The Quantum Theory of Measurement

Lecture Notes in Physics New Series m: Monographs Editorial Board H. Araki, Kyoto, Japan E. Br6zin, Paris, France J. Eh...

Author: Paul Busch | Pekka J. Lahti | Peter Mittelstaedt

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Lecture Notes in Physics New Series m: Monographs Editorial Board

H. Araki, Kyoto, Japan E. Br6zin, Paris, France J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Ztirich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Kippenhahn, G6ttingen, Germany H. A. Weidenmtiller, Heidelberg, Germany J. Wess, Mtinchen, Germany I. Zittartz, K61n, Germany Managing Editor

W. Beiglb6ck Assisted by Mrs. Sabine Landgraf c/o Springer-Verlag, Physics Editorial Department II Tiergartenstrasse 17, D-69121 Heidelberg, Germany

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Tokyo

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Paul Busch

Pekka J. Lahti

Peter Mittelstaedt

The Quantum Theory of Measurement Second Revised Edition

Springer

Authors Paul Busch Department of Applied Mathematics The University of Hull Kingston upon Hull HU6 7RX, United Kingdom Pekka J. Lahti Department of Physics University of Turku SF-2o5oo Turku, Finland Peter Mittelstaedt Institute for Theoretical Physics University of Cologne D-5o937 Cologne, Germany Cataloging-in-Publication Data applied for. Die D e u t s c h e B i b l i o t h e k - C I P - E i n h e i t s a u f n a h m e Busch, Pa.l: The q u a n t u m theory of m e a s u r e m e n t / Paul Busch ; Pekka J. Lahti ; Peter M i t t e l s t a e d t . - 2. ed. - Berlin ; H e i d e l b e r g ; New Y o r k ; Barcelona ; Budapest ; H o n g Kong ; L o n d o n ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1996 (Lecture notes in physics : N.s. M, Monographs ; 2) ISBN 3-540-61355-2 NE: Lahti, Pekka J.:; Mittelstaedt, Peter:; Lecture notes in p h y s i c s / M ISBN 3-54o-61355-2 2nd Edition Springer-Verlag Berlin Heidelberg New York ISBN 3-54o-54334-1 1st Edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991,1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by authors Cover Design: Design ¢~ Production, Heidelberg SPIN: 1o54o939 55/3142-54321o - Printed on acid-free paper

P r e f a c e to t h e S e c o n d E d i t i o n Since the first edition of The Quantum Theory of Measurement appeared nearly five years ago, research into numerous areas of the foundations of quantum mechanics has been carried on at a breathtaking pace. The strangeness of the quantum world has continued to be a source of creativity, and it is becoming clear that new applications are to be expected in a not too distant future in fields such as 'quantum' communication, computation, or cryptography; applications that could not have been anticipated on the basis of classical physics. These developments, together with the increasing recognition of the ubiquity and, in many respects, fundamental importance of imperfect, or nonideal measurements, have made the need for a thorough understanding of the problems of measurement and an elaboration of measurement theory as an applied discipline ever more pressing. In the meantime this book has been accompanied by a treatise, coauthored by two of us, that takes up the latter demand: Operational Quantum Physics develops the theory and various applications of unsharp observables and their measurements. In view of this new text it was felt desirable that The Quantum Theory of Measurement should be made available again, but not without a revision of its scope in the context of the new developments. As a result, the central chapters, II, III and IV, have been substantially rewritten. The definition of the concept of objectivity has been reformulated so as to make clearly visible the distinction of its formal and interpretational components (Chap. II). This separation is then extended to all theorems which are now stated as formal results in the first instance. In this way, we hope, the whole theory is made accessible also to those readers who do not fully (or at all) share our philosophical inclinations. The most important and significant changes concern the elucidation of the various necessary objectification requirements - among them the pointer valuedefiniteness and pointer mixture conditions. A thorough understanding of their implications for the structures of measurement schemes has been achieved; and this has led to a completely comprehensive formulation of an insolubility theorem for the objectification problem, which includes unsharp object observables and unsharp pointers. As one consequence, the idea of unsharp objectification has now assumed a fairly sharp contour (Chap. III). The review of the various approaches and interpretational attempts at dealing with the objectification problem (Chap. IV) has been rearranged according to the simplified logical classification scheme offered by the insolubility theorem. As in the first edition, we have refrained from entering into detailed comparisons and evaluations of the various 'schools'. In fact it is becoming more and more obvious

vi

Preface to the Second Edition

in each of them that a lot of 'internal' questions are still open, so that it is too early for conclusive judgements. The kind of formal investigations needed for answering these questions is sketched out in some cases; apart from that we have restricted ourselves to short indications of the current state of the art, referring our readers to recent expert accounts of the various approaches. Many colleagues have encouraged us with their comments to embark on the work for this new edition. In particular we are indebted to Heinz-Jiirgen Schmidt and Reinhard Werner for posing critical questions about the previous edition. Sincere thanks go to our friends Gianni Cassinelli and Abner Shim0ny: it was in our collaborations with them that we envisaged the full scope of generality that has now been achieved.

Cologne, Huff, Turku April 1996

Paul Busch Pekka Lahti Peter Mittelstaedt

P r e f a c e to t h e First E d i t i o n The present treatise is concerned with the quantum mechanical theory of measurement. Since the development of quantum theory in the 1920s the measuring process has been considered a very important problem. A large number of articles have accordingly been devoted to this subject. In this way the quantum mechanical measurement problem has been a source of inspiration for physical, mathematical and philosophical investigations into the foundations of quantum theory, which has had an impact on a great variety of research fields, ranging from the physics of macroscopic systems to probability theory and algebra. Moreover, while many steps forward have been made and much insight has been gained on the road towards a solution of the measurement problem, left open nonetheless are important questions, which have induced several interesting developments. Hence even today it cannot be said that the measurement process has lost its topicality and excitement. Moreover, research in this field has made contact with current advances in high technology, which provide new possibilities for performing former Gedanken experiments. For these reasons we felt that the time had come to develop a systematic exposition of the quantum theory of measurement which might serve as a basis and reference for future research into the foundations of quantum mechanics. But there are other sources of motivation which led us to make this effort. First of all, in spite of the many contributions to measurement theory there is still no generally accepted approach. Much worse, a considerable fraction of even recent publications on the subject is based on an erroneous or insufficient understanding of the measurement problem. It therefore seems desirable to formulate a precise definition of the subject of quantum measurement theory. This should give rise to a systematic account of the options for solving the problem of measurement and allow for an evaluation of the various approaches. In this sense the present work may be taken as a first step towards a textbook on the quantum theory of measurement, the lack of which has been pointed out by Wheeler and Zurek (1983). In view of the difficulties encountered in the quantum theory of measurement many distinguished authors have considered the possibility that quantum mechanics is not a universally valid theory. In particular, the question has been raised whether macroscopic systems, such as measuring devices, are beyond the scope of this theory. Adopting this point of view would allow one to reformulate, and possibly solve, the open problems of quantum mechanics within the framework of more general theories. Such far-reaching conclusions should, however, be substantiated by means of a close chain of arguments. We shall try to spell out some of the arguments that endeavour to prove the limitations of quantum mechanics in the

viii

Preface to the First Edition

context of measurement theory. The resulting no-go theorems naturally entail a specification of the various modifications of quantum mechanics which might lead to a satisfactory resolution. At the same time they contribute to an understanding of those interpretations maintaining the universal validity of quantum mechanics. Next, we are not aware of the existence of a review of the measurement problem which takes into account the developments in the foundations of quantum mechanics over the past two decades. The operational language based on the notions of effects and operations, and the ensuing general concepts of observables and state transformers have proved extremely useful not only in foundational issues (as documented in the monographs of Ludwig (1983a,1987), Kraus (1983), or Prugove~ki (1986)), but also in applications of quantum physics in areas like quantum optics or signal processing (as represented by the books of Davies (1976), Helstrom (1973), or Holevo (1982)). These concepts must be regarded as the contemporary standards for the rigorous formulation of physical problems. They will be employed here for the precise definitions of operational and probabilistic concepts needed for uniquely fixing the notion of measurement in quantum mechanics and developing a formulation of the quantum theory of measurement general enough to cover the present scope of applications. The introduction of general observables has shed new light on the problem of macroscopic quantum systems and the question of the (quasi-) classical limit of quantum mechanics, thus providing a redefinition of the notion of macroscopic observables. In this way a new approach to the measurement p r o b l e m - unsharp objectification- has emerged in the last few years and will be sketched out in the course of our review. The failure of the quantum theory of measurement in its original form has led several authors to propose a modified conception of dynamics, incorporating stochastic elements into the SchrSdinger equation or taking into account the influence of the environment of a quantum system. In both cases the measuring process can no longer be described in terms of a unitary dynamical group. Hence the traditional theory of measurement should also be extended to cover nonunitary state transformations. The preceding remarks suggest that the incorporation of general observables and nonunitary dynamics into quantum measurement theory necessitates, and makes possible, an entirely new approach to this theory. We shall try to bring into a systematic order the new results obtained in the course of many detailed investigations, recovering the known results as special cases. In this way we shall hope to have established a systematic description of the quantum mechanical measurement process together with a concise formulation of the measurement problem. In our view the generalised mathematical and conceptual framework of quantum mechanics referred to above allows for the first time for a proper formulation of many aspects of the measurement problem within this theory, thereby opening up new options for its

Preface to the First Edition

ix

solution. Thus it has become evident that these questions, which were sometimes considered to belong to the realm of philosophical contemplation, have assumed the status of well-defined and tractable physical problems.

Cologne, June 1991

Paul Busch Pekka Lahti Peter Mittelstaedt

Contents I. 1. 2.

Introduction

..........................................................

The P r o b l e m of M e a s u r e m e n t in Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . .

1 1

Historical Account: I n t e r p r e t a t i o n s a n d Reconstructions of Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3.

Decision Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

II.

Basic Features of Quantum Mechanics .............................

7

1.

Hilbert Space Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.1. Basic F r a m e w o r k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

7

1.2. Tensor P r o d u c t and C o m p o u n d Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.3. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Probability S t r u c t u r e of Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1. States as Generalised Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2. Irreducibility of Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3. Superselection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4. Nonobjectivity of Observables

18

.....................................

2.5. Nonunique Decomposability of Mixed States . . . . . . . . . . . . . . . . . . . . . . .

21

2.6. E n t a n g l e d Systems and Ignorance I n t e r p r e t a t i o n for Mixed States ..

22

III. T h e Q u a n t u m T h e o r y o f M e a s u r e m e n t

...........................

