From here, we start relaxing the constraints that we had imposed in the earlier chapters for simplicity, and start moving towards more general scenarios. First, measurements with inadequate resolution, employing insufficiently selective measurement devices have been introduced. In this book they have been termed imprecise measurements. Here, we admit for the first time that all measurements do not prepare quantum states. We argue how the description of imprecise observables requires the use of Hermitian operators with degenerate eigenvalues: degeneracy encodes insufficient selectiveness. The general law of state preparation, the standard jargon for which is “collapse postulate”, is described in its full glory here. We then move on to introduce the idea of compatibility of observables and demonstrate how certain groups of imprecise measurements can collaborate to prepare states. This leads to the important idea of “complete set of compatible observables” and the related concepts of “complete and incomplete measurements”. Subsequently, the association of compatible observables and commuting operators is established which also connects complete set of compatible observables and complete set of commuting operators (both of which are accronymed CSCO, without ambiguity). We also give an explicit illustration of the algebraic construction of a CSCO.
From here, we start relaxing the constraints that we had imposed in the earlier chapters for simplicity, and start moving towards more general scenarios. First, measurements with inadequate resolution, employing insufficiently selective measurement devices have been introduced. In this book they have been termed imprecise measurements. Here, we admit for the first time that all measurements do not prepare quantum states. We argue how the description of imprecise observables requires the use of Hermitian operators with degenerate eigenvalues: degeneracy encodes insufficient selectiveness. The general law of state preparation, the standard jargon for which is “collapse postulate”, is described in its full glory here. We then move on to introduce the idea of compatibility of observables and demonstrate how certain groups of imprecise measurements can collaborate to prepare states. This leads to the important idea of “complete set of compatible observables” and the related concepts of “complete and incomplete measurements”. Subsequently, the association of compatible observables and commuting operators is established which also connects complete set of compatible observables and complete set of commuting operators (both of which are accronymed CSCO, without ambiguity). We also give an explicit illustration of the algebraic construction of a CSCO.
This chapter employs the following notation and basic formulae of quantum mechanics; the units are such that h = c = 1. Observables are represented in quantum mechanics by Hermitian operators in Hilbert space. These operators are analogues of finite-dimensional Hermitian operators, but the infinite dimensionality requires some mathematical explanation. The physical fact that not all measurements (preparations) are compatible with each other corresponds to the fact that not all operators commute. The ‘state’ of a physical system is fully determined in one way or another if a complete set of compatible measurements has been carried through such that no further measurement exists which is compatible with, and which is not simply a calculable function of, those measurements already made. Since not all independent measurements are compatible, there exist several non-compatible ‘complete sets of measurements’ for a given physical system, which lead to different aspects of that system.
From this chapter we start our exposition of quantum mechanics. First, we lay down the typical setting of a quantum mechanical problem. The fundamental feature of quantum indeterminism and the probabilistic nature of quantum mechanics is emphasized. We proceed by constructing, from the single notion of measurement, all the essential conceptual ingredients of quantum mechanics (i.e., the concepts of observable, state and system). Then we formulate the target questions that quantum mechanics aims to solve. Next, we proceed to state the specific laws that constitute the logical premise of the subject. Finally, we demonstrate how the laws, most dominantly, the superposition principle (the fact that quantum probabilities interfere) provides an improved recipe for obtaining the answers to the target questions of quantum mechanics. This also leads to a more refined definition of the quantum state. In order to be able to quickly introduce the basic framework of quantum mechanics, we restrict the discussion to a simple scenario of “instantaneously subsequent measurements” (where there is no time evolution), and where states can be prepared by a single measurement (in the standard jargon, this would be called, “quantum mechanics without degeneracy”; these are descriptions involving CSCOs comprising just one observable).
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.
Basic scientific theories usually start with a set of hypotheses or “postulates.” There is generally no logical reason, apart from internal consistency, that can be given to justify such postulates absolutely. They come from ‘revelations’ in the minds of ‘geniuses,’ most likely with hints from Nature based on extensive careful observations. Their general validity can only be established through experimental verification. If numerous rigorously derived logical consequences of a very small set of postulates all agree with experimental observations without exception, one is inclined to accept these postulates as correct descriptions of the laws of Nature and use them confidently to explain and predict other natural phenomena. Quantum mechanics is no exception. It is based on a few postulates. For the purpose of the present discussion, we begin with three basic postulates involving: the “state functions,” “operators,” and “equations of motion.” In this chapter, this set of basic postulates and some of the corollaries and related definitions of terms are introduced and discussed. We will first simply state these postulates and introduce some of the related mathematical tools and concepts that are needed to arrive at their logical consequences later. To those who have not been exposed to the subject of quantum mechanics before, each of these postulates taken by itself may appear puzzling and meaningless at first. It should be borne in mind, however, that it is the collection of these postulates as a whole that forms the foundations of quantum mechanics.
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.
In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean. BASIC FEATURES OF QUANTUM OBSERVABLES This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.