In the last chapter, we considered the possibility that an object such as the pointer of an apparatus designed to measure H/V polarisation (see Figure 7.1) might in principle be forbidden from being in a quantum superposition if it was large enough. A microscopic system, such as a photon polarised at 45° to the horizontal, would then collapse into an h or a v state as a consequence of such a measurement. To test this idea, we considered how we might detect a macroscopic object in such a superposition and we found that this was very difficult in practice. The oscillating pointer of Figure 7.1 is very sensitive to random thermal disturbances and as a result is almost certainly going to swing in one direction or the other, rather than being in a superposition of both. Macroscopic superpositions have indeed been observed in SQUIDs (see the last section in Chapter 7), but only after great care has been taken to eliminate similar thermal effects.

In the context of the last chapter, the randomness associated with thermal motion was considered as a nuisance to be eliminated, but might it instead be just what we are looking for? Perhaps it is not that thermal effects prevent us observing quantum superpositions, but rather that such states are impossible in principle when thermal disturbances are present. This could provide another way out of the measurement problem.

In the last chapter we saw how the measurement problem in quantum theory arises when we try to treat the measurement apparatus as a quantum system. We need more apparatus to measure which state the first apparatus is in, and we have a measurement chain that seems to go on indefinitely. There is, however, one place where this apparently infinite sequence certainly seems to end and that is when the information reaches us. We know from experience that when we look at the photon detector we see that either it has recorded the passage of a photon or it hasn't. When we open the box and look at the cat either it is dead or it is alive; we never see it in the state of suspended animation that quantum physics alleges it should be in until its state is measured. It might follow, therefore, that human beings should be looked on as the ultimate measuring apparatus. If so, what aspect of human beings is it that gives them this apparently unique quality? It is this question and its implications that form the subject of the present chapter.

Let us examine more closely what goes on when a human being observes the quantum state of a system. We return to the set-up described in the last chapter, where a 45° photon passes through a polarisation analyser that moves a pointer to one of two positions (H or V) depending on whether the photon is horizontally or vertically polarised. At least that is what would happen if the analyser and pointer behaved as a measuring apparatus. If however we treat them as part of the quantum system, the pointer is placed in a superposition of being at H and at V until its state is measured. We now add a human observer who looks at the pointer (it would be possible to imagine the observer using one of his other senses, for example hearing a particular sound caused by a change in the pointer’s position, but it is clearer if we think of a visual observation).

The previous chapter surveyed part of the rich variety of physical phenomena that can be understood using the ideas of quantum physics. Now that we are beginning the task of looking more deeply into the subject we shall find it useful to concentrate on examples that are comparatively simple to understand but which still illustrate the fundamental principles and highlight the basic conceptual problems. Some years ago most writers discussing such topics would probably have turned to the example of a ‘particle’ passing through a two-slit apparatus (as in Figure 1.2), whose wave properties are revealed in the interference pattern. Much of the discussion would have been in terms of wave–particle duality and the problems involved in position and momentum measurements, as in the discussion of the uncertainty principle in the last chapter. However, there are essentially an infinite number of places where the particle can be and an infinite number of possible values of its momentum, and this complicates the discussion considerably. We can illustrate all the points of principle we want to discuss by considering situations where a measurement has only a small number of possible outcomes. One such quantity relating to the physics of light beams and photons is known as polarisation. In the next section we discuss it in the context of the classical wave theory of light, and the rest of the chapter extends the concept to situations where the photon nature of light is important.

Albert Einstein's comment that ‘God does not play dice’ sums up the way many people react when they first encounter the ideas discussed in the previous chapters. How can it be that some future events are not completely determined by the way things are at present? How can a cause have two or more possible effects? If the choice of future events is not determined by natural laws does it mean that some supernatural force (God?) is involved wherever a quantum event occurs? Questions of this kind trouble many students of physics, though most get used to the conceptual problems, say ‘Nature is just like that’ and apply the ideas of quantum physics to their study or research without worrying about their fundamental truth or falsity. Some, however, never get used to the, at least apparent, contradictions. Others believe that the fundamental processes underlying the basic physics of the universe must be describable in deterministic, or at least realistic, terms and are therefore attracted by hidden-variable theories. Einstein was one of those. He stood out obstinately against the growing consensus of opinion in the 1920s and 30s that was prepared to accept indeterminism and the lack of objective realism as a price to be paid for a theory that was proving so successful in a wide variety of practical situations. Ironically, however, Einstein's greatest contribution to the field was not some subtle explanation of the underlying structure of quantum physics but the exposure of yet another surprising consequence of quantum theory.