25

S u r v e y - T h e Notion of M e a s u r e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.

...................................

27

.

General Description of M e a s u r e m e n t

1.1. The P r o b l e m of Isolated Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.2. M e a s u r e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Premeasurements ......................................................

31

2.1. P r e m e a s u r e m e n t s and State Transformers . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.2. U n i t a r y P r e m e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.3. Calibration Condition a n d Probability Reproducibility . . . . . . . . . . . . .

34

2.4. Reading of Pointer Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.5. Discrete Sharp Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.6. The S t a n d a r d Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

xii 3.

4.

Contents Measurement and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Two P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43

3.2. M e a s u r e m e n t Statistics I n t e r p r e t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.3. Statistical Ensemble I n t e r p r e t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Probabilistic C h a r a c t e r i s a t i o n s of M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . .

49

4.1. Statistical D e p e n d e n c e a n d Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2. S t r o n g Correlations Between Observables . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.3. S t r o n g Correlations Between Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.4. S t r o n g Correlations Between Final C o m p o n e n t States . . . . . . . . . . . . . . 4.5. First K i n d M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58

4.6. R e p e a t a b l e M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Ideal M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60

4.8. R ~ s u m ~ - A Classification of P r e m e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . .

63

I n f o r m a t i o n Theoretical Aspects of M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . 5.1. T h e C o n c e p t of E n t r o p y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 65

5.2. T h e C o n c e p t of I n f o r m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.3. I n f o r m a t i o n a n d C o m m u t a t i v i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. T h e Objectification P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73

6.2. Insolubility of t h e Objectification P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Classical P o i n t e r Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 77

6.4. R e g i s t r a t i o n a n d R e a d i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

7.

Measurement Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. T h e P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. A n Inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82

8.

L i m i t a t i o n s on Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

8.1. R e p e a t a b l e M e a s u r e m e n t s a n d Continuous Observables . . . . . . . . . . . . . 8.2. C o m p l e m e n t a r y Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 86

8.3. M e a s u r a b i l i t y a n d Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.

6.

9.

P r e p a r a t i o n a n d D e t e r m i n a t i o n of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

9.1. S t a t e P r e p a r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

9.2. S t a t e D e t e r m i n a t i o n Versus S t a t e P r e p a r a t i o n . . . . . . . . . . . . . . . . . . . . . .

90

Contents IV. Objectification and Interpretations 1. 2. 3. 4.

5.

V.

of Quantum

Mechanics

xiii .....

91

Routes Towards Solving the Objectification Problem . . . . . . . . . . . . . . . . . . . . Historical P r e l u d e - Copenhagen Interpretations . . . . . . . . . . . . . . . . . . . . . . . . Ensemble and Hidden Variable Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . Modifying Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 95 102 105

4.1. Operational Approaches and the Quantum-Classical Dichotomy . . . . 4.2. Classical Properties of the A p p a r a t u s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Moclified Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Changing the Concept of Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Many-Worlds Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Modal Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Decoherence via Environment-Induced Superselection . . . . . . . . . . . . . . 5.4. Algebraic Theory of Superselection Sectors . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Unsharp Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 107 109 111 113 116 123 125 127

Conclusion

...........................................................

131

.............................................................

139

Bibliography Author Notation

Index and References

..........................................

..................................................................

Subject Index

...........................................................

145 177 179

I. I n t r o d u c t i o n 1.1. T h e P r o b l e m of M e a s u r e m e n t in Q u a n t u m M e c h a n i c s An understanding of quantum mechanics in the sense of a generally accepted interpretation has not yet been attained. The ultimate reason for this difficulty must be seen in the irreducibly probabilistic structure of quantum mechanics which is rooted in the nonclassical character of its language. An operational analysis of the peculiarities of quantum mechanics shows that the interpretational problems are closely related to the difficulties of the quantum theory of measurement. It is the purpose of this review to spell out in detail these connections. The task of the quantum theory of measurement is to investigate the semantical consistency of quantum mechanics. Phrased in general terms, quantum mechanics, as a physical theory, and the quantum theory of measurement as a part of it, are based on a 'splitting' of the empirical world into four 'parts" (1) object systems S (to be observed), (2) apparatus ,4 (preparation and registration devices), (3) environments g (the 'rest' of the physical world which one intends to ignore), and (4) observers O. Depending on the type of interpretation in question, observers or environments may or may not be neglected in the description of the measuring process within the quantum theory of measurement. Providing that quantum mechanics is considered as a t~eory of individual objects, the most important questions to be answered by measurement theory are: (1) how it is possible for objects to be prepared, that is, isolated from their environments and brought into well defined states; (2) how the measurement of a given observable is achieved; and (3) how objects can be reestablished after measurements, that is, be separated from the apparatus. The underlying common issue is the objectit~cation problem; that is, the question of how definite measurement outcomes are obtained. We shall try to elucidate the status and the precise form of these questions. In Chap. II basic features of quantum mechanics are summarised which may be regarded as the root of the objectification problem. Chapter III is devoted to a systematic exposition of the quantum theory of measurement. Various solutions to the measurement problem proposed within a number of current interpretations of quantum mechanics will be reviewed in Chap. IV. Chapter V closes the treatise with our general conclusions. In the present chapter a decision tree will be formulated as a guide to a systematic evaluation of the various interpretations of quantum mechanics. A brief historical overview of these interpretations may serve as a first orientation, showing, in passing, the origins of the present approach.

2

I. Introduction

1.2. Historical Account: Interpretations and Reconstructions of Quantum Mechanics One may distinguish four or five overlapping phases in the development of research in the foundations of quantum mechanics. Early discussions among the pioneers (1927-1935) led to the well-known versions of the so-called Copenhagen interpretation. In the discussions between Bohr and Heisenberg [Bohr 28, Heis 27] and Bohr and Einstein [Bohr 49] the quantum theory of measurement was touched upon only in an informal way. It is only in the monographs of von Neumann [yon Neumann 1932] and Pauli [Pauli 1933] that one finds the first rigorous and explicit formulations of measurement problems in the manner in which they axe the subject of the present treatise. Reconsiderations of interpretational questions extending essentially from the 1950s to the 1970s were mainly motivated by attempts to explore the possibilities of establishing realistic interpretations of quantum mechanics considered as a universally valid theory. Much of this was anticipated in and taken up from the early works of von Neumann Iron Neumann 1932], Einstein, Podolsky, and Rosen [Eins 35], Schrhdinger [Schr5 35,36] and others. The London-Bauer [Lon 39] theory of measurement and its critique through the story of Wigner's [Wig 61] 'friend' are concerned with the possibility already pointed out by von Neumann and Pauli that the observer's consciousness enters in an essential way into the description of quantum measurements. Other denials of the possibility of realistic interpretations are formulated in the position that only a statistical interpretation of quantum mechanical probabilities is tenable [Bal 70, Eins 36, Maxg 36]. In this view quantum mechanics refers only to ensembles of measurement outcomes or of physical systems but does not lead to statements about properties of individual systems. On the other hand, hidden variable approaches aimed at restoring classical realism in quantum mechanics. These, again, are forced to render quantum mechanics as a statistical theory. Many of such attempts were refuted by a number of no-gotheorems like those by Gleason [Glea 57], Kochen and Speaker [Koch 67], or Bell [Bell 66] (see, e.g., [Giuntini 1991, Peres 1993]), leaving open up to now only nonlocal, contextual theories such as those of de Broglie [de Broglie 1953], Bohm [Bohm 52], or Bohm and Vigier [Bohm 54]. The 'many-worlds interpretation' developed by Everett [Eve 57], DeWitt and Graham [DeWitt and Graham 1973] offers one way of taking seriously quantum mechanics as a universal theory. We shall be very brief with our subsequent discussions of the early developments (Chap. IV) and refer the interested reader to the monographs of [Jammer 1966, 1974], and to the collection of papers edited by Wheeler and Zurek [Wheeler and Zurek 1983]. Reconstructions and generalisations of quantum mechanics (pursued systematically since the 1960s) have aimed at an understanding of the role of Hilbert space in quantum mechanics. One may distinguish three groups of approaches. (1) The quantum logic approach aims at an operational justification of the - gen-

1.2 Historical Account

3

erally non-Boolean - structure of the lattice of the propositions of the language of a physical theory [Beltrametti and Cassinelli 1981, Jauch 1968, Mackey 1963, Mittelstaedt 1978, Piron 1976, Varadarajan 1985]. Measurement theory enters this approach only in an informal way in terms of postulates characterising propositions as properties of physical systems. In order to establish the formal language of quantum physics, one assumes that elementary propositions are value definite, that is, that there exists an experimental procedure - a measuring process - which shows whether the proposition is true or false. An essential presupposition is that this measuring process will lead to a complete objectification. The importance of the quantum logic approach for the present work lies in the fact that it supports the attempts at formulating a consistent realistic interpretation of quantum mechanics. (2) The operational approach takes as its starting point the convex structure of the set of (statistical) states representing the preparations of physical experiments. Measurement theoretical aspects are investigated primarily on the object level in terms of the notion of operation representing state changes induced by measurements [Davies 1976, Fou 78, Holevo 1982, Kraus 1983, Ludwig 1983]. The quantum theory of measurement presented in Chap. III is formulated in the spirit of the operational approach. (3) The algebraic approach emphasizes the algebraic structures of the set of observables and it exploits the formal analogy between classical mechanics and quantum mechanics, aiming, in particular, at convenient 'quantisation' procedures. One of its advantages is the great formal flexibility which allows for an elegant incorporation of superselection rules and other structural changes generalising quantum mechanics. Hence this approach offers a mathematical language for a discussion of the measurement problem in more general terms. As a survey and rather exhaustive literature guide the reader may wish to consult the monograph [Primas 1983]. Each of the so-called axiomatic approaches has deepened our understanding of the mathematical and conceptual structures of quantum mechanics. However, none of them led to a thorough justification of the ordinary Hilbert space quantum mechanics. Due to this fact, but also due to the success of Hilbert space quantum mechanics, many recent investigations in quantum mechanics have been done directly within the Hilbert space formulation of quantum mechanics. The present work is also written entirely within this framework. The recent revival of interest in foundational issues was encouraged during the 1980s due partly to advances in the formal and conceptual structures of quantum mechanics and also to new experimental possibilities and technological demands. This went hand in hand with new ideas on interpretations and on proposals for solving the objectification problem (see Chap. IV). Fundamental experiments have been performed and these have contributed to bringing the quantum theory of measurement closer to empirical testability. Quantum optical and neutron interferometry experiments on the wave-particle dualism, Einstein-Podolsky-Rosen and

4

I. Introduction

delayed choice experiments, macroscopic tunnelling, and mesoscopic quantum effects are some examples. Instead of trying to survey these important developments here, we shall simply refer to the many recent conferences devoted to them such as those in Baltimore 1994, Castiglioncello 1992, Cologne 1984, 1993, Erice 1989, 1994,

Gdafisk 1987, 1989, 1990, Helsinki 1992,1994, Joensuu 1985, 1987, 1990, Munich 1981, New York 1986, 1992, Nottingham 1994, Paris 1990, Prague 1994, Rome 1989, Tokyo 1983, 1986, 1989, 1992 or Vienna 1987. Various aspects of these developments are reflected in the monograph Operational Quantum Physics coauthored by two of the present writers [Busch, Grabowski, Lahti 1995] as well as in the recent books by Peres, Quantum Theory: Concepts and Methods, 1993, and Schroeck, Quantum Mechanics on Phase Space, 1996.

