Quantum Measurement

544 Pages·2016·6.111 MB·English

by  Paul Busch, Pekka Lahti, Juha-Pekka Pellonpää, Kari Ylinen (auth.)

Theoretical and Mathematical Physics
Paul Busch
Pekka Lahti
Juha-Pekka Pellonpää
Kari Ylinen
Quantum
Measurement
Quantum Measurement
Theoretical and Mathematical Physics
Theseriesfoundedin1975andformerly(until2005)entitledTextsandMonographsin
Physics (TMP) publishes high-level monographs in theoretical and mathematical phy-
sics.ThechangeoftitletoTheoreticalandMathematicalPhysics(TMP)signalsthatthe
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The books, written in a didactic style and containing a certain amount of elementary
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monographs. They can thus serve as basis for advanced studies, not only for lectures
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Editorial Board
W. Beiglböck, Institute of Applied Mathematics, University of Heidelberg,
Heidelberg, Germany
P. Chrusciel, Gravitational Physics, University of Vienna, Vienna, Austria
J.-P. Eckmann, Département de Physique Théorique, Université de Genéve,
Geneve, Switzerland
H. Grosse, Institute of Theoretical Physics, University of Vienna, Vienna, Austria
A. Kupiainen, Department of Mathematics, University of Helsinki, Helsinki, Finland
H. Löwen, Institute of Theoretical Physics, Heinrich-Heine-University of Düsseldorf,
Düsseldorf, Germany
M. Loss, School of Mathematics, Georgia Institute of Technology, Atlanta, USA
N.A. Nekrasov, IHÉS, Bures-sur-Yvette, France
M. Ohya, Tokyo University of Science, Noda, Japan
M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg,
Heidelberg, Germany
S. Smirnov, Mathematics Section, University of Geneva, Geneva, Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USA
J. Yngvason, Institute of Theoretical Physics, University of Vienna, Vienna, Austria
More information about this series at http://www.springer.com/series/720
Paul Busch Pekka Lahti
(cid:129)
ää
Juha-Pekka Pellonp Kari Ylinen
(cid:129)
Quantum Measurement
123
PaulBusch Juha-Pekka Pellonpää
DepartmentofMathematics,YorkCentrefor Department ofPhysics andAstronomy,
QuantumTechnologies TurkuCentrefor Quantum Physics
University of York University of Turku
York Turku
UK Finland
Pekka Lahti KariYlinen
Department ofPhysics andAstronomy, Department ofMathematics andStatistics
TurkuCentrefor Quantum Physics University of Turku
University of Turku Turku
Turku Finland
Finland
ISSN 1864-5879 ISSN 1864-5887 (electronic)
Theoretical andMathematical Physics
ISBN978-3-319-43387-5 ISBN978-3-319-43389-9 (eBook)
DOI 10.1007/978-3-319-43389-9
LibraryofCongressControlNumber:2016946315
©SpringerInternationalPublishingSwitzerland2016
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Preface
Quantum Measurement is a book on the mathematical and conceptual foundations
ofquantummechanics,withafocusonitsmeasurementtheory.Ithasbeenwritten
primarily for students of physics and mathematics with a taste for mathematical
rigour and conceptual clarity in their quest to understand quantum mechanics. We
hopeitwillalsoserveasausefulreferencetextforresearchersworkinginabroad
range of subfields of quantum physics and its foundations.
The exposition is divided into four parts entitled Mathematics (Chaps. 2–8),
Elements (Chaps. 9–13), Realisations (Chaps. 14–19), and Foundations
(Chaps. 20–23). An overview of each part is given in the Introduction, Chap. 1,
and each chapter begins with a brief non-technical outline of its contents.
A glance through the table of contents shows that different chapters require
somewhat different backgrounds and levels of prerequisite knowledge on the part
of the reader. The material is arranged in a logical (linear) order, so it should be
possible to read the book from beginning to end and gain the relevant skills along
the way, either from the text itself or occasionally from other sources cited.
However, the reader should also be able to start with any part or chapter of her or
his interest and turn to earlier parts where needed.
PartIisdesignedtobeaccessibletoareaderpossessinganundergraduatelevelof
familiarity with linear algebra and elementary metric space theory. Chaps. 2 and 3
canbereadasanintroductiontothepartofHilbertspacetheorywhichdoesnotneed
measure and integration theory. The latter becomes an essential tool from Chap. 4
onwards, so we give a summary of the key concepts and some relevant results.
StartingwithSect.4.10,andmoreessentiallyfromChap.6on,weoccasionallyneed
thebasicnotionsofgeneraltopologyandtopologicalvectorspaces.Elementsofthe
theoryofC(cid:1)-algebrasandvonNeumannalgebrasarebrieflysummarisedinChap.6,
buttheirrole isvery limited inthe sequel.
While prior study of quantum mechanics might be found useful, it is not a
prerequisite for a successful study of the book. The essence of the work is the
development of tools for a rigorous approach to central questions of quantum
mechanics,whichareoftenconsideredinamoreintuitiveandheuristicstyleinthe
v
vi Preface
literature.Inthiswaytheauthorshopetocontributetotheclarificationofsomekey
issues in the discussions concerningthe foundations and interpretation ofquantum
mechanics.
