The reduction of the state vector is rightly held as one of the most puzzling aspects of quantum theory. Before adding a few more pages to the inflatory literature on the subject, I wish to voice a somewhat sobering remark. Namely, there is a striking contrast between the amount of theoretical discussions about the nature and significance of this notion, and its practical import. Indeed, the so-called “axiom” of reduction of the state vector has a rather limited use in the effective working of quantum theory. There are very few situations where we really need to assert that, after a measurement, the state vector has been reduced to the eigenvector of the physical property corresponding to the eigenvalue resulting from the measurement (“observed”). Most of the time, in effect, the system under investigation is forgotten or discarded after the measurement - or even destroyed by it (photons). We rarely make successive observations of the same system, so that we do not usually need to know what the state of a system is after a measurement. One might argue that the reduction of the state vector is operationally used mainly to describe the preparation of a system before acting upon it, that is, to ensure that we are indeed dealing with spin up electrons or velocity selected protons such as we wish to experiment about. Anyway, the statement of reduction is to be considered more as a kind of consistency condition to be required at the beginning or end of a theoretical analysis than as an effective tool of this analysis - such as, for instance, the statement of unitary evolution, which leads to the concrete construction of the evolution operator, hence the S-matrix, cross-sections, etc. In other words, the practical status of the state vector reduction, even if it is considered as an axiom of quantum theory, is different enough from that of the constructive axioms of the theory.
The reduction of the state vector is rightly held as one of the most puzzling aspects of quantum theory. Before adding a few more pages to the inflatory literature on the subject, I wish to voice a somewhat sobering remark. Namely, there is a striking contrast between the amount of theoretical discussions about the nature and significance of this notion, and its practical import. Indeed, the so-called “axiom” of reduction of the state vector has a rather limited use in the effective working of quantum theory. There are very few situations where we really need to assert that, after a measurement, the state vector has been reduced to the eigenvector of the physical property corresponding to the eigenvalue resulting from the measurement (“observed”). Most of the time, in effect, the system under investigation is forgotten or discarded after the measurement - or even destroyed by it (photons). We rarely make successive observations of the same system, so that we do not usually need to know what the state of a system is after a measurement. One might argue that the reduction of the state vector is operationally used mainly to describe the preparation of a system before acting upon it, that is, to ensure that we are indeed dealing with spin up electrons or velocity selected protons such as we wish to experiment about. Anyway, the statement of reduction is to be considered more as a kind of consistency condition to be required at the beginning or end of a theoretical analysis than as an effective tool of this analysis - such as, for instance, the statement of unitary evolution, which leads to the concrete construction of the evolution operator, hence the S-matrix, cross-sections, etc. In other words, the practical status of the state vector reduction, even if it is considered as an axiom of quantum theory, is different enough from that of the constructive axioms of the theory.
From a fundamental point of view we must acknowledge that, since about 1935, our conceptual understanding of quantum mechanics has not progressed so much. Really new ideas are few and far between – except of course the major line initiated by the contribution of Bell [6]. This is in big contrast with the rest of physics, where new theoretical and experimental discoveries in many fields have flourished, very often with the help of the tools of quantum mechanics. The fantastic evolution of the experimental techniques has completely changed the situation. At the beginning of quantum mechanics, the observation of the tracks of single particles in Wilson chambers [348] played the essential role in the introduction of the postulate of state vector reduction, but otherwise it was impossible to observe continuously a single electron, atom, or ion; the experiments that theorists were proposing in discussions on the principles of quantum measurement were therefore “thought experiments” (“Gedanken Experiment”), as for instance in the famous Solvay meetings [1, 21]. But nowadays, after almost of century of experimental progress, experiments that were then unthinkable have become a reality. A huge number of contemporary experiments involves the laws of quantum mechanics in general; several books would not be sufficient to describe all of them. Nevertheless, in the majority of experiments, what is really observed is the sum over a very large number of particles of one individual microscopic observable (sum of atomic dipoles for instance), which is accurately described by the average value of this observable.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
Conceptual difficulties still remain in quantum mechanics, even if they had already been identified by its inventors. This does not mean that the theory is not successful! The reality is quite the opposite: in fact, independently of these difficulties, quantum mechanics is certainly one of the most successful theories of all science. One can even consider that its ability to adapt to new situations is one of its most remarkable features. It continues to give efficient and accurate predictions while new experiments are performed, even in situations that the founding fathers had no way to imagine. Actually, there are very few theories that have been verified with the same accuracy in so many situations. Nevertheless, it remains true that conceptual difficulties subsist, and their discussion is the object of this chapter. As we will see, most of them relate to the process of quantum measurement, in particular to the very nature of the random process that takes place on this occasion. We have seen that, in most cases, the wave function evolves gently, in a perfectly predictable and continuous way, according to the Schrödinger equation; in some cases only (when a measurement is performed), unpredictable changes take place, according to the Von Neumann postulate of state vector reduction. Obviously, having two different postulates for the evolution of the same mathematical object is very unusual in physics. The notion was a complete novelty when it was introduced, and still remains unique, but also the source of difficulties – in particular logical difficulties related to the compatibility between the two different postulates.