1.3. D e c i s i o n Tree

In the minimal interpretation, quantum mechanics is regarded as a probabilistic physical theory, consisting of a language (propositions about outcomes of measurements), a probability structure (a convex set of probability measures representing the possible distributions of measurement outcomes) and probabilistic laws. In addition, probabilities are interpreted as limits of relative frequencies of measurement outcomes, that is, in the sense of an epistemic statistical interpretation. It is well-known that the minimal interpretation has not been the only one proposed. Other interpretations were formulated earlier. We shall try to give a fairly systematic list of them along with a sequence of decisions to be made concerning the goals quantum mechanics could be desired to serve. The first decisive question to be answered is the one about the referent of quantum mechanics: measurement outcomes (the epistemic option) or object systems (the ontic, or realistic option)? The ontic answer maintains that quantum mechanics deals with individual objects and their properties. It is only here that the measurement problem arises. Following this branch, the second decisive question is the completeness of quantum mechanics, that is, the question of whether or not all elements of physical reality can be described by quantum mechanics. The first option leads to, and is motivated by, the consideration of hidden variable theories underlying the allegedly incomplete theory of quantum mechanics, which then is interpreted as a mere statistical theory about ensembles of objects. In the other option, that of maintaining the completeness of quantum mechanics and following a realistic interpretation, one is facing the phenomenon of nonobjectivity. Accordingly, quantum mechanical probabilities are objective in the sense of propensities, or potentialities, expressing tendencies in the behaviour of individual objects. Again, in the incompleteness interpretations there is no measurement problem: objectification is not at issue at all, since all properties are considered as real throughout but not as subject to quantum mechanics. However, as mentioned ear-

1.3 Decision Tree

5

lier, there is not much room for hidden variable theories, and the only ones that survived the known no-go statements do not really go beyond the formalism of quantum mechanics. Turning to the realistic interpretations maintaining the completeness of quantum mechanics, it must be said that these have not up to now produced generally accepted solutions to the measurement problems. Thus one is forced into a third decision about the range of validity of quantum mechanics: is quantum mechanics universally valid or only of limited validity? Some authors have concluded that quantum mechanics, originally devised as a theory for microsystems, cannot be extrapolated in a straightforward way to larger systems, such as measuring devices. It is argued either that more general theories need to be developed which allow for certain macroscopic quantities to be classical observables, or that the time evolution is not correctly described by the SchrSdinger equation. The reductionistic conviction is given up in these views. In some sense, the more challenging route is that which maintains the universality of quantum mechanics. It forces one to carefully reconsider the concepts of objectivity and objectification, a decision that is made in the many worlds interpretation, the modal interpretations, the decoherence theories and the unsharp objectification proposal. In our opinion no conclusive decision between these two options can be made at present. We shall therefore be content to provide a short systematic review of the various approaches to the measurement problem in Chap. IV, guided by the above discussion as summarised in the decision tree of Table 1.

6

I. Introduction

Table 1: DecisionTree: Interpretations of quantum mechanics and approaches to the objectification problem.

Quantum Mechanics Minimal Interpretation relative frequency of measurement outcomes (

REFERENT? ~

[ Statistical Interpretation only measurement outcomes

Realistic Interpretation properties of individual systems

- objectification problem excluded

C OBJECTIVITY/ OMPLETENESS ? ~

Incompleteness all properties objective hidden variables ensemble of objects

Completeness nonobjectivity

- objectification problem excluded

~BJECTIFICATION/~ UNIVERSALITY ? J

Limited Validity objectification

searched by modifying quantum mechanics:

superselection rules modified dynamics

Universal Validity cha//enging the concept of objectification: many-worlds interpretation modal interpretations decoherence approach unsharp objectification

II. Basic Features of Q u a n t u m M e c h a n i c s II.1. Hilbert Space Q u a n t u m Mechanics This section summarises the basic elements and results of quantum mechanics which are relevant to the quantum theory of measurement. It also serves to define our notations and terminology. The standard results quoted here can be found, for example, in the following monographs [Beltrametti and Cassinelli 1981, Davies 1976, Jauch 1968, Kraus 1983, Ludwig 1983, yon Neumann 1932]. We are also using freely the well-known results of the Hilbert space operator theory, as presented, for instance, in the book [Reed and Simon 1980].

II.l.1 Basic Framework The basic concepts of quantum mechanics are the dual notions of states and observables, both being defined in their most general forms in terms of operators acting on a Hilbert space. a) M a t h e m a t i c a l structures. Let 7-/be a complex separable Hilbert space with the inner product ('l'). An element ~ e 7-/is a unit vector if (~]~) - I I ~ II2 -- 1, and the vectors ~, ¢ E 7-/are orthogonal if (~I¢) = 0. A set ( ~ i ) C 7-/is orthonormal if the vectors ~i are mutually orthogonal unit vectors. If ( ~ i ) C T/is a basis, that is, a complete orthonormal set, then any ¢ E 7-/can be expressed as the Fourier series ¢ = ~'~(~i]¢)~i with ]1 ¢ II2 -- ~ ](~ill~))l 2" Any unit vector ~ E 7-/ determines a one-dimensional projection operator P[~] through the formula P[~]¢ - (~]¢)~ for ¢ e 7-/. We also use the bracket notation ]~)(~I for this projection. If {~oi} is a basis of 7-/, then the projection operators P[~i] are mutually orthogonal and P[~i] - I, where I is the identity operator on 7-/. Let/:(7-/) denote the set of bounded operators on 7-/. An operator A E/:(7-/) is positive, A _ O, if (~IA~) _ 0 for all vectors ~ E 7-/. Then the relation A > B, defined as A - B >_ 0, is an ordering on the subset of self-adjoint bounded operators. Let ~ be a nonempty set and ~" a a-algebra of subsets of ~ so that (~,~') is a measurable space. A normalised positive operator valued (POV) measure E " ~" --. /:(7-/) on (~,~') is defined through the properties: i) E(X) >_ 0 for all X e ~" (positivity); ii) if (Xi) is a countable collection of disjoint sets in ~ then E(UXi) E(X~), the series converging in the weak operator topology (a-additivity); iii) E ( ~ ) - I (normalisation). For any e o v measure E " ~ --, L(7-/) the following two conditions are equivalent: i) E ( X ) 2 - E ( X ) f o r all X E ~'; ii) E ( X M Y) = E ( X ) E ( Y ) for all X, Y E ~. Thus a positive operator valued measure is a projection operator valued (ev) measure exactly when it is multiplicative. Further, if the measurable space ( ~ , ~ ) is the real Borel space (R, B(R)), or a subspace of

8

II. Basic Features of Q u a n t u m Mechanics

it, then E determines a unique self-adjoint operator fR ~dE in 7-/. Here t denotes the identity function on R. Conversely, according to the spectral theorem, each self-adjoint operator A in 7-/defines a unique PV measure E : B(R) ~ / : ( 7 ~ ) such that A - f R tdE. If E is a PV measure on (R, B(R)) it shall be denoted as E A in order to explicate the unique self-adjoint operator A associated with it. The set of trace class operators on 7-/will be denoted as T(7-/), and T(7~) + consists of the positive trace one operators on 7-/. The trace, T ~ tr[T], is a positive linear functional on T(~/). The one-dimensional projections P[~] are positive operators of trace one. They are the extremal elements of the set T(7~) +. Indeed T(7-/) + is a convex set (with respect to the linear structure of T(7-/)), so that an element T e T(7-/) + is extremal if the condition T = wT1 + ( 1 - w ) T 2 , with T1,T2 e T(7-l) +, and 0 < w < 1, implies that T = T1 = T2. But T E T(7-/) + is extremal if and only if it is idempotent (T 2 = T), which is the case exactly when T is of the form P[7~] for some unit vector 7~ E 7-/. The set of extremal elements of T(7-/) + exhausts the whole set T(7-/) + in the sense that any T e T(7-/) + can be expressed as a a-convex combination of some extremal elements (P[~i]): T = ~ i wiP[7~i], where (wi) are suitable weights, that is, 0 < wi

(w) =

(5)

ijk

Here [~i)(~ojIis the bounded linear operator on ?-/sgiven by [~i)TE~ 1> -~ E:/2TE:/2,

(9)

(where E~/2 denotes the square root of Si) so that the mixture (8) can also be written as

Thus all the relations (8)-(10) hold whenever the effects El, E2, • .. are objective in the state T.

II.2.5 Nonunique Decomposability of Mixed States The objectivity of an observable in a state T is linked to the possibility of decomposing this state in'a certain way. In addition this decomposition is required to admit an ignorance interpretation. However the set of states S(7-/) possesses a fundamental structural feature which makes the ignorance interpretation of mixed states highly problematic: any mixed state T of 8(7/) admits infinitely many decompositions into vector states P[~] (cf. [Beltrametti and Cassinelli 1981]). To see how 'bad' the situation is, one may ask which vector states P[~o] can occur as components in some decomposition of T, that is, for which P[~] there exist w E (0, 1) and T t E 8(7-/) such that

T = wP[~o] + (1 - w)T ~.

(11)

The answer is as follows: it is precisely the unit vectors ~ in the range of the square root of T which give rise to such a decomposition [Had 81]. The nonunique decomposability of mixed states in quantum mechanics is quite in contrast to classical probability theory, where a decomposition of any probability measure into extremal elements is unique. The reason for this difference is that in the classical case all extremal elements of the set of probability measures are {0, 1}-valued measures and therefore mutually disjoint: for two {0, 1)-valued measures to be distinct, there must exist a set on which one assumes the value 1 and the other one the value 0. By contrast, in 8(7-/) there are plenty of pairs of vector states which are not mutually orthogonal, so that there is room for convex decompositions of mixed states into nonorthogonal extremal elements, besides the orthogonal decomposition(s) induced by the spectral resolution.

