Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
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.
This short chapter contains basics of the mathematical formalism for thequantum measurement theory. In this book we proceed mainly withthe von Neumann measurement theory in which observables are given byHermitian operators and the state update by projections. However, we alsomention the measurement formalism based on quantum instruments, sinceit gives the general framework for quantum measurements. This formalismis used only in Chapters 10 and 18. The latter chapter is devoted to quantum-likemodeling – the applications of the mathematical formalism and methodologyof quantum mechanics (QM) to cognition, psychology, and decision making.Surprisingly, in such applications even the simplest effects can’t be described bythe von Neumann theory. One should use quantum instruments (compare withquantum physics where the main body of theory can be presented solelywithin the von Neumann measurement theory).
The quantum mechanical measurement process is analysed in detail, especially from the group theoretical point of view. The first three chapters review the results obtained before briefly. In §1, the classical theory of measurement, originated by von Neumann, is birefly described. The importance of considering more general schemes of measurements than the original one is, however, emphasized. In §2, the role of conservation laws, in imposing limitations on the possibility of measurements, is discussed. Accurately measurable quantities are defined in this context. In §3, the role of conservation laws is further analysed. It is shown that the conserved quantity of the apparatus cannot be used for a “pointer reading.” The following two chapters discuss the nature of the exactly measurable quantities. In §4, exactly measurable quantities are explicitly determined. It is shown that the measurement of projection operators in the Hilbert space of the object which commute with the operators of the restricted Poincaré group give all the information on the states of the object that can be obtained by accurate measurements. In §5, the measurement on objects which are composed of more than one particle is discussed. The last chapter, §6, deals with approximately measurable quantities, but no complete estimate of the unavoidable inaccuracy of the measurement of these is obtained. It also contains some concluding remarks.
We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.
Abstract The object of this paper is to reformulate the principles of the quantum theory as a sequence of propositions which shall be either summary statements of standard experimental procedure or hypotheses concerning the results of experiment and having an immediate physical interpretation. It is shown that the standard process in micro-physics is a generalised spectral analysis, whose properties are simply expressible in symbolic form by means of projective or “idempotent” operators (Einzeloperatoren). It appears that only two hypotheses need be made and that these relate to the existence and properties of transition probabilities. From these fundamental principles, which have a direct physical significance, it is possible to deduce the subsidiary principles which form the accepted basis of the mathematical analysis of the quantum theory and which deal with the representation of quantum states and physical quantities by vectors and linear operators respectively. In this paper the emphasis is laid on the experimental process determining a state of a system and on the associated operators rather than on the state itself or the vector representing it in the system space. Projective operators, which represent actual processes of measurement, and unitary operators, which represent actual transformations of systems of measurement, are given priority over the (statistical) operators which represent physical variables. This method of representation makes the physical meaning of the theory fundamental, instead of leaving it to be extracted from a purely mathematical system of non-commutative algebra or differential equations.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.