In this chapter we ask if there is some aspect of the nature of the thermodynamic changes that occur in measuring processes that could clearly distinguish them from the kind of process for which reversibility is a possibility and to which pure quantum theory can be applied. This idea has been suggested on a number of occasions, and was developed in the 1980s by Ilya Prigogine, who won a Nobel prize in 1977 for his theoretical work in the field of irreversible chemical thermodynamics. The starting point of the approach by Prigogine and his co-workers is a re-examination of the validity of the ergodic principle, which leads to the idea of the Poincaré recurrence. We made the point in the last chapter that, in the simple case of a single particle confined to a rectangle, unless the starting angle has a special value, the particle trajectory will fill the whole rectangle and the particle will sooner or later revisit a state that is arbitrarily close to its initial state. By this we meant that, although the initial and final states cannot in general be precisely identical, we can make the difference between the initial and final positions and velocities as small as we like by waiting long enough. The implicit assumption is that the future behaviour of the system will not be significantly affected by this arbitrarily small change in state; its behaviour after the recurrence will then be practically the same as its behaviour was in the first place.

One of the main purposes of a physical theory is to describe and predict events observed in physical experiments. In this chapter we introduce the key physical concepts used in the description of quantum experiments and present their mathematical formalization as Hilbert space operators.

Duality of states and effects

Fundamental in quantum theory is the concept of a state, which is usually understood as a description of an ensemble of similarly prepared systems. An effect, on the other hand, is a measurement apparatus that produces either ‘yes’ or ‘no’ as an outcome. The concept of an effect was coined by Ludwig [96] and made popular by Kraus [89]. The duality of states and effects means, essentially, that when a state and an effect are specified we can calculate a probability distribution. This can be compared with the outcomes of real experiments, and it gives a physical meaning to the mathematical Hilbert space machinery.

Basic statistical framework

A basic situation in physics is the following: we have an object system under investigation, and we are trying to learn something about it by doing an experiment. As a result, a measurement outcome is registered. A statistical theory, such as quantum theory, does not in general predict which individual outcomes will occur in any particular measurement; it merely predicts their probabilities of occurrence. Hence, we take the output of an experiment (which consists of many measurement runs) to be a probability distribution on a set Ω of the possible measurement outcomes.

In Chapters 2 and 3 we assumed a model according to which an experiment is split into two parts – preparation and measurement. This led us to the associated concepts of states and observables, respectively. On the basis of this picture we can think about two types of apparatus, which can be characterized abstractly as follows.

• Preparators are devices producing quantum states. No input is required, but a quantum output is produced.

• Measurements are input–output devices that accept a quantum system as their input and produce a classical output in the form of measurement outcome distributions.

This setting is summarized in Figure 2.2. Furthermore, in subsection 3.5.2 we discussed coarse-graining relations; coarse-graining can be depicted as a box with a classical input and a classical output. This line of thought indicates that at least one type of input–output box is so far missing from our discussion – a device taking a quantum input and producing a quantum output. Hence, we say that

•quantum channels are input–output devices transforming quantum states into quantum states.

Each quantum channel can be placed between arbitrary preparation and measurement devices (Figure 4.1). Channels, and a slightly more general concept, that of operations, are the topics of this chapter.

Transforming quantum systems

In this section we discuss the mathematical definition of operations and channels. The notion of an operation was introduced by Haag and Kastler [70] within the framework of algebraic quantum field theory. Later Kraus [88], Lindblad [93] and Davies [51] discussed the requirement for operations to be completely positive.

In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B, is associated with a tensor product space HA? HB. Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems.

As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement, which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing.

Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al. [80] and Plenio et al. [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.

Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device.

In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments, will be investigated in this chapter.

There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process. A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements. Some other authors use the term indirect measurements.

Three levels of description of measurements

It is convenient to think of three levels of description when considering measurements.

Quantum theory, in its conventional formulation, is built on the theory of Hilbert spaces and operators. In this chapter we go through this basic material, which is central for the rest of the book. Our treatment is mainly intended as a refresher and a summary of useful results. It is assumed that the reader is already familiar with some of these concepts and elementary results, at least in the case of finite-dimensional inner product spaces.

We present proofs for propositions and theorems only if the proof itself is considered to be instructive and illustrative. This gives us the freedom to present the material in a topical order rather than in the strict order of mathematical implication. Good references for this chapter are the functional analysis textbooks by Conway [45], Pedersen [113] and Reed and Simon [121]. These books also contain the proofs that we skip here.

Hilbert spaces

As an introduction, before a formal definition is given, one may think of a Hilbert space as the closest possible generalization of the inner product spaces cd to infinite dimensions. Actually, there are no finite-dimensional Hilbert spaces (up to isomorphisms) other than cd spaces. The crucial requirement of completeness becomes relevant only in infinite-dimensional spaces. This defining property of Hilbert spaces guarantees that they are well-behaved mathematical objects, and many calculations can be done almost as easily as in c d.