The three main issues in the foundations of quantum theory are measurement, non‐locality, and identical particles (quantum statistics). All of these pertain to the relations between different parts of a composite system. Hence this chapter covers tensor products, reduction of the density matrix (reduced states), and certain aspects of Hugh Everett's relative state formalism. This is then utilized in an extended discussion of measurement as a quantum‐mechanical process.
The three main issues in the foundations of quantum theory are measurement, non‐locality, and identical particles (quantum statistics). All of these pertain to the relations between different parts of a composite system. Hence this chapter covers tensor products, reduction of the density matrix (reduced states), and certain aspects of Hugh Everett's relative state formalism. This is then utilized in an extended discussion of measurement as a quantum‐mechanical process.
In this final chapter we discuss composite systems in Quantum Mechanics. First, we lay out the necessary conditions that the state space of a composite system must satisfy by considering the trivial case of two juxtaposed systems. Then we point out the features that the interesting case of a composite system, whose parts interact with each other, must have. We then move to demonstrate how such a system can be naturally described (quantized) by employing tensor product spaces. The definition and the most relevant properties of tensor product spaces are introduced. Then the meaning of “interactions” in quantum mechanics is explored and the most nontrivial aspects of correlation and entanglement are discussed. Finally, in the last section, we introduce systems of identical particles. We describe how to represent the state space of such systems in terms of one-particle states, number operators and occupation number representations. Classification of identical particles into Bosons and Fermions is explained. Finally, introducing symmetric and antisymmetric states, the traditional quantization scheme using the symmetrization postulate is described. The permutation operator is introduced which is the main tool for dealing with identical particle systems in this language. We wind up by indicating how exchange degeneracy is removed which embodies the signature of indistinguishability.
This chapter is concerned with the measure of quantum states . This requires one to introduce the subtle notion of quantum measurement , an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, starting with Dirac notation , a formalism that is not only very elegant but is relatively simple to handle. The introduction of Dirac notation makes it possible to become familiar with the inner product for quantum states, as well as different properties for operators and states concerning projection , change of basis , unitary transformations , matrix elements , similarity transformations , eigenvalues and eigenstates , spectral decomposition and diagonal representation , matrix trace and density operator or matrix . The concept of density matrix makes it possible to provide a very first and brief hint of the analog of Shannon's entropy in the quantum world, referred to as von Neumann's entropy , to be further developed in Chapter 21. Once we have all the required tools, we can focus on quantum measurement and analyze three different types referred to as basis-state measurements , projection or von Neumann measurements , and POVM measurements. In particular, POVM measurements are shown to possess a remarkable property of unambiguous quantum state discrimination (UQSD), after which it is possible to derive “absolutely certain” information from unknown system states. The more complex case of quantum measurements in composite systems described by joint or tensor states is then considered.
In most textbooks, measurement does not receive the full attention it deserves and sometimes is even not treated at all, apart a brief and cryptic mention of the “reduction of the wave packet.” However, in the last decades, the situation has profoundly changed and it is time to consider measurement a fundamental part of quantum mechanics, even, to a certain extent, an important generalization of the traditional theory (see also Chs. 14–15). This chapter consists of three major parts. In the first block (Secs. 9.1–9.4) we develop the main physical features of the measurement process: the heart of the argument is here represented by the aspects related to the interpretation. In the second part (Secs. 9.5–9.8) we discuss several special (and partly interdependent) topics of measurement: the heart here is represented by experimental aspects. In the third part (Secs. 9.9–9.12) we deal with the measurement process on a more formal plane, making use, in particular, of the generalization represented by the concepts of effect and positive operator valued measure (POVM). As we have said, the measurement problem is one of the most fundamental issues in the conceptual structure of quantum mechanics (as described in Sec. 9.1) and has a long history that will be examined in Sec. 9.2. In this context, the existence of apparently paradoxical quantum states comes about: the so-called Schrödinger cat states (see Sec. 9.3).
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.
We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If α and β are entities, we define what it means for α to be a part of β. This relationship is essentially equivalent to α being a function of β and in this case β can be employed to measure α. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map α^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.
This chapter reviews the quantum theoretical formalism and sketches its foundational problems. The main function of this chapter is to establish common ground for the interpretive discussions which follow, not to provide a comprehensive introduction to quantum theory. 6 Its first section recapitulates the essential aspects of the Hilbert space formalism of quantum mechanics, the second section covers the textbook account of measurement and measurement collapse, and the third and final section outlines the quantum theoretical treatment of many-component systems and explains the concept of quantum entanglement. Further introductory material is provided in the Appendices A, B, and C, all of which cover important no-go theorems for the interpretation of the theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.