The problem of quantum state reduction The fundamental problem of quantum mechanics, as that theory is presently understood, is to make sense of the reduction of the state vector (i.e. collapse of the wave function ), denoted here by R . This issue is usually addressed in terms of the ‘quantum measurement problem’, which is to comprehend how, upon measurement of a quantum system, this (seemingly) discontinuous R -process can come about. A measurement, after all, merely consists of the quantum state under consideration becoming entangled with a more extended part of the physical universe, e.g. with a measuring apparatus. This measuring apparatus – together with the observing physicist and their common environment – should, according to conventional understanding, all also have some quantum description. Accordingly, there should be a quantum description of this entire quantum state, involving not only the original system under consideration but also the apparatus, physicist, and remaining environment – and this entire state would be expected to evolve continuously, solely according to the Schrödinger equation (unitary evolution), here denoted by the symbol U . Numerous different attitudes to R have been expressed over many years, ever since quantum mechanics was first clearly formulated. The most influential viewpoint has been the ‘Copenhagen interpretation’ of Niels Bohr, according to which the state vector │ψ〉 is not to be taken seriously as describing a quantum-level physical reality, but is to be regarded as merely referring to our (maximal) ‘knowledge’ of a physical system, and whose ultimate role is simply to provide us with a means to calculate probabilities when a measurement is performed on the system.
All physical systems register bits of information, whether it be an atom, an electrical current, the location of a billiard ball, or a switch. Information can be classical, quantum, or a hybrid of both, depending on the system. For example, an atom or an electron or a superconducting system can register quantum information because the quantum theory applies to each of these systems, but we can safely argue that the location of a billiard ball registers classical information only. These atoms or electrons or superconducting systems can also register classical bits because it is always possible for a quantum system to register classical bits. The term information , in the context of information theory, has a precise meaning that is somewhat different from our prior “everyday” experience with it. Recall that the notion of the physical bit refers to the physical representation of a bit, and the information bit is a measure of how much we learn from the outcome of a random experiment. Perhaps the word “surprise” better captures the notion of information as it applies in the context of information theory. This chapter begins our formal study of classical information. Recall that Chapter 2 reviewed some of the major operational tasks in classical information theory. Here, our approach is somewhat different because our aim is to provide an intuitive understanding of information measures, in terms of the parties who have access to the classical systems. We define precise mathematical formulas that measure the amount of information encoded in a single physical system or in multiple physical systems. The advantage of developing this theory is that we can study information in its own right without having to consider the details of the physical system that registers it. We first introduce the entropy in Section 10.1 as the expected surprise of a random variable. We extend this basic notion of entropy to develop other measures of information in Sections 10.2–10.6 that prove useful as intuitive informational measures, but also, and perhaps more importantly, these measures are the answers to operational tasks that one might wish to perform using noisy resources. While introducing these quantities, we discuss and prove several mathematical results concerning them that are important tools for the practicing information theorist.