22

II. Basic Features of Quantum Mechanics

II.2.6 Entangled Systems and Ignorance Interpretation for Mixed States The nonunique decomposability of mixed states bears severe implications for the interpretation of such states in quantum mechanics. In fact, generally a mixed state T, with a decomposition T = ~ w~P[~], does not admit an ignorance interpretation according to which the system S prepared in state T would actually be in one of the component states P[~i] with the subjective probabilities wi. The above result (11) at once makes such an interpretation problematic. However, this question deserves to be studied in greater detail since it is of foremost importance within measurement theory. In order to decide on this issue, let us review the possibilities of preparing mixed states in quantum mechanics. Consider a sequence of vector states ~i, i - 1, 2,..-, together with a sequence of weights wi, 0 0), and a straightforward calculation shows that

p(7~s(P[U(~ ® ¢)]), ~ A (P[U(~o ® ¢)]); P[U(~ ® ¢)]) = 1.

(23)

Hence these states are always strongly correlated. III.4.5 First K i n d M e a s u r e m e n t s The notion of a first kind measurement was discussed by Pauli [Pauli 1933]. He gave two definitions which he considered equivalent. The first definition says that a measurement is of the first kind if it leads to the same result upon repetition. The second definition says that a measurement is of the first kind if the probability of obtaining a particular result is the same both before and after the measurement. In the general context of measurement theory these two definitions are not equivalent. We shall adopt the first definition as the definition of repeatability (Sect. 4.6), and the second as the definition of first kind measurements. A premeasurement M of an observable E and its associated state transformer is said to be of t h e / i r s t kind if

= P'Rs(V(T®T.a))(X)

(2a)

for all X E ~" and T E S(7-/s). In Sect. 4.2 we pointed out necessary and sufficient conditions under which a first kind measurement is a strong observable-correlation measurement We also observed that for unsharp observables the first-kind property is not sufficient to entail strong observable-correlations. In order to further illustrate the connections between first kind and strong value-correlation measurements, let us consider once more a Liiders measurement of the two-valued observable wi ~ Ei, with the ensuing state transformer being given by the operations ~i(T) = E~/2TE~/2, T e S(7-ls). A direct computation gives

p(Ei,Pi; V(T ® TA)) 2 = tr[TE2] - tr[TEi]2 tr[TEi] - tr[TEi] 2 "

(25)

III.4. Probabilistic Characterisations

59

Hence the strong value-correlation occurs only if t r [ T E 2] = tr[TEi] for all T (for which 0 ~ tr[TEi] # 1), that is, Ei is a projection operator. In that case the final component state Ts(i,T) - ~i(T)/tr[TEi] is a 1-eigenstate of Ei whenever 0. A large class of first kind measurements are given by the standard model of Sect. 2.6. Indeed the coupling U - e i X A ® s defines always a first kind measurement scheme.

III.4.6 Repeatable Measurements Repeatable measurements are an important class of measurements. In fact repeatability was already pointed out to be fundamental to the value objectification. Intuitively a measurement of an observable is repeatable if its repeated application yields the same results. This idea can be formalised systematically within the theory of sequential measurements (see, for example, [Bus 90b, Dav 70]). A premeasurement A4 is repeatable if its repetition does not lead - from the probabilistic point of view - to a new result, that is, if

tr[Z~(Y)(ZM(X)(T))]

= tr[Z~(Y NX)T]

(26)

for all X, Y E ~" and for all T E 8(7-/s). This condition may be written in the following equivalent ways:

tr[Ts(X,T)E(X)] - 1

(whenever tr[TE(X)] ~ 0),

E ( X ) T s ( X , T) = Ts(X, T),

(27 ) (27b)

which are to hold for all X E 7" and all T E 8(7-ls). As an immediate observation we note once more that a repeatable premeasurement is always of the first kind. We say that an observable E admits a repeatable measurement if there is a premeasurement of E which is repeatable. It is an old issue in the quantum theory of measurement, dating from von Neumann's work [yon Neumann 1931], as to whether observables which admit repeatable measurements are necessarily discrete. It was always anticipated that this must be the case. On the basis of important contributions by Stinespring [Sti 55], Davies [Davies 1976] and others the problem was finally solved by the results of Ozawa [Oza 84] and Luczak [Lucz 86]: THEOREM 4.6.1. If an observable E admits a repeatable premeasurement 2~4, then

E is discrete. We note an immediate corollary to the correlation theorems. COROLLARY 4.6.2. Let M be a premeasurement of an observable E and 7~ be any reading scale. The following statements hold true. a) If All n is repeatable then it produces strong observable-correlations and strong

value-correlations.

60

III. The Quantum Theory of Measurement

b) Let T£ be finite. M ~ is repeatable whenever it is a strong observable-correlation premeasurement. c) Let ¢~4~ satisfy the pointer value-definiteness condition and let E be a a sharp observable. 2~4~ is repeatable whenever it is a strong value-correlation premeasurement. An observable need not be a sharp one in order to admit a repeatable unitary premeasurement. As an illustration, any collection of effects Ei, i E I, which all have eigenvalue 1 and which sum up to the unit operator I, constitute an observable admitting a repeatable unitary premeasurement. Indeed, choosing a 1-eigenstate Ti for each i, EiTi = Ti, the state transformer, with the operations ¢ i ( T ) : = tr[TEi]Ti, T E S(7-ls), is repeatable and completely positive (cf. Sect. 2.2), although the Ei need not be projection operators. We noted already that repeatable measurements are of the first kind, but a first kind measurement need not be repeatable. The standard measurement of an unsharp position (Sect. 2.6) or the Liiders measurement of a simple observable E, introduced in Equation (17), are of the first kind but in general not repeatable; the latter being repeatable if and only if El, and thus also E2, is a projection operator. Besides the first kindness, there is another probabilistic weakening of the notion of repeatability. We say that a measurement A~i of E is value reproducible if for any X E 9r and T E S(7-ls) the following implication holds true: if p E ( X ) = 1,

then

p ET s ( n , T ) ( X ) - 1.

(28)

Clearly any first kind measurement is also value reproducible, but the converse inclusion does not hold. In the case of sharp observables the notions of first kind, value reproducible and repeatable measurement coincide [Bus 95a]. THEOREM 4.6.3. Let E be a sharp observable and Z any of its associated state

transformers. Z is repeatable ff and only ff it is of the first kind, if and only if it is value reproducible. The von Neumann measurements of a discrete sharp observable A are repeatable, but not every repeatable measurement of A is a von Neumann measurement. Indeed the A-compatible state transformer Z ( { a i } ) ( T ) = tr[TEA({ai})]Ti is repeatable whenever tr[TiEA({ai})] - 1 for each i - 1 , 2 , . . . ,N. Yet it is not a von Neumann state transformer if some of the eigenvalues ai of A are degenerate.

III.4.7 Ideal M e a s u r e m e n t s No measurement that is capable of providing some probabilistic information can leave unchanged all the states of the measured system. Hence it is important to investigate to what extent state changes are necessary in a measurement. The value reproducibility and first kind properties introduced in Sects. 4.5 and 4.6 are probabilistic nondisturbance features which quantum mechanical meas-


61

urements may or may not possess. A more stringent nondisturbance property representing a kind of minimal disturbance is that of ideality [Beltrametti and Cassinelli 1981]. A measurement is ideal if it alters the measured system only to the extent that is necessary for obtaining a measurement result: all the properties which are real in the initial state of the object system and which are coexistent with the measured observable remain real also in the final state of the system. In the case of sharp observables A this intuitive conception of ideality leads to the following probabilistic formulation, called here p-ideality. An A-measurement A4 and its induced state transformer Z ~ are called p-ideal if for any state T E S(7~s) and for any sharp property P E C(7-/s), which is coexistent with A, the following implication holds true: if tr[TP] = 1, then tr [Z~ (R)(T)P] = 1.

(29)

When applied to the measured observable A itself, the p-ideality of A,[ implies that for any state T and for all value sets X the following implication holds true: if

pAT(X) =

1, then

pA~(R)(T)(X ) -- 1.

(30)

For sharp observables this condition is equivalent to the repeatability of Az[. Thus we have the following result [Lah 91]: THEOREM 4.7.1.

A p-ideal premeasurement of a sharp observable is repeatable.

The yon Neumann measurements of a sharp discrete observable are always repeatable and satisfy (28). However these measurements are not p-ideal except when they are Liiders measurements. As a corollary to the above theorem we note that a sharp observable A is discrete whenever it admits a p-ideal measurement (Theorem 4.6.1). In the case of a discrete sharp observable the p-ideality of an A-measurement AJ implies the following condition: for all i - 1, 2 , . . . and all T, if

tr[TEA({ai})]

= 1,

then

Z~({ai})(T)=T.

(31)

Indeed if ~ is a vector state for which (~lEA({ai})~) = 1, then, by p-ideality, tr[ZA4(R)(P[~])(P[~])] : 1. But in this case the support projection of the s t a t e / : ~ ( R ) ( P [ ~ ] ) is contained in P[~], which is possible only if = P[~o]. Since P[~] is a vector state with tr[ZA4({ai})(P[~])] - 1, we finally have ZA4({ai}) (P[~o]) : P[~]. By linearity the argument extends to arbitrary states T for which tr[TEA({ai})] = 1. We shall see below that condition (31) is in fact equivalent to the p-ideality of Z ~ . Condition (31) admits an immediate generalisation to arbitrary discrete observables E. Since this generalisation will turn out to be important for the objectification problem we formulate it as the definition of the d-ideality of a measurement, d referring to the discreteness-assumption.

