, an electrical conductor, is also a thermal conductor, while the handle, an electrical insulator, is also a thermal insulator. To take an example more directly related to information theory, the behavior of the transistors etched on the chip inside your computer could not have been imagined by Bardeen, Brattain, and Shockley in 1947 were it not for their knowledge of quantum physics. Although your computer is not a quantum computer, it does function according to the principles of quantum mechanics!

This quantum behavior is also a collective behavior. Let us give two examples. First, if the value 0 of a bit is represented physically in a computer by an uncharged capacitor while the value 1 is represented by the same capacitor charged, the passage between the charged and uncharged states amounts to the displacement of 104 to 105 electrons.

We are still at the very beginnings of physical implementations of quantum computers. The devices listed below have been used successfully to entangle two qubits (at most!), except for NMR which has gone up to 7 qubits. It is still premature to try to predict which device will prove most effective for building a quantum computer capable of dealing with several hundred qubits (if such a computer someday exists); perhaps it will be something new, not on this list at all. In any case, it would be as foolish to predict that such a computer will not be available by 2050 as to predict the contrary.

The storage and processing of quantum information requires physical systems possessing the following properties (di Vincenzo criteria):

1. (i) they must be scalable, that is, capable of being extended to a sufficient number of qubits, with well defined qubits;

2. (ii) they must have qubits which can be initialized in the state |0〉;

3. (iii) they must have qubits which are carried by physical states of sufficiently long lifetime, so as to ensure that the quantum states remain coherent throughout the calculation;

4. (iv) they must possess a set of universal quantum gates: unitary transformations on individual qubits and a cNOT gate, which are obtained by controlled manipulations;

5. (v) there must be an efficient procedure for measuring the state of the qubits at the end of the calculation (readout of the results).

This book originated in a course given first to computer scientists at the University of Nice–Sophia Antipolis in October 2003, and later to second-year graduate students at the same University.

Quantum information is a field which at present is undergoing intensive development and, owing to the novelty of the concepts involved, it seems to me it should be of interest to a broad range of scientists beyond those actually working in the field. My goal here is to give an elementary introduction which is accessible not only to physicists, but also to mathematicians and computer scientists desiring an initiation into the subject. This initiation includes the essential ideas from quantum mechanics, and the only prerequisite is knowledge of linear algebra at the undergraduate level and some physics background at the high school level (except in Chapter 6 where I discuss the physical implementations of quantum computers and more physics background is required). In order to make it easier for mathematicians and computer scientists to follow, I give an elementary presentation of the Dirac notation. Quantum physics is an extremely large subject, and so I have attempted to limit the introductory concepts to the barest minimum needed to understand quantum information.

In Chapter 2, I introduce the concept of the quantum bit or qubit using the simplest possible example, that of the photon polarization. This example allows me to present the essential ideas of quantum mechanics and explain quantum cryptography. Chapter 3 generalizes the concept of qubit to other physical systems like spin 1/2 and the two-level atom, and explains how qubits can be manipulated by means of Rabi oscillations.

In this chapter we shall present the basic postulates of quantum physics, generalizing the results obtained in the preceding chapter for the two special cases of photon polarization and spin 1/2. In general, the space of states will a priori have any dimension N, which may even be infinite, rather than only two dimensions. The postulates which we present in this chapter fix the general conceptual framework of quantum mechanics and do not directly provide the tools necessary for solving specific problems. The solution of a specific physical problem always involves a modeling stage, where the system to be studied is simplified, the approximations to be used are defined, and so on, and this modeling stage inevitably rests on more or less heuristic arguments which cannot be derived within the general framework of quantum physics. In Section 3.2.5 we gave an example of a heuristic procedure leading to the solution of a specific problem, that of the motion of a spin 1/2 in a magnetic field.

Other sets of postulates can be used. For example, another approach is to state the postulates of quantum mechanics in terms of path integrals. As is often the case, the same physical theory can be dressed in various different mathematical clothes. Finally, it should be emphasized that the postulates of quantum physics give rise to some difficult epistemological problems which are still largely under debate and which we do not discuss in this book. The interested reader may consult, for example, the book by Isham [1995].