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An application of operator *-algebras to the theory of quantum measurement

Juan Sotelo-Campos

Published Jan 1, 1987• 0 citations

Abstract

In this thesis we adapt the Davies and Lewis’ operational approach to quantum probability to the Op*-algebra framework. In Chapter I we introduce our formalism of quantum mechanics. We define the space of pure states U as a nuclear-Frechet space generated by a self-adjoint operator M whose inverse is nuclear. The algebra of observables is taken to be L+, the adjoint-stable set of linear maps from U into itself. This algebra and its dual under the uniform topology form a dual pair. We derive several algebraic and topological properties of this dual pair and show that it forms a suitable structure for the adaptation of Davies and Lewis’ theory to the Op*-algebra framework. In Chapter II we set the basis of the Davies and Lewis’ operational approach to quantum probability within the framework developed in chapter I. Here we define the fundamental concepts of an expectation and an instrument and discuss their relationship. We show that instruments are bounded Radon measures and that the compose of two of them is not in general a stable operation on the universe of all instruments. We conclude the chapter discussing the general Robertson-Heisenberg uncertainty relations. In Chapter III we build a special class of instruments to measure Q, P and H on S; these are based on Davies’ approximate position measurements. In particular we show that the compose of any two, hence any finite number, of these instruments is an instrument.

DOI: 10.21954/ou.ro.0000f940

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In this thesis we adapt the Davies and Lewis’ operational approach to quantum probability to the Op*-algebra framework. In Chapter I we introduce our formalism of quantum mechanics. We define the space of pure states U as a nuclear-Frechet space generated by a self-adjoint operator M whose inverse is nuclear. The algebra of observables is taken to be L+, the adjoint-stable set of linear maps from U into itself. This algebra and its dual under the uniform topology form a dual pair. We derive several algebraic and topological properties of this dual pair and show that it forms a suitable structure for the adaptation of Davies and Lewis’ theory to the Op*-algebra framework. In Chapter II we set the basis of the Davies and Lewis’ operational approach to quantum probability within the framework developed in chapter I. Here we define the fundamental concepts of an expectation and an instrument and discuss their relationship. We show that instruments are bounded Radon measures and that the compose of two of them is not in general a stable operation on the universe of all instruments. We conclude the chapter discussing the general Robertson-Heisenberg uncertainty relations. In Chapter III we build a special class of instruments to measure Q, P and H on S; these are based on Davies’ approximate position measurements. In particular we show that the compose of any two, hence any finite number, of these instruments is an instrument.

A Theory of Quantum Measurement Based on the CCR Algebra L

^+(\mathcal W)

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.

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.

Our basic structure is a finite-dimensional complex Hilbert space

H

. We point out that the set of effects on

H

form a convex effect algebra. Although the set of operators on

H

also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.

Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states and to transitions between quantum effects. We also present their dependence on a measured operation or instrument. We begin by defining our concepts on a general quantum effect algebra. These concepts are illustrated using Holevo operations and instruments. We then present transition probabilities in the special case of the Hilbert space formulation of quantum mechanics. We show that for pure states and particular types of operations the transition probabilities reduce to their usual form. We give examples in terms of Lüders operations and instruments.

Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.