:r~(R)(P[~])

62


Let E be a discrete observable with the generating (nonzero) effects E ({wit), i - 1, 2,.... An E-measurement A4, or an E-compatible state transformer Z, is d-ideal if it does not change the state of S whenever a particular result is certain from the outset: if tr[TE((wi))] : 1,

then

Z~((wi))(T):

T

(32)

for all i = 1, 2 , . . . and for any T E S(7-/s). For general discrete observables a d-ideal measurement need not be repeatable nor first kind. Indeed the Liiders state transformer T ~ E~/2TE~/2 of a discrete observable E (with the generating effects E1 and E2) is d-ideal but never repeatable unless the Ei are projection operators. Suppose the Ei have eigenvalue 1 with associated spectral projections E~, and let U " 7-/s ~ 7-/s be a unitary mapping which acts as an identity on the eigenspaces E~(Hs). Then the E-compatible state transformer T ~ UE~/2TE:/2U-1 is still d-ideal but not first kind, unless U commutes with the Ei. The question of the structure of ideal, repeatable measurements has been a major issue since von Neumann's work [yon Neumann 1932]. These properties are crucial for the realistic interpretation of quantum mechanics insofar as the existence of measurements with these properties ensure the interpretation of possible measurement results as potential properties of the system. The repeatability of a premeasurement A4 requires the measured observable E to be discrete (Theorem 4.6.1). The d-ideality or the repeatability of a measurement A4 of a discrete observable E do not imply that E is an sharp observable. In order to afford this conclusion, one needs to postulate an additional property of the measurement, its nondegeneracy. A measurement A4 of E is nondegenerate if the set of all possible final component states ( T s ( X , T ) " X e 3c, T e S(7-ls)) separates the set of effects; that is, for any B E £(T/s), if tr[Ts(X, T)B] = 0 for all X e 3~, T e S(7-/), then S = O.

(33)

The following theorem then holds true [Davies 1976]: THEOREM 4.7.2. A discrete observable E admits a repeatable, d-ideal, nondegencrate measurement/f and only if E is a sharp observable. In that case the premeas-

urement f14 is equivalent to a Liiders measurement of E; that is, the induced state transformer Zj~ is of the form Z~(X)(T)

-

Z

E((wi))TE((wi))

(34)

w~EX

for all T E S(7-ls) and for all X E 3c. It can be shown that a d-ideal measurement of a discrete sharp observable is nondegenerate [Bus 90b]. Furthermore, a result similar to Theorem 4.7.1 shows that also d-ideality implies repeatability [Lah 91]. Hence the considerations of this section give rise to the following statement.


63

4.7.3. The d-ideal premeasurements of a sharp discrete observable are exactly the Lfiders measurements. COROLLARY

A Liiders measurement of a sharp observable is p-ideal. Thus the concepts of p-ideality and d-ideality are equivalent in the case of sharp discrete observables. From now on we shall only consider the d-ideality as given by (32), and we refer to it as the ideality of a premeasurement. III.4.8 R@sum@- A Classification of P r e m e a s u r e m e n t s We summarise the main relationships between the various probabilistic characterisations of premeasurement discussed so far. While all of the notions to be listed can be formulated solely in terms of premeasurements, their consideration is actually motivated by the idea of measurements. We therefore drop the prefix 'pre' in the sequel. In the following summary we are using obvious abbreviations for the terms ideality (ID), repeatability (REP), first kind property (FK), value reproducibility (VR), strong observable-correlation (SOC), strong value-correlation (SVC), and strong state-correlations (SSC). It is convenient to consider four successive classes of measurements, each containing the subsequent one. A) A r b i t r a r y observables. There are measurements which are value reproducible but not of the first kind, and there are first kind measuremets which are not repeatable. But repeatable measurements are always of the first kind, and first kind measurements are always value reproducible. Repeatable measurements yield strong correlations, but there exist first kind measurements which produce neither strong observable nor strong value correlations. Hence we have the implications:

(REP) ~

(FK)==~ (VR);

(REP) ~

(SOC);

(REP) ~

(SVC).

If the reading scale is finite then one has: (REP) ~

(SOC).

An observable admitting a repeatable measurement is discrete. We recall also that the ideality and repeatability of a measurement of some discrete observable do not yet require this observable to be a sharp one. B) S h a r p o b s e r v a b l e s In this case the important notions of repeatability, first kind property, and value reproducibility are the same: (REP) ~

(FK) ~

(VR).

64


Furthermore, if the measurement fulfils the pointer value-definiteness condition (which is the case if the pointer observable is sharp), then (REP) ~

(SVC)

with respect to any reading scale. C) D i s c r e t e s h a r p observables (ID) ~

(REP)

In the case of minimal unitary measurements ~4~ one has, in addition: (REP) ==~ (SSC). As mentioned above, both the ideality and the repeatability requirements imply the discreteness of a sharp observable. For an arbitrary observable E, besides ideality and repeatability, yet another property, the nondegeneracy, of state transformers is needed in order to ensure that E is an sharp observable. Finally we recall that in general a von Neumann measurement is not ideal (though repeatable). In fact the ideal measurements of discrete sharp observables are precisely the Liiders measurements (LU)" oo

(ID) ~

(LU).

D) Nondegenerate, discrete sharp observable Here we note the additional fact: (ID) ~

(REP).

We emphasise once more that the notions of repeatability and ideality are crucial for the realistic interpretation of quantum mechanics. We have seen that quantum mechanics does allow for such measurements. But it is also important to realise that to any observable E belongs a large (infinite) class of E-measurements. This fact opens up the possibility of accounting for real (imperfect) measurements.

III.5. Information Theoretical Aspects of Measurements The incorporation of the concepts of entropy and information into quantum probability theory runs up against severe difficulties, which are mainly due to the existence of noncommuting observables. That this problem is of considerable interest in various branches of quantum physics is demonstrated, for instance, in the monographs of Helstrom [Helstrom 1976], Lindblad [Lindblad 1983] and Whirring [Thirring 1980]. It took a long time until a rigorous and comprehensive framework has been set up. An account of the present state of art is presented in [Ohya and Petz 1993]. Important landmarks of the earlier developments with some bearings

III.5. Information Theoretical Aspects

65

on quantum measurement theory are given, for example, by the works of von Neumann [von Neumann 1932], Everett [Eve 57], Ingarden [Ing 76], or Lindblad [Lin 73]. The application of information theoretical concepts within the quantum theory of measurement has recently attracted increasing attention, as documented in [Santa Fe 1989]. Information theoretical characterisations of measurements will be the subject of the present section. Any premeasurement A/[ of an observable E induces the following transformations

T ~ Ts (~t, T), E

(1)

(2)

where T is the initial state of the object system S. Moreover, any reading scale 7~ defines a natural decomposition of Ts(~, T):

Ts(f~, T) = ~pi Ts(i, T),

(3)

where Pi - pE(xi). Properties of the transformations (1) and (2) reflect the properties of A/I, and the decomposition (3) is the one for which one might tend to apply the ignorance interpretation. Various probabilistic features of the transformations (1) and (2) were studied in the preceding section. We next come to apply the concepts of entropy and information in order to further characterise these transformations. In particular, we are able to characterise those premeasurements which lead to an optimal separation of the final component states Ts(i, T) of the object system S (Sect. 5.1). We also specify the conditions under which the deficiency of information (implicit in the probability measure pTE) for predicting a certain measurement outcome can be identified with a potential information gain (associated with P~s(~,T)) obtained by reading the actual measurement result (Sect. 5.2). Some information theoretical aspects of von Neumann and Liiders measurements will be pointed out as illustrations of the general relations obtained. Finally an information theoretical characterisation of the commutativity of discrete sharp observables will be presented (Sect. 5.3). III.5.1 T h e C o n c e p t of E n t r o p y Following [von Neumann 1932], the entropy of a state T is defined as the (nonnegative) number S ( T ) " - -tr[Tln(T)]. Using the spectral decomposition T tiE T ({ti }) of T, one obtains the expression

S(T) = - ~ t i l n ( t i ) tr[E T({ti})].

(4)

Since all the spectral projections E T ({ti)) (with ti ¢ 0) are finite dimensional, we may also write S(T) = - ~ t i l n ( t i ) (5)

66


whenever T is nondegenerate, or if we allow each term tiln(ti) to appear in the series as many times as indicated by the degeneracy of the eigenvalue ti. S(T) is finite if Tln(T) is a trace class operator. In particular, if the dimension of the range of T is finite, say n, then 0 _ S(T) < ln(n). The case S(T) - 0 occurs exactly when T is a vector state, whereas S(T) - ln(n) holds if and only if T is totally degenerate (maximally mixed)[Feinstein 1958]. Consider any decomposition of a state T into some other states Ti, T - ~ wiTi, with 0 < wi

(15)

is finite for each ~o if and only if M ~ is a Liiders measurement. In this case one

has S(P[7~][TL(A, 7~)) = S({pi}).


69

III.5.2 The Concept of Information The notion of average information of a probability measure, as introduced by Shannon, can be employed to yield characterisations of measurements. Consider a measurement A4 of an observable E and fix a reading scale T~. Then for any state T the average deticiency of information of the discrete probability measure Xi ~, pE (Xi) = Pi is given by the quantity

H(E,T; Tt) := S((pi)).

(16)

In accordance with the minimal interpretation, H(E,T; T~) can be interpreted as the (average) deticiency of information for predicting a measurement outcome Xi in the initial state T of the system S. In particular, the case H(E, T; :R) - 0 of vanishing deficiency of information occurs exactly when T is an eigenstate of E, that is, if E(Xi)T = T holds for some Xi E T~. Similarly, the state Ts(~t, T) of S after the measurement gives the probability E (Xi) = qi, with the corresponding deficiency of information measure Xi ~ PTs(f~,T)

H(E, Ts(f~,T); T~) = S({qi}).

(17)

Assuming that value objectification takes place in the E-measurement A/I, then this number could be interpreted as the potential in[ormation gain upon reading the result Xi E T~. On this assumption it becomes interesting to compare the initial deficiency of information, given by H(E, T; T~), with the potential information gain, H(E, Ts(fl, T); 7~) and to ask under what conditions they are the same. E Noting that Ts(~, T) = ~ piTs(i, T) and hence PTs(n,T) = ~ pipETs(i,T), one has the basic inequalities [Lin 73]

~piH(E, Ts(i,T); n) _O. Hence for repeatable measurements the deficiency of information S(Ts(~2, T)) about the actual final state of the system is never less than the potential information gain H(E, Ts(~, T); T~) upon reading the actual result. The quantities coincide exactly when all component states Ts(i,T) are vector states. For arbitrary repeatable measurements this need not be so. We can illustrate the above results as characterisations of the von Neumann and Liiders measurements of a discrete sharp observable A. Since these measurements are repeatable, the relations (21) may be applied. For a Liiders measurement of A one obtains S(TL(A, 7~)) = H(A,P[7~]) = H(A, TL(A, 7~)) (23)


71

for any initial vector states ~. On the other hand, for a (maximal) von Neumann measurement of A one has only

S(T N(A.