Thebibliographyisfairlyextensive,butitdoesnotclaimtobecomprehensivein
any sense. It contains many works on general background and key papers in the
developmentofthefieldofquantummeasurement.Naturally,especiallymostofthe
morerecentreferencesrelatetothetopicscentraltothisbook,inwhichtheauthors
and their collaborators have also had their share.
The reader will notice that the word measure is used in a variety of meanings,
which should, however, be clear from the context. A measure as a mathematical
conceptisasetfunctionwhichcanbespecifiedbygivingthevaluespace:we talk
about (positive) measures, probability measures, complex measures, operator
measures, etc. We also speak about the measures of quantifiable features such as
accuracy, disturbance, or unsharpness. The etymologically related word measure-
mentmaybetakentorefertoaprocess,butitisalsogivenaprecisemathematical
content that can be viewed as an abstraction of this process.
Muchofthematerialinthisbookhasbeenextractedanddevelopedfromvarious
series of lecture notes for graduate and postgraduate courses in mathematics and
theoretical physicsheld over many years attheuniversities ofHelsinki,Turkuand
York. In its totality, however, the work is considerably more comprehensive than
the union of these courses. It reflects the development of its subject from the early
daysofquantummechanicswhiletheselectionoftopicsisinevitablyinfluencedby
theauthors’researchinterests.Infact,thebookemergedinitspresentshapefroma
decade-long collective effort alongside our investigations into quantum measure-
menttheoryanditsapplications.Atthispointwewishtoexpressourdeepgratitude
and appreciation to the many colleagues, scientific friends and, not least, our stu-
dents with whom we have been fortunate to collaborate and discuss fundamental
problems of quantum physics.
York, UK Paul Busch
Turku, Finland Pekka Lahti
Juha-Pekka Pellonpää
Kari Ylinen
Contents
1 Introduction... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Background and Content . .... .... .... .... .... ..... .... 1
1.2 Statistical Duality—an Outline. .... .... .... .... ..... .... 5
References. .... .... .... ..... .... .... .... .... .... ..... .... 9
Part I Mathematics
2 Rudiments of Hilbert Space Theory. .... .... .... .... ..... .... 13
2.1 Basic Notions and the Projection Theorem.... .... ..... .... 13
2.2 The Fréchet–Riesz Theorem and Bounded Linear Operators ... 17
2.3 Strong, Weak, and Monotone Convergence of Nets
of Operators .. .... ..... .... .... .... .... .... ..... .... 20
2.4 The Projection Lattice PðHÞ .. .... .... .... .... ..... .... 22
2.5 The Square Root of a Positive Operator.. .... .... ..... .... 24
2.6 The Polar Decomposition of a Bounded Operator .. ..... .... 26
2.7 Orthonormal Sets .. ..... .... .... .... .... .... ..... .... 27
2.8 Direct Sums of Hilbert Spaces . .... .... .... .... ..... .... 31
2.9 Tensor Products of Hilbert Spaces .. .... .... .... ..... .... 32
2.10 Exercises. .... .... ..... .... .... .... .... .... ..... .... 34
Reference . .... .... .... ..... .... .... .... .... .... ..... .... 36
3 Classes of Compact Operators. .... .... .... .... .... ..... .... 37
3.1 Compact and Finite Rank Operators. .... .... .... ..... .... 37
3.2 The Spectral Representation of Compact
Selfadjoint Operators .... .... .... .... .... .... ..... .... 40
3.3 The Hilbert–Schmidt Operator Class HSðHÞ.. .... ..... .... 45
3.4 The Trace Class TðHÞ... .... .... .... .... .... ..... .... 47
3.5 Connection of the Ideals TðHÞ and HSðHÞ
with the Sequence Spaces ‘1 and ‘2. .... .... .... ..... .... 49
3.6 The Dualities CðHÞ(cid:1) ¼TðHÞ and TðHÞ(cid:1) ¼LðHÞ . ..... .... 52
vii
viii Contents
3.7 Linear Operators on Hilbert Tensor Products
and the Partial Trace..... .... .... .... .... .... ..... .... 55
3.8 The Schmidt Decomposition of an Element of H (cid:3)H .. .... 59
1 2
3.9 Exercises. .... .... ..... .... .... .... .... .... ..... .... 61
4 Operator Integrals and Spectral Representations:
The Bounded Case . .... ..... .... .... .... .... .... ..... .... 63
4.1 Classes of Sets and Positive Measures ... .... .... ..... .... 63
4.2 Measurable Functions.... .... .... .... .... .... ..... .... 65
4.3 Integration with Respect to a Positive Measure .... ..... .... 65
4.4 The Hilbert Space L2ðΩ;A;μÞ. .... .... .... .... ..... .... 68
4.5 Complex Measures and Integration.. .... .... .... ..... .... 69
4.6 Positive Operator Measures ... .... .... .... .... ..... .... 71
4.7 Positive Operator Bimeasures.. .... .... .... .... ..... .... 75
4.8 Integration of Bounded Functions with Respect
