The photon in the interferometer

This chapter introduces many of the ideas of quantum theory by exploring three specific “case studies” of quantum systems. Each is an example of a qubit, a generic name for the simplest type of quantum system. The concepts we develop will be incorporated into a rigorous mathematical framework in the next chapter. Our business here is to provide some intuition about why that mathematical framework is reasonable and appropriate for dealing with the quantum facts of life.

Interferometers

In Section 1.2 we discussed the two-slit interference experiment with a single photon. In that experiment, the partial waves of probability amplitude were spread throughout the entire region of space beyond the two slits. It is much easier to analyze the situation in an interferometer, an optical apparatus in which the light is restricted to a finite number of discrete beams. The beams may be guided from one point to another, split apart or recombined as needed, and when two beams are recombined into one, the result may show interference effects. At the end of the interferometer, one or more sensors can measure the intensity of various beams. (A beam is just a possible path for the light, so there is nothing paradoxical in talking about a beam of zero intensity.) Figure 2.1 shows the layout of a Mach–Zehnder interferometer, which is an example of this kind of apparatus.

Note: this introduction is written in an intuitive style, so a scientifically oriented non-mathematician might get something out of it. It is the only part of the book that requires no mathematical expertise.

Question: What is ergodic theory?

Let's start with two examples.

Example 1: Imagine a potentially oddly shaped billiard table having no pockets and a frictionless surface. Part of the table is painted white and part of the table is painted black. A billiard ball is placed in a random spot and shot along a trajectory with a random velocity. You meanwhile are blindfolded and don't know the shape of the table. However, as the billiard ball careens around, you receive constant updates on when it's in the black part of the table, and when it's in the white part of the table. From this information you are to deduce as much as you can about the entire setup: for example, whether or not it is possible that the table is in the shape of a rectangle.

Example 2: (This example is extremely vague by intention.) Imagine you are receiving a sequence of signals from outer space. The signal seems to be in some sense random, but there are recurring patterns whose frequencies are stationary (that is, do not alter over time). We are unable to detect a precise signal but we can encode it by interpreting five successive signals as one signal: unfortunately, this code loses information. Furthermore, we make occasional mistakes. We wish to get as much knowledge as possible about the original process.

Measure preserving transformations. The subject matter encompassing the previous two examples is called ergodic theory.

1. Discussion. In this opening chapter, we offer a review of the basic facts we need from measure theory for the rest of the book (it doubles as an introduction to our pedagogic method). For readers seeking a true introduction to the subject, we recommend first perusing, e.g. Folland (1984); experts meanwhile may safely jump to Chapter 2.

2. Comment. When an exercise is given in the middle of a proof, the end of the exercise will be signaled by a dot:

The conclusion of a proof is signaled by the box sign at the right margin, thus:

Basic definitions

In this subchapter, we discuss algebras, σ-algebras, generation of a σ-algebra by a family of subsets, completion with respect to a measure and relevant definitions.

3. Definition. Let Ω be a set. An algebra of subsets of Ω is a non-empty collection A of subsets of Ω that is closed under finite unions and complementation. A σ-algebra is a collection A of subsets of Ω that is closed under countable unions and complementation.

4. Comment. Every algebra of subsets of Ω contains the trivial algebra {∅, Ω}.

5. Exercise. Let Ω be a set and let C be a family of subsets of Ω. Show that the intersection of all σ-algebras of subsets of Ω containing C is itself a σ-algebra.

Hilbert space

The prototype qubit systems of the last chapter are very simple, but they can be generalized to more complicated versions. We can send a photon through an interferometer with three, four or more distinct beams. We can perform experiments on particles with higher intrinsic angular momentum than the spin-½ particles we have discussed. And we can analyze atomic systems in situations that involve more than two different energy levels. For these cases and others, we will need a more general version of quantum theory.

That theory will include two pieces. First, we will have a general mathematical structure that is applicable to many kinds of system. Here the qubit case will be our guide, since many of the basic concepts for other quantum systems are already present in the qubit case. Second, we will have to describe how to apply the quantum formalism to specific physical situations. Though the quantum systems we discuss will appear quite various, they share strong family resemblances that are expressed in the common mathematical framework. Keeping the framework in mind will help us understand specific examples; keeping the examples in mind will help us understand the framework.

The states of a quantum system are described by kets |ψ〉, which obey the principle of superposition. This means that the kets are elements of an abstract vector space ℋ called a Hilbert space.

Introduction

Digital communication systems have been studied for many decades, and they have become an integral part of the technological world we live in. Many excellent books in recent years have told the story of this communication revolution, and have explained in considerable depth the theory and applications. Since the late 1990s particularly, there have been a number of significant contributions to digital communications from the signal processing community. This book presents a number of these recent developments, with emphasis on the use of filter bank precoders and equalizers. Optimization of these systems will be one of the main themes in this book. Both multiple-input multiple-output (MIMO) systems and single-input single-output (SISO) systems will be considered. It is assumed that the reader has had some exposure to digital communications and signal processing at the introductory level. Many text books cover this prerequisite, and some are mentioned at the beginning of Sec. 1.5.

Before we describe the contents of the book we first give an introductory description of analog and digital communication systems in the next few sections. The scope and outline of the book will be described in Sec. 1.5.