- H(A. P[(p]) = S(T.N(A. cp)) - H(A. TvN(A. = Ep,

(2a)

>_ O.

so that the deficiency of information on the actual state of the system after the measurement is not less than the potential information gain upon reading the result. As in the case of state entropy, there are several alternative information theoretical characterisations of the probability changes pE ~-~ P TEs ( ~ , T ) associated with an E-measurement j~4. In addition to the above considerations we may mention the concept of average relative information, which can be used to compare the initial and final probability measuresp E and PTs(~,T)'E Assume that the final probability measE E ure PTs(~,T) is absolutely continuous with respect to pE, that is, pTs(~2,T)(X) -- 0 E whenever p E ( x ) -- O. Then the average relative information of PTs(~,T) with re-

spect to pE is defined as the nonnegative number [Jan 72]" dpETs(n,T) dpET In

dp E

dp E.

(25)

Consider an E-measurement Az[ with a given reading scale TO. Assume that the final probability measure xi ~ p TEs ( g t , T ) ( X i ) -- qi is absolutely continuous with respect to the initial one, Xi ~ pE(Xi) = Pi, that is, qi - 0 whenever Pi = O. Then one obtains

H ({qi}I(pi})= ~ - ~ q i l n ( q ~ p i " - ' \ ] P iPi

(26)

= - Z q i l n ( p i ) - H(E, Ts(~, T); 7~). This relation can be used, in particular, for a characterisation of repeatable unitary measurements M ~ of A. Assuming that the final probability measure is absolutely continuous with respect to the initial probability measure for any initial (vector) state of S, it follows that the A4~-generating set {¢ij } (cf. Sect. 2.5) is a complete set of eigenvectors of A. Hence A/Iff is repeatable so that one has H(pAs(n,T) IpA) -0 for any state T. The notion of average information (16) of a discrete probability measure is not the only one ever proposed in the context of quantum mechanics. Let A be a discrete sharp observable such that the degree of degeneracy n(i) of any eigenvalue ai is finite. Then the formula

H*(A,T) "- - - Z p A ( a i ) ln (pA!?:) ~ \ n(.t) /

(27)

72


defines an information functional considered by Everett [Eve 57]. Obviously we have H* (A, T) = H(A, T) + ~pAT(ai ) ln(n(i)) _ H(A, T). (28) In order to give a first intuitive comparison of the two concepts of information H and H*, we shall consider a situation in which the outcome ak is known before the measurement, that is, pA(a~) = 5ik. It follows that H(A,T) = 0 so that there is no ignorance left about the measuring result. On the other hand, we find H*(A,T) = ln(n(k)), that is, there is still some deficiency of information. This result can be explained in the following way. Even if the result ak is known, nothing is known about the pure state 7~kj of S which lies in the subspace with dimension n(k). There are n(k) orthogonal states of this kind with probability w~j -- n (1k ) , that 1 is, the system with eigenvalue ak is described by the state Tk = ~ ( k 1) n(k) P[7~kJ]• In order to determine the pure state ~kj of S, one could measure some nondegenerate observable Sk = ~(=k) bjP[7~kj]. The deficiency of information about the value and thus about the state 7~kj is then given by

H(Bk, Tk) = --~~jwkjln(wkj) = ln(n(k))

b

(29)

in accordance with the formula for H*(A, T). Hence H*(A, T) describes not only the deficiency of information about the value ak but also about the pure state of the system S. This consideration leads to the following more general remark. Let A, B be two sharp discrete observables such that B is a refinement of A, so that A - f(B) for some function f. It follows that the potential information gain with respect to B is larger than that for A:

H(B, T) >_H (f(B), T).

(30)

For a given state T, there exists a maximal (nondegenerate) refinement B -- A0 ~_,~j aijP[cpij] of A such that the probabilities pAT°(aij) are independent of the degeneracy index j , that is, leads to

pAT°(aij) -- ~ 1 pAT(ai) • This T-dependent choice of A0 H(Ao,T) = H*(A,T)

which again illustrates the meaning of the entity equality (28) is seen to be a special case of (30). III.5.3

Information

and

(31)

H*(A, T). Furthermore, the in-

Commutativity

The commutativity of two bounded sharp discrete observables A and B can be given an operational meaning by means of Liiders measurements. If T is an initial state of S, then TL (A, T) is the state of S after a Liiders measurement of A. For a measurement of another discrete sharp observable B we may (c~) measure B directly

III.6. Objectification

73

on the system S in the state T, or (fl) first perform a Liiders premeasurement of A on the system in the state T and then perform a B-measurement on ,~ in the state TL(A, T). Even if the operators A and B commute, the outcome of a single B-measurement will in general be different in the two cases (a) and (/3) since in the case (/3) the A-measurement produces changes of the state of the system, which may influence the result of the succeeding B-measurement. However, in the statistical average these differences disappear. This is the content of the following theorem due to Liiders [Liid 51]. THEOREM 5.3.1. Bounded discrete sharp observables A and B commute//and only if t r [ T S ] - tr[TL(A, T)B] for all states T. This theorem provides an operational meaning of the concept of commutativity: if for any preparation T a Liiders measurement (without reading) of A has no influence on the expectation value of B, then it follows that A and B commute, and vice versa. It is essential for the theorem that only Liiders measurements of the discrete observable A are considered. However, it is straightforward to generalise the theorem for continuous observables in the sense of referring to all possible discretised versions of them. The intimate connection between Liiders measurements and the concept of information makes it possible to characterise the commutativity also in terms of information theory. The deficiencies of information about B which correspond to the measuring procedures (a) and (~) are H(B,T) and H(B, TL(A,T)) respectively. An information theoretical characterisation of the commutativity can then be formulated in the following way ITS1 87]: THEOREM 5.3.2. Discrete sharp observables A and B commute ff and only ff H(B, P[~]) - H(B, TL(A, ~)) for all vector states qo. This means that A and B commute if and only if for any state P[~] the deficiency of information about B does not depend on whether a Liiders measurement (without reading) of A has been performed or not. A Liiders measurement (without reading) of an observable A which does not commute with B may thus change the deficiency of information about B. It is remarkable that the initial ignorance H(B, ~) about B can be either increased or decreased in this way ITS1 87].

III.6. Objectification III.6.1 The Objectification Problem A premeasurement of an observable is a measurement if it satisfies the objectification requirement. This condition rests on the idea that a measurement leads to a definite result. Objectification has proved to be the key problem of the quantum theory of measurement. If a premeasurement j~4 of an observable E is performed on the object system S in the state T then it leads to the final state V(T ® TA) of S +.A. The final states

74

Ill. The Quantum Theory of Measurement

of S and ,4 are the reduced states

Ts(f~,T)

:=

T,a(f~, T) :=

Tis(V(T ® T~4)), 7~,4(V(T ® T.4)).

(1) (2)

In order to formulate the idea of definite measurement results we refer to the notion of the objectivity of an observable (Sects. II.2.4 and III.2.4). In fact pointer objectification will have been achieved exactly when the pointer observable PA is objective in the final apparatus state (2). If in addition the measured observable E is objective in the final object state (1), then also value objectification will have been reached. Since from the outset the pointer observable need not be a discrete sharp observable and the pointer function f need not be an identity function, some care is required in the elaboration of these conditions. Let 7~ be the reading scale with respect to which the registration and reading of a result will be carried out and let T~ denote the final apparatus state conditioned by this reading scale,

T~'=

EpE(Xi)TA(i,T).

(3)

The objectivity of the coarse-grained pointer observable P ~ in the state (2) requires the equivalence of this state with the state (3) and the validity of the pointer value-definiteness condition (2.11). We collect the implications of previous theorems on the structure of the premeasurement M if these objectivity conditions are to be fulfilled. According to Theorem 2.4.1, a sufficient condition for both the pointer mixture condition [TA(f~, T) = T~] and the pointer value-definiteness condition [tr[TA(i,T)Pi] = 1 for all i e I with pET(Xi) ~ 0] is furnished by the mutual orthogonality of the object states Ts(i,T), i E I. Moreover for unitary measurements the condition Ts(i, T). Ts(j, T) = O, i ~ j, to hold for for all vector state preparations T - P[7~] of S, is also necessary for the equality of (2) and (3). (In such measurements the pointer value-definiteness condition is automatically fulfilled since the pointer is a sharp observable). In case the final component states Ts(i, T) and T~4(i, T), i e I, are vector states, the equality T,4(~, T) = T~ is given exactly when M is a strong state-correlation measurement (Theorem 4.4.1). The objectivity of P ~ in the state T.a(f~, T) is achieved if and only if i) the states (2) and (3) are equivalent, ii) Pi is real in T,4(i,T), and iii) the ignorance interpretation can be applied to the particular decomposition given in (3). If this is the case, then the 'definite' value f - l ( x k ) , say, of PA can be determined without changing the apparatus state in any way. Due to the reality of Pi in TA(i, T) the decomposition (3) of T~ is an orthogonal decomposition. The applicability of the ignorance interpretation to the state (3) also requires that the states V(T ® T~4) and ~ I ® P:/2V(T ® T.4)I ® p:/2 are equivalent. This corresponds to the fact that the objectivity of P ~ entails that of I ® P ~ in the


75

present situation (cf. Sect. II.2.6). We conclude therefore that the equivalence

V(T ®

I®

V(T ® TA) I ®

(4)

is a necessary condition for the objectivity of P ~ in the final apparatus state TA(~, T) and also for the objectivity of I ® P ~ in the final state V(T ® Tx) of S + A. Obviously, condition (4) is not satisfied in general. A sufficient condition for (4) is that P ~ is a classical observable of A. In the case of a minimal unitary measurement A/[~ of a discrete sharp observable the classical nature of the pointer observable will be seen to be even necessaryfor (4) (provided that the object system is a proper quantum system). If the pointer objectification with respect to 7~ had been obtained, one would still face the question of whether the registered value f-l(Xk) of PA indicates a real property of the object system S. One could consider the requirement that the premeasurement A/t produces strong correlations between the final component states Ts(i,T) and Tx(i,T) of S and ,4. But even if the state-correlation conditions p(Ts(i,T),TA(i,T); V(T @ TA)) - 1 are realised, it could still happen that tr[Ts(i,T)Ei] ~ 1, so that E~ is not real in the state Ts(i,T). Thus, to allow for the objectivity of E n, one could stipulate further that the premeasurement A/In would produce also strong value-correlations: p(Ei, Pi; V(T @ Tx)) - 1 for all i e I and all T e S ( ~ s ) (with 0 ~ pE (X i) ~ 1). For a sharp observable E 7¢ strong value-correlation is achieved only if the measurement is repeatable (Theorem 4.3.1). Then the relation tr[Ts(i,T)Ei] = 1 holds for each i e I and for all T e S(7-/s) such that tr[TEi] ~ 0. Under these conditions the measurement A4 leads to value objectification with respect to the reading scale T~, since the objectivity of P ~ is then inherited by E n (Sect. II.2.6). Similarly, in the more general case of a strong state-correlation measurement, the objectivity of the pointer is transferred to that of those properties which are real in some of the (mutually orthogonal) states Ts(i, T).