to a Positive Operator Measure. .... .... .... .... ..... .... 80
4.9 The Connection Between (Semi)Spectral Measures
and (Semi)Spectral Functions.. .... .... .... .... ..... .... 83
4.10 A Riesz–Markov–Kakutani Type Representation Theorem
for Positive Operator Measures. .... .... .... .... ..... .... 85
4.11 The Spectral Representation of Bounded Selfadjoint
Operators .... .... ..... .... .... .... .... .... ..... .... 86
4.12 The Spectrum of a Bounded Operator ... .... .... ..... .... 93
4.13 The Spectral Representations of Unitary and Other Normal
Operators .... .... ..... .... .... .... .... .... ..... .... 94
4.14 Exercises. .... .... ..... .... .... .... .... .... ..... .... 96
References. .... .... .... ..... .... .... .... .... .... ..... .... 99
5 Operator Integrals and Spectral Representations:
The Unbounded Case ... ..... .... .... .... .... .... ..... .... 101
5.1 Elementary Notes on Unbounded Operators... .... ..... .... 101
5.2 Integration of Unbounded Functions with Respect
to Positive Operator Measures . .... .... .... .... ..... .... 104
5.3 Integration of Unbounded Functions with Respect
to Projection Valued Measures. .... .... .... .... ..... .... 107
5.4 The Cayley Transform ... .... .... .... .... .... ..... .... 110
5.5 The Spectral Representation of an Unbounded Selfadjoint
Operator. .... .... ..... .... .... .... .... .... ..... .... 113
5.6 The Support of the Spectral Measure of a Selfadjoint
Operator. .... .... ..... .... .... .... .... .... ..... .... 115
5.7 Applying a Borel Function to a Selfadjoint Operator ..... .... 117
5.8 One-Parameter Unitary Groups and Stone’s Theorem..... .... 119
5.9 Taking Stock: Hilbert Space Theory and Its Use
in Quantum Mechanics... .... .... .... .... .... ..... .... 123
Contents ix
5.10 Exercises. .... .... ..... .... .... .... .... .... ..... .... 125
References. .... .... .... ..... .... .... .... .... .... ..... .... 126
6 Miscellaneous Algebraic and Functional Analytic Techniques. .... 127
6.1 Normal and Positive Linear Maps on LðHÞ... .... ..... .... 127
6.2 Basic Notions of the Theory of C(cid:1)-algebras and Their
Representations.... ..... .... .... .... .... .... ..... .... 130
6.3 Algebraic Tensor Products of Vector Spaces .. .... ..... .... 134
6.4 Completions.. .... ..... .... .... .... .... .... ..... .... 135
6.5 Exercises. .... .... ..... .... .... .... .... .... ..... .... 135
References. .... .... .... ..... .... .... .... .... .... ..... .... 136
7 Dilation Theory.... .... ..... .... .... .... .... .... ..... .... 137
7.1 Completely Positive Linear Maps... .... .... .... ..... .... 137
7.2 A Bilinear Dilation Theorem .. .... .... .... .... ..... .... 139
7.3 The Stinespring and Naimark Dilation Theorems... ..... .... 144
7.4 Normal Completely Positive Operators
from LðHÞ into LðKÞ.... .... .... .... .... .... ..... .... 147
7.5 Naimark Projections of Operator Integrals .... .... ..... .... 152
7.6 Operations and Instruments.... .... .... .... .... ..... .... 153
7.7 Measurement Dilation.... .... .... .... .... .... ..... .... 159
7.8 Exercises. .... .... ..... .... .... .... .... .... ..... .... 161
References. .... .... .... ..... .... .... .... .... .... ..... .... 162
8 Positive Operator Measures: Examples.. .... .... .... ..... .... 163
8.1 The Canonical Spectral Measure and Its Fourier-Plancherel
Transform.... .... ..... .... .... .... .... .... ..... .... 163
8.2 Restrictions of Spectral Measures... .... .... .... ..... .... 166
8.3 Smearings and Convolutions... .... .... .... .... ..... .... 168
8.4 Phase Space Operator Measures.... .... .... .... ..... .... 172
8.5 Moment Operators and Spectral Measures .... .... ..... .... 175
8.6 Semispectral Measures and Direct Integral Hilbert Spaces . .... 178
8.7 A Dirac Type Formalism: An Elementary Approach ..... .... 182
8.8 Exercises. .... .... ..... .... .... .... .... .... ..... .... 185
References. .... .... .... ..... .... .... .... .... .... ..... .... 186
Part II Elements
9 States, Effects and Observables .... .... .... .... .... ..... .... 191
9.1 States ... .... .... ..... .... .... .... .... .... ..... .... 192
9.2 Effects... .... .... ..... .... .... .... .... .... ..... .... 196
9.3 Observables .. .... ..... .... .... .... .... .... ..... .... 200
9.4 State Changes. .... ..... .... .... .... .... .... ..... .... 208
9.5 Compound Systems ..... .... .... .... .... .... ..... .... 213
9.6 Exercises. .... .... ..... .... .... .... .... .... ..... .... 221
References. .... .... .... ..... .... .... .... .... .... ..... .... 223