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Positive Operator-Valued Measures and Phase-Space Representations Positive operator-valued measures and phase-space representations Citation for published version (APA): Beukema, R. (2003). Positive operator-valued measures and phase-space representations. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR561990 DOI: 10.6100/IR561990 Document status and date: Published: 01/01/2003 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 26. Sep. 2021 Positive Operator-Valued Measures and Phase-Space Representations PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 17 februari 2003 om 16.00 uur door Robbert Beukema geboren te Groningen Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. J. de Graaf en prof.dr. F.W. Sluijter Copromotor: dr. W.M. de Muynck CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Beukema, Robbert Positive operator-valued measures and phase-space representations / by Robbert Beukema. - Eindhoven : Technische Universiteit Eindhoven, 2003. Proefschrift. - ISBN 90-386-0662-1 NUR 921 Subject headings : functional analysis / quantum mechanics 2000 Mathematics Subject Classification : 46G10, 81S30, 81S99 Gedrukt door: Universiteitsdrukkerij, Technische Universiteit Eindhoven Contents Chapter 1. Introduction 7 1. Context 7 2. Probability distributions on the outcome space 8 3. Motivation 8 4. Results 10 5. Outline 11 Chapter 2. Preliminaries 13 1. Conventions, terminology and notation 13 2. Tensor product of two Hilbert spaces 13 3. Spaces of operators on a Hilbert space 14 4. Topologies on B∞(H) and L∞(µ) 14 Chapter 3. Positive operator-valued measures 17 1. Introduction 17 2. Conventions, terminology and notation 17 3. Preliminaries 18 4. Naimark’s theorem 19 5. Separability 21 6. Operator densities 23 7. Integration with respect to a POVM 26 8. Dominance of POVMs 29 9. Four remarks about the definition of ← 29 10. Dominance between PVMs 32 11. Isomorphic POVMs 33 12. Unitary equivalence 34 13. Image measures 35 14. Cones generated by POVMs 35 15. Maximal PVMs 37 16. Examples of maximal PVMs 39 17. Maximal POVMs 39 18. Bargmann POVM 46 19. Susskind-Glogower phase POVM 46 20. Density of the span of the range of a POVM 47 21. Prototypical POVMs 50 22. Multiplicity theory 51 23. Bounded subnormal operators 55 24. Restriction algebra 56 25. Constructing POVMs from PVMs 57 26. POVMs on Federer metric spaces 61 27. Integration of unbounded functions with respect to a POVM 64 Chapter 4. Wigner and Husimi representations, and in between 69 3 4 CONTENTS 1. Introduction 69 2. Conventions and notation 69 3. Wigner representation 70 4. A family of representations interpolating between the Wigner and Husimi representations 72 5. The range of Ws 74 6. The spaces B+ and B− 75 7. Characterization of Ws(B+) and Ws(B−) 76 Chapter 5. Phase-space analogues for quantum mechanical expectation values 79 1. Introduction 79 2. Wigner representation 80 3. Existence of phase-space analogues for quantum mechanical expectation values 80 4. Non-uniformity of approximation of expectation values 81 5. A method to approximate expectation values 82 Chapter 6. Extended Bargmann space 87 1. Introduction 87 2. Hermite polynomials 87 3. Laguerre polynomials 89 4. Relation between Hermite and Laguerre polynomials 92 5. Analytic and Harmonic functions 94 6. Extended Bargmann space 95 Appendix A. Miscellaneous results 99 1. Measurable covers and partitions 101 2. Hilbert-Schmidt operators 101 3. Trace-class operators 101 4. Convergence and σ-additivity of measures 101 5. Monotone convergence theorem and a partial converse 102 6. Radon-Nikodym theorem 102 7. Strong topological dual of L1(Ω, Σ, µ) 103 Appendix B. A criteria for the density of subspaces 105 Appendix C. Weak-star topology on dual Banach spaces 107 Appendix D. Bounded sesquilinear forms on Hilbert spaces 111 Appendix E. Hilbertian semi-norms 113 Appendix F. Bargmann space 117 Appendix G. C∗-algebras 119 1. Von Neumann algebras 119 2. Commutative von Neumann algebras 119 3. Maximal commutative von Neumann algebras 119 Appendix H. Operator ranges 121 Appendix I. Measurable Banach space-valued functions 123 Appendix J. Carleman operators 127 1. Generating vectors 127 CONTENTS 5 Appendix K. Application of Gauss’s theorem 129 1. On the real line 129 2. On the complex plane 130 m 3. Gaussian convolution on Lp(R ) 132 Appendix L. Integral operators with a Gaussian kernel 135 1. Calculations 135 2. The integral operators Spqr 136 1/2 Appendix M. Gelfand-Shilov space S1/2. 