III.6.2 Insolubility of the Objectification Problem The objectification requirement becomes problematic if the measurement coupling is taken to be unitary and both S and ,4 are proper quantum systems. In fact for a unitary measurement A/Iv of a sharp discrete observable A, the state U(~ ® ¢) is a superposition of eigenstates of the pointer observable I ® AA whenever ~ is a superposition of A-eigenstates. Thus, as long as A is not real prior to measurement, the pointer cannot be objective after the interaction. This simple model presentation of the objectitication problem rests on the idealising assumptions that the measured observable is a (discrete) sharp observable, the apparatus is initially in a pure state, and the pointer is a sharp observable. Neither of these assumptions is, however, necessary for the conclusion reached. Thus the potential objection that the measurement problem might result only from some

76


excessive idealisations, and that it would disappear if a more realistic account were given, is not tenable. The following result, the most general one to be conceived in the present formulation of the measurement theory, is obtained as the finishing touch [Bus 96c,96d] to a long development, marked by contributions of von Neumann [von Neumann 1932], Wigner [Wig 63], and many others, eg., [Fine 70, Shi 74, Bro 86]. THEOREM 6.2.1. Let E be a (nontrivial) observable of a proper quantum system 8. There is no premeasurement (7-/A, P.a,T~4, U, f} of E, with ,4 being a proper quantum system and U a unitary measurement coupling, that satisfies the pointer value-definiteness condition for ,4 and the pointer mixture condition for S + ,4,

U(T ® TA)U*

= E

I ® p:/2 U(T ® T.a)U* I ®

p1/2

(5)

for some (nontrivial) reading scale T~ and all initial states T of,9. The insolubility theorem shows the assumptions that are left open for challenge. a) Apparatus/s a proper quantum system. As mentioned in Sect. 6.1, assuming the pointer to be a classical observable would be sufficient to grant the objectification. In the next subsection it will be shown that some classical features of the apparatus are necessary for objectification. The difficulties arising with this demand are discussed in Sects. 7.2 and IV.4.1-2. b) Measurement coupling is unitary. New possibilities arise indeed if this assumption is given up. In fact, regarding (tacitly) both S and ,4 as proper quantum systems, Wigner [Wig 63] concluded that the linearity of the quantum mechanical dynamics cannot be maintained if objectification is to be achieved. This argument is taken up in various recent attempts to introduce modified quantum dynamics, thus giving room for spontaneous, autonomous processes leading to the objectivity of some macroscopic observables (cf. Sect. IV.4.3).

c) Measurement scheme provides pointer value-definiteness and pointer mixture property for 8 + ~4. Some approaches try tO dissolve the objectification problem by redefining the notion of objectivity in such a way that the pointer mixture property for S + ,4 need not be stipulated in order to be allowed to attribute definite values to the pointer. This poses the task to formulate new interpretations of quantum mechanics (cf. Sect. IV.5.1-2). Alternatively, it may turn out that pointers, being macroscopic quantities (Sect. IV.4.1), are genuinely unsharp observables; that is, the pointer value-definiteness may only be approximately realisable such that the probability pP~(x,,T)(f-I(xi)) is never equal to unity but rather equal to l - e for some perhaps small but nonzero number e. Hence there may be room for a resolution of the quantum measurement problem by relaxing the requirement of objectification into unsharp objectification. This idea will be explored in Sect. IV.5.5.


77

III.6.3 Classical P o i n t e r O b s e r v a b l e If the assumption is given up that the apparatus is a pure quantum system, it becomes possible to fulfill the necessary objectification condition (4). This will then imply that the apparatus has some classical properties. We illustrate this consideration for the case of a minimal unitary premeasurement M ~ of a discrete sharp observable A. If ~ is an initial (vector) state of S, then the final states of S + A, S, and A are

A T~s(P[U(~ ® ¢)1) - Zp~(ai)P[Til 7~A(P[U(T ® ¢)]) = Z

~/pA(ai)pA(ak)(TilTk)lCk)(¢i[

(6b) (6c)

respectively. A natural reading scale is 7~ = {ai}ieI, so that An - A~, and the reading scale conditioned final apparatus state is A

T~ = Z Pv (ai)P[¢i].

(7)

It will be immediately observed that the states (6c) and (7) are the same for all if and only if the vectors 7i are pairwisely orthogonal (for each ~), that is, the U-defining set {¢ij } is orthonormal. In general this is not the case. The necessary condition (4) for the pointer objectification now reads

P[U(~ ® ¢)] ~ Z

(I ® P[¢i]) P[U(~ @ ¢)] (I ® P[¢i]).

(8)

Let R E/~(7-/A). If (8) is to hold for S being a proper quantum system, that is, (U(~o ® ¢)[(P ® R ) U ( ~ ® ¢)) = Z

(U(~ ® ¢)[(P ® P[¢i]RP[¢i])U(~o®¢)) (9)

for all P E P(7-ls) and for all ~o E 7"/s, then it follows that R must commute with all P[¢i]. Obviously the commutativity of R with An is also sufficient for (8). This shows that, in the present situation, the commutativity of the pointer observable with any other observable of the apparatus is a necessary and sufficient condition for the pointer objectification. Since classical observables are always objective, we can formulate this result as follows [Belt 90]. THEOREM 6.3.1. Let Az['~ be a minimal unitary premeasurement of an observable A performed on a proper quantum system S. The pointer objectification is obtained

if and only if the pointer observable AA is a classical observable. This theorem can be extended to more general measurement situations, as, for example, to an A-measurement induced by a measurement A4~ of a refinement B

78

III. The Q u a n t u m Theory of Measurement

of A = f ( B ) . The equivalence (4) still assumes the form (8), but now it only follows that the degenerate pointer observable f ( A A ) is a classical observable of ,4. In the general case of a unitary measurement Jk4v, the objectification requirement leads to the conclusion that the apparatus must have some classical properties, excluding, in particular, superpositions of 1-eigenstates of the pointer observable corresponding to different readings [Bus 90c]. The proof given in [Bus 90c] can be easily extended to the case where the pointer observable is not sharp but the measurement scheme provides pointer value-definiteness. We assume now that the pointer observable Ax of the premeasurement 2~4~ of A is indeed classical. A maximal pointer observable Ax being classical implies that any other observable of ,4 is a function of Ax and hence that that ,4 is a classical system. In that case the only pure states of the measuring apparatus ,4 are the eigenstates P[¢k] of the pointer observable. The pointer objectification condition (8) is fulfilled. In particular, the final apparatus state (6c) is equivalent to (7). The decomposition (7) is the only decomposition of T~ into pure states A of ,4. This means, in particular, that the pointer probabilities p~(ai) as well as the final state of ,4 allow an ignorance interpretation: when the apparatus ,4 is in the state T~, then it is actually in one of the pure states P[¢k], the coefficients p~A (a~) describing our knowledge about the actual state of A. The actual value of the pointer observable AA can be read without changing the actual state of ,4. The final state of the object system ,S is not directly affected by the assumption that Ax is classical. However, if the measurement A4~ produces strong correlations between the component states P['Yi] and P[¢i] (in which case the "Yi are pairwise orthogonal), then the ignorance interpretation can be applied to the final object state as well. Assuming that this is the case, if P[¢k] is the actual final state of ,4, then P['Yk] is the actual final state of S. Still, the probability (~/klEA({ak})'yk) need not be 1. If, in addition, the measurement M ~ is a strong value-correlation (or repeatable) measurement, then the U-generating vectors ¢ij, and thus ~i, are eigenvectors of A, and A is objective in the actual final state of 8. The value objectification is thereby achieved. In the present approach the classical nature of the pointer observable in a measurement M ~ of a discrete sharp observable is a sufficient (as in the general case) but also a necessary condition for the pointer objectification. The difficult problem of how to realise classical (pointer) observables within quantum mechanics cannot be tackled here. Nevertheless the above result gives an illustration of what is needed for ensuring the consistency of the minimal interpretation with the assumption that quantum mechanics is a complete theory of individual objects. In addition to the question of how to explain the existence of a classical pointer observable, this solution of the objectification problem bears with itself some further problems. First, the assumption that the unitary measurement coupling U represents an observable H of S + ,4 via the relation U = e iH cannot be reconciled with


79

the classical nature of A,4. In other words, if H commutes with I ® AA, then there is no measurement, that is, the measured observable is trivial: E(X)= p ~ ( X ) I . Since on the other hand the classical nature of AA is inevitable for the pointer objectification and thus for the measurement, one arrives at the surprising conclusion that the unitary operator U represents a measuring coupling only if H is not an observable (Sect. 7.2). Second, if AA is a discrete maximal observable, then its classical nature forces the apparatus jt to be a discrete classical system. But such a system cannot be a carrier of the Galilei covariant canonical position and momentum observables [Mackey 1989, Piron 1976, Varadarajan 1985]. Thus a measuring apparatus cannot be a Galilei invariant quantum mechanical system and at the same time have a classical pointer. Obviously this conclusion invalidates some of the presuppositions of a universally valid quantum mechanics (Sects. 1.3, III.l.1). With these problems arising from the fulfilment of the objectification requirement, we are facing the following important conclusions, which were anticipated in Table 1 (Sect. 1.3). If it is possible to explain the origin (and to ensure the existence) of classical (pointer) observables within quantum mechanics, then the incorporation of fundamental symmetries (Galilei covariance) still requires a framework more general than that furnished by a single separable Hilbert space. Hence quantum mechanics cannot be a universal theory. If, on the other hand, quantum mechanics is unable to account for classical (pointer) observables, then there are two possible conclusions. Either one believes that (continuous) superselection rules do exist in a strict sense, which in turn means that one has to give up the universal validity of quantum mechanics. Or one accepts that quantum mechanics properly accounts for the fact that superselection rules and classical observables are only approximately realised in nature. Then quantum mechanics may be regarded as a universal theory, but at the price of a fundamentally weakened conception of reality, perhaps even an intrinsically unsharp reality.

