In these early decades of the information age, the flow of information is becoming more and more central to our daily lives. It has therefore become important that information transmission be protected against eavesdropping (as, for example, when one sends credit card information over the Internet) and against noise (which might occur in a cell phone transmission, or when a compact disk is accidentally scratched). Though most of us depend on schemes that protect information in these ways, most of us also have a rather limited understanding of how this protection is done. Part of the aim of this book is to introduce the basic concepts underlying this endeavor.

Besides its practical significance, it happens that the subject of protecting information is intimately related to a number of central ideas in mathematics and computer science, and also, perhaps surprisingly, in physics. Thus in addition to its significance for society, the subject provides an ideal context for bringing ideas from these disciplines together. This interdisciplinarity is part of what has attracted us to the subject, and we hope it will appeal to the reader as well.

Among undergraduate texts on coding or cryptography, this book is unusual in its inclusion of quantum physics and the emerging technology of quantum information. Quantum cryptography, in which an eavesdropper is detected by his or her unavoidable disturbance of delicate quantum signals, was proposed in the 1980s and since then has been investigated and developed in a number of laboratories around the world.

The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics.

There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.

Recall that in the Bennett–Brassard key distribution scheme, after Alice and Bob have obtained bit strings that ideally are supposed to be identical, they have to do some further processing to make sure they end up with strings that really are identical. Our first goal in this chapter is to show how they can do this.

As you might expect, we will use error-correcting codes of the sort we have been discussing in the preceding chapter. However, the way we use error-correcting codes in quantum key distribution is not quite the same as in classical communication. Normally, one corrects errors by encoding one's message into special codewords that are sufficiently different from each other that they will still be distinguishable after passing through a noisy channel. But in quantum key distribution the “noise” of the channel might actually be the effect of an eavesdropper who is free to manipulate the signals sent by Alice. Ordinary error correction is not designed for such a setting. So instead of using codewords to encode the original transmission, we wait until all the bits have been sent and then use an error-correcting code after the fact. In this respect error correction in quantum key distribution is similar to the use of an error-correcting code in the “hat problem” discussed at the end of Chapter 4. There also, the error-correcting code is applied only after all the data – in that case the vector of hat colors – has been generated and conveyed to the participants.

In the preceding chapter we mentioned the inevitable errors that occur when one tries to send quantum signals over, say, an optical fiber, even when there is no eavesdropper. But errors in transmission are not a problem just for quantum cryptography. For this entire chapter we forget about sending quantum information and instead focus on simply transmitting ordinary data faithfully over some kind of channel. Moreover, we assume that the data either is not sensitive or has already been encrypted. Unfortunately, many methods for transmitting data are susceptible to outside influences that can cause errors. How do we protect information from these errors? Error-correcting codes provide a mathematical method of not only detecting these errors, but also correcting them. Nowadays error-correcting codes are ubiquitous; they are used, for example, in cell-phone transmissions and satellite links, in the representation of music on a compact disk, and even in the bar codes in grocery stores.

The story of modern error-correcting codes began with Claude Shannon's famous paper “A Mathematical Theory of Communication,” which was published in 1948. Shannon worked for Bell Labs where he specialized in finding solutions to problems that arose in telephone communication. Quite naturally, he started considering ways to correct errors that occurred when information was transmitted over phone lines. Richard Hamming, who also worked at Bell Labs on this problem, published a groundbreaking paper in 1950 on the subject.

As we have said, quantum mechanics has become the standard framework for most of what is done in physics and indeed has played this role for three-quarters of a century. For just as long, physicists and philosophers have, as we have already suggested, raised and discussed questions about the interpretation of quantum mechanics: Why do we single out measurement as a special kind of interaction that evokes a probabilistic, irreversible response from nature, when other kinds of interaction cause deterministic, reversible changes? How should we talk about quantum superpositions? What does entanglement tell us about the nature of reality? These are interesting questions and researchers still argue about the answers. However, in the last couple of decades researchers have also been thinking along the following line: Let us accept that quantum objects act in weird ways, and see if we can use this weirdness technologically. The two best examples of potential uses of quantum weirdness are quantum computation and quantum cryptography.

One of the first quantum cryptographic schemes to appear in the literature, and the only one we consider in detail in this book, was introduced by Charles Bennett and Gilles Brassard in 1984. Their idea is not to use quantum signals directly to convey secret information. Rather, they suggest using quantum signals to generate a secret cryptographic key shared between two parties. Thus the Bennett–Brassard scheme is an example of “quantum key distribution.”

Introduction

In an ordinary computer, information is stored in a collection of tiny circuits each of which is designed to have two stable and easily distinguishable configurations: each represents a bit. In our study of quantum cryptography, we have seen how it can be useful to express information not in ordinary bits but in qubits. Whereas a bit can have only two values, say 0 and 1, a qubit can be in any quantum superposition of ∣0〉 and ∣1〉. Moreover, a qubit can be entangled with other qubits. Thus one might wonder whether a quantum computer, in which the basic elements for storing and processing information are qubits, can outperform an ordinary (classical) computer in certain ways. This question was addressed by researchers starting in the 1980s. In terms of practical consequences, perhaps the most dramatic answer has been given by Peter Shor in his 1994 factoring algorithm for a quantum computer, an algorithm that is exponentially faster than any known classical algorithm. As we have seen in Chapter 1, the difficulty of factoring a product of two large primes is the basis of the security of the RSA cryptosystem. So if one could build a large enough quantum computer – and there is no reason in principle why this could not be done – the RSA system would be rendered ineffective. In this chapter we present the basics of quantum computation and then focus on Shor's factoring algorithm.

Fields

Fields are important algebraic structures used in almost all branches of mathematics. Here we only cover the definitions and theorems needed for the purposes of this book.

Definition. A field F is a set along with two operations (denoted with addition and multiplication notation) on pairs of elements of F such that the following properties are satisfied.

1. For all a and b in F, we have that a + b ∈ F.

2. For all a, b, and c in F, we have that (a + b) + c = a + (b + c).

3. There exists an element 0 in F satisfying a + 0 = a for all a ∈ F.

4. For every a ∈ F there exists a b in F such that a + b = 0.

5. For all a and b in F we have that a + b = b + a.

6. For all a and b in F we have that ab ∈ F.

7. For all a, b, and c in F we have that (ab)c = a(bc).

8. There is an element 1 in F satisfying 1a = a for all a ∈ F.

9. For every a ∈ F with a ≠ = 0, there exists a b ∈ F such that ab = 1.

10. For every a and b in F we have that ab = ba.

11. For every a, b, and c in F we have that a(b + c) = ab + ac.