139 1. Introduction 139 2. The spaces H+ and H−. 139 (s) (s) 3. The spaces H+ and H− 140 (s) (s) 4. Duality between H− and H+ 141 5. Approximation in H− 141 Appendix N. Estimates 143 1. Estimates for Bessel functions on (0, ∞) 143 2. Estimates for the Gamma function 146 3. Estimates for Laguerre polynomials on (0, ∞) 146 Appendix O. Cambell-Baker-Hausdorff formulas 149 1. Introduction 149 2. e[A,·] if A is nilpotent 149 3. A generalization 150 4. If operators A and B are skew-adjoint 151 5. If operators A and B commute with [A, B]. 151 Bibliography 153 Index 157 Acknowledgement 159 Samenvatting 161 Curriculum Vitae 163 CHAPTER 1 Introduction 1. Context In the standard formalism of quantum mechanics as formulated by Dirac [36] and von Neumann [106], a measurement with outcome space Ω corresponds to a spectral measure M concentrated on Ω and acting on a complex separable Hilbert space H. States of a quantum mechanical system are described mathematically by density operators ρ, which are non-negative trace-class operators on H with trace 1. The set of all density operators, the ‘state space,’ is denoted by S(H). The following is postulated: (P) The number Tr(ρM(∆)), i.e. the trace of the composition of operators ρ and M(∆), is to be interpreted as the probability that in state ρ the outcome of the measurement lies in ∆. Two spectral measures E and F acting on the same Hilbert space, and defined on the Borel subsets of outcome spaces Ω1 and Ω2 respectively, can be combined to get a spectral measure E⊗F, defined on the Borel subsets of Ω1 ×Ω2 by E⊗F (∆1 ×∆2) = E(∆1)F (∆2) (and unique extension), if and only if E and F commute. Von Neumann shows (Section III.3 in [106]) that this condition is necessary and sufficient in order for the quantities corresponding to E and F to be simultaneously measurable with arbitrary high accuracy. Such measurements are not always desirable because they are very incomplete in the 0 sense that to each density operator ρ corresponds a large class [ρ]M of states ρ satisfying Tr((ρ − ρ0)M(∆)) = 0 for all ∆. In other words: The range of M, considered as a subspace of the vector space of bounded self-adjoint operators, has a large ‘orthogonal complement’ in the space of self-adjoint trace-class operators. More recent investigations [74, 73], [31] (among others) have led to the so-called ‘operational approach to quantum measurement.’ This theory is about the compound system formed by object and measurement instrument, and the interaction between these two parts plays an essential role. (See e.g. [58], [55], [30], [15] and [17].) A measure- ment is described mathematically by an operator-valued measure M, defined on a σ-field Σ of subsets of a set Ω and taking values in the set of non-negative bounded operators on a separable complex Hilbert space, and normalized by the condition M(Ω) = I, the identity operator. A measure M with these properties, is called a normalized positive operator-valued measure, which we abbreviate to ‘POVM.’ A POVM M whose range consists of projection operators only, is called a projection-valued measure, which we abbreviate to ‘PVM.’ Measurements corresponding to POVMs are called generalized measurements, and measurements corresponding to PVMs (spectral measures in par- ticular) are called simple measurements. Many experiments performed in practice are generalized measurements: For example [83].
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