III.6.4 Registration and Reading Registration and reading are the final steps in a measuring process. First, after the measurement interaction, the apparatus reaches a stage at which it records some outcome; that is, the pointer assumes some value on the reading scale. In this sense the process of registration is nothing but the pointer objectification, so that the respective apparatus state TA(f~, T) ~- T~ admits an ignorance interpretation with respect to its components TA(i, T). The remaining step, the reading, is performed by the observer, who in this way eliminates his ignorance and changes the description of the apparatus state according to the registered outcome. In the preceding considerations the reading scale was considered discrete. In fact it was defined as a partition of the value space of the pointer observable. There exist various types of arguments indicating that this discreteness is mandatory. a) Pragmatic need. Any physical experiment is designed to yield definite outcomes out of a collection of alternatives. These outcomes must be described by

80


essentially finite means, either by digital recordings, or by estimating a pointer position in terms of a rational number on an apparently continuous scale. b) Statistics requirement. The statistical evaluation of experimental outcomes is based on counting frequencies of mutually exclusive events out of a countable collection. This again requires the fixing of a partition of the value space of the pointer observable. c) Pointer objectitication. The fact that the pointer ultimately assumes a definite position is to be interpreted as a repeatable measurement of the pointer observable. Hence either this observable itself or one of its (actually measured) coarse-grained versions must be discrete. These arguments for the need for discrete reading scales can be substantiated in formal terms. First, the pragmatic argument a) is well illustrated by means of the information concept. Let E be a continuous sharp observable on B(R), meaning that for all X E B(R) there exists Y E B(R) such that Y C X and 0 ~ E(Y) < E(X). Further, let T~a, 7~2, ... be a sequence of increasingly finer reading scales on B(R) such that T~n+l consists of partitions of the elements of T£n. As shown in Sect. 5, we have

H(E,T;nn+x) >_H(E,T;TCn)

(10)

for all states T and for all n. Assume that the partitions approach points so that the maximum size of the partition intervals X} n), i - 1, 2, ..., of 7~n tends to zero in the following sense: for each state T, the sequence sup{tr[TE(X}n))] • i - 1, 2, ...} converges to zero. Then it follows that the sequence of numbers H(E,T; T~n) increases indefinitely for all T. Hence if there were a continuous reading scale then an E-measurement would have to lead to an infinite increase in information. Argument b) refers to the intended empirical content of the minimal interpretation of the probability measures pS. if a measurement of E in a state T had been repeated n times, and the result X had occurred v times, then limn-.oo v/n - p~ (X). A formal justification of such a statistical interpretation was reviewed in Sect. III.3. Here we recall only that the measurement statistics interpretation of the probabilities BE(x) , X E ~', T E 8(7-/s), with respect to a given E-measurement 2~4, induces a family of discretised pointer observables P ~ , 7~ a reading scale, from which the relevant probabilities

T~A(V(T®TA))

(11)

---- P T C ~ ( V ( T ® T ~ ) )

for each 7~ and any Xi (in T~) are obtained as relative frequencies; that is, for each T and each T~ there is a sequence FT,T¢ of P~-outcomes such that for all i,

relf(X~T¢) FT,~Z)="P~ ({i}) ~"R.A(V(T®T.4)) '

"

(12)

III.7. Measurement Dynamics

81

The third argument c) referring to repeatability is based on Theorem 4.6.1, applied here to the apparatus system. This argument can be carried further if also the value objectification is taken into account, which also requires repeatability (Sect. 6.1). Therefore it can be achieved only if the measured observable is discrete. The discrete nature of the reading scale entails that a given measuring apparatus allows only a measurement of a discrete version of the observable under consideration. With this we do not, however, deny the operational relevance of continuous observables. On the contrary, their usefulness as idealisations shows itself in the fact that they represent the possibility of indefinitely increasing the accuracy of measurements by choosing increasingly refined reading scales.

III.7. Measurement Dynamics III.7.1 T h e P r o b l e m An important problem of the quantum theory of measurement related both to the possibility of premeasurements as well as to the objectification requirement is the question of the physical realisability of the appropriate measurement couplings, that is, the state transformations V in the measurement schemes (7-/x, PA, TA, V, f}. Within the conventional description of the dynamics of isolated quantum systems, there are logically two possibilities: (i) the system S + jt consisting of object and apparatus can be considered as an isolated system; or (ii) the influence of the environment E on S + ,4 cannot be neglected. In the case (i) the usual description of dynamics applies, and V should be in the range of the mapping t ~ Ht, t E R, the dynamical group of S + ,4 (cf. Sect. II.1.3.). More explicitly, S and A should be dynamically independent before and after the measurement, that is, before a time t = 0 and after some time t = T > 0. This implies that the Hamiltonian H, which generates the dynamics lit, t E R, coincides with the free Hamiltonian Hs + HA before and after the measurement, while in the time interval 0 < t < r the measurement interaction comes into play:

H = Hs + HA + Hint.

(1)

The mapping V should be identified with Hr. But then the unitary dynamics t Ut = e x p ( - ~i Ht) is either discontinuous, or a time-dependent interaction Hint(t) is to be introduced. Both possibilities are, however, excluded by the continuity and the group properties of the dynamics t ~/act. This problem of incorporating V into the dynamics t ~ Ht of S + ,4 seems to allow only solutions in the sense of some approximations. The interaction part Hint of the total Hamiltonian H should be negligible before and after the measuring process. If the actual duration of the interaction is of no concern, then the canonical approach is that of describing measuring processes as scattering processes, so that V is identified with the respective scattering operator [Ludwig 1987]. However,

82


it may be desirable to account explicitly for the finite times of preparation and registration. In this case the finite duration of the interaction, that is, the apparent time dependence of Hint, needs to be explained. This can be achieved by regarding the relative motion of S and ,4 as the relevant 'clock' determining approximately the times of turning on and off the interaction. In [Aha 61] it is indicated by means of a simple model how in this way an effectively time-dependent Hamiltonian is obtained from a unitary dynamical group if the system ,4 is 'large' in some suitable sense. Another aspect of the problem of measurement dynamics is the limited number of interactions available. This contingent fact leads to a 'natural' restriction of the set of operators which correspond to actually observable quantities. In particular, the semiboundedness of the Hamiltonian entails that the probability reproducibility may be achieved, in general, only in an approximative way [Grab 90]. Turning to the second option, case (ii) above, the measurement coupling V should result from a family Pt, t E R, of linear state transformations representing the reduced dynamics of S + ,4 derived from the unitary dynamics t ~ Lit of the isolated system S + ,4 + g. But treating S + ,4 + g as an isolated system entails exactly the same problems as those encountered in case (i).

III.7.2 An Inconsistency Besides the question of realising the measurement coupling V as a part of the dynamics t ~ Pt, the objectification requirement poses additional constraints on the measurement interactions. Apart from the fact that the measurement coupling V should give rise to suitable correlations in the final state V(T ® T,4) of S + ,4, the apparatus should assume a definite pointer value at the end of the measurement. As shown in Sect. 6, this can be achieved by assuming that the pointer observable is a classical observable. This state of affairs forces one to consider a modification of quantum mechanics; in particular, it implies constraints on the dynamics of ,4 as well as of S q-,4 [Beltrametti and Cassinelli 1981]. The assumption that the pointer observable is classical implies a further puzzling feature, namely the measurement coupling cannot represent an observable of S-b A [Belt 90]. To discuss this problem, consider a unitary measurement A4u of a discrete sharp observable A, and assume that the pointer observable AA is in fact classical. The pointer observable Ax can be interpreted as an observable I ® AA of the compound system S + A. As the von Neumann algebra L:(7-/s ® 7-/,4) of bounded operators on 7-/s @ ?/,4 is generated by the operators of the product form B ® C, B E £(7~s), C E L:(7~x), one recognises that I ® AA is a classical observable of S + ,4 as well [Takesaki 1979]. This fact has an important consequence. Indeed, writing the unitary measurement coupling as U - exp(iH), and assuming that H commutes with A,4, one then has =

P~®¢

=

Pu(~®¢)

=

P

(P[u(~®¢)])

=

P~

(2)

III.8. Limitations on Measurability

83

for any unit vector ~o E ~ , since I ® Act commutes with U. Since p ~ does not depend on the initial state of the object system, Equation (2) is incompatible with the probability reproducibility condition unless A is a trivial (constant) observable. Hence we have established the following result. THEOREM 7.2.1. Let a measurement scheme (7-/ct, Act, ¢, U) be a candidate for a unitary premeasurement M u of a discrete sharp observable A. If Act is a classical observable and if the coupling U is generated by an observable o[ ,~ + A, then

(7-lct, Act, ¢, U) cannot ful~l the probability reproducibility condition. We conclude that if Act is a classical pointer observable associated to a premeasurement M u of A, then H cannot be an observable of S + A, and vice versa. As shown in Sect. 6.3, the classical nature of the pointer observable Act is a consequence of the objectification requirement in the case of a minimal unitary premeasurement M ~ . Therefore such a measurement cannot be realised by means of a unitary dynamical group/4t, the generator of which being an observable of S + ,4. If, on the other hand, only dynamical groups of this kind are available - as it is usually assumed - then no premeasurement j~4~ can serve as a measurement, since the objectification is impossible. Finally, insisting on b o t h - classical pointer Act and unitary measurement dynamics b/t - forces one into the strange conclusion that the generator H, the Hamiltonian of ,S + ,4, is no observable. The problem of incorporating the measurement coupling between 8 and j[ as a part of the dynamics of S + ,4, or $ + ,4 + g, and the inconsistency between the classical nature of the pointer observable and the physical realisability of a unitary measurement coupling lead to the consideration of modified descriptions of dynamics. It has been shown that a suitable additional term in the von Neumann-Liouville equation leads to a spontaneous pointer localisation and thus to the effective classical nature of pointer observables [Ghi 86]. Still the nonunique decomposability of mixtures would require an explicit description of dynamics as a stochastic process on the level of the Gemenge representation of states. This, in its turn, suggests considering stochastic, nonlinear modifications of the SchrSdinger equation [Gis 84] (Sect. IV. 4.3). III.8. L i m i t a t i o n s o n M e a s u r a b i l i t y

In investigating the measurement possibilities of quantum mechanics, the quantum theory of measurement also reveals the limitations on measurability. There are two types of such limitations, those implied by the theory itself, and those which may arise when the theory is supplemented by some further assumptions. The question of practical limitations in the sense of what actually can be measured in a laboratory is outside the scope of the present account. In the first group there are limitations such as 'only discrete observables admit repeatable measurements' and 'complementary observables cannot be measured

84


together