Change is built into the structure of a quantum state. State collapse is a very particular and distinctive process, but it is not the only way that quantum states can change. For most of the time, various quantum systems that exist in nature wander about the universe without being involved in experiments, yet clearly, things can and do change. Eigenstates describe systems as having a definite fixed value of a given physical property. Such states are very important. If we know that an object is in an eigenstate, we can predict with absolute certainty what a measurement is going to produce. It’s very important to be able to tell if a state is a mixed state or an eigenstate with reference to a given measurement. There are many different operators in quantum theory, which have different jobs to do. Perhaps the most important ones are the operators that represent the process of measuring a physical property.
Change is built into the structure of a quantum state. State collapse is a very particular and distinctive process, but it is not the only way that quantum states can change. For most of the time, various quantum systems that exist in nature wander about the universe without being involved in experiments, yet clearly, things can and do change. Eigenstates describe systems as having a definite fixed value of a given physical property. Such states are very important. If we know that an object is in an eigenstate, we can predict with absolute certainty what a measurement is going to produce. It’s very important to be able to tell if a state is a mixed state or an eigenstate with reference to a given measurement. There are many different operators in quantum theory, which have different jobs to do. Perhaps the most important ones are the operators that represent the process of measuring a physical property.
State collapse is a very peculiar process, and I will have a lot more to say about it through this book. However, it is not the only way that quantum states can change. For most of the time, the various quantum systems in nature wander about the universe not being involved in experiments, yet clearly things can and do change. Any self-respecting theory must be capable of describing the ordinary business of a world in which things interact with one another, change, and develop. Part of our dealings in this chapter will be to develop the machinery that quantum theory uses to portray the changes of ordinary development and the, rather more mysterious, changes due to measurement.
In these notes, based on lectures given as part of the Les Houches summer school on Quantum Optics and Nanophotonics in August, 2013, I have tried to give a brief survey of some important approaches and modern tendencies in quantum measurement. I wish it to be clear from the outset that I shy explicitly away from the "quantum measurement problem," and that the present treatment aims to elucidate the theory and practice of various ways in which measurements can, in light of quantum mechanics, be carried out; and various formalisms for describing them. While the treatment is by necessity largely theoretical, the emphasis is meant to be on an experimental "perspective" on measurement -- that is, to place the priority on the possibility of gaining information through some process, and then attempting to model that process mathematically and consider its ramifications, rather than stressing a particular mathematical definition as the {\it sine qua non} of measurement. The textbook definition of measurement as being a particular set of mathematical operations carried out on particular sorts of operators has been so well drilled into us that many have the unfortunate tendency of saying "that experiment can't be described by projections onto the eigenstates of a Hermitian operator, so it is not really a measurement," when of course any practitioner of an experimental science such as physics should instead say "that experiment allowed us to measure something, and if the standard theory of measurement does not describe it, the standard theory of measurement is incomplete." Idealisations are important, but when the real world breaks the approximations made in the theory, it is the theory which must be fixed, and not the real world.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Most of the physicians that I know are chary of Physics, especially the theory of elementary particles. Even if in Isaac Newton's mathematically precise universe, everything follows clear-cut laws and predictions are easy given appropriate starting conditions. Perhaps, however, physicians coping with this uncertainty will perhaps feel more in tune with Heisenberg's uncertainty law. Heisenberg (1901–76) described a theory of quantum mechanics, published in 1925, when he was only 23 years old. The uncertainty principle describes an imprecision in nature resulting in a limit to what we can know about the behaviour of quantum particles and, hence the smallest scales of nature. Heisenberg was awarded the Nobel Prize for Physics for 1932 for this theory and its application that resulted amongst other things the discovery of allotropic forms of hydrogen. His theory stated that we cannot always give an electron a position in space at a given time, nor follow it in its orbit, so that we cannot assume that the planetary orbits proposed by Niels Bohr exist. Mechanical quantities, such as position and velocity, should be represented, not by ordinary numbers, but by abstract mathematical structures called ‘matrices’ and he formulated his new theory in terms of matrix equations. The principle of uncertainty, says that the determination of the position and momentum of a mobile particle necessarily contains errors the product of which cannot be less than the quantum constant h and that, although these errors are negligible on the human scale, they cannot be ignored in studies. We cannot measure the position (x) and the momentum (p) of a particle with absolute precision. The more accurately we know one of these values, the less accurately we know the other. There are many ways of looking at this and in medicine we need to take account of uncertainty. Uncertainty in medicine can be resolved in many ways and can include Occam's razor. In medicine, all presenting symptoms in a patient may be explained by a few possible causes. In medicine, all presenting symptoms in a patient may be due to several disease and social processes at the same time. How can this be relevant to a working physician? The whole of medical practice has parallels with the certainties of Newton and Bohr. The object of a clinical consultation is to give a sign or complaint a diagnostic named description to be followed by a treatment regime which resolves the problem. Yet every physician knows and automatically models his understanding of a clinical problem perhaps unknowingly on a model based on Heisenberg's uncertainty principle. A core process, which is complicated by factors understood or not. The same principles are relevant to so many other features of life. There are certain fixed points in the diagnostic and clinical process. Gender, race, age, weight to height ratio, privileged and social life and intelligence. Some people age at different rates to their age cohort? Similarly, obesity does not always matter. The patient′s family history is another variable, complicated occasionally when the putative father with a family health trait may not be the actual father. The social story is a minefield, with the possibility of every phase of a person’s life being of relevance. A further element is how well the physician and the patient and their caring team relate. The clinical examination then gives a further array of data. From all of this comes a diagnosis which gives a pathway to a treatment schedule the ability of the patient to manage their treatment regime becomes important. There is a tendency that as soon as the diagnosis is made then the other elements fade away from the care plan, though of course it persists in the person's life. We are all consciously or unconsciously aware of the complex fabric of clinical care. It is intriguing to appreciate that other disciplines have been presented with the same multifactorial uncertainties. The era in Physics when Heisenberg introduced his view of nuclear physics was one when a precise physical model was being completed and gave certainty to a complex problem. His principle while obviously reintroducing uncertainty gave freedom of thinking and allowed further fruitful progress. We physicians do not enjoy uncertainty, and to have it formalized is uncomfortable. So we instinctively identify the major problem and treat that. By and large this is a useful process. It is however important to see this as a convenience. A convenience that may not always suit the patient. Applying the Heisenberg principles gives a broader view of a patient's concerns and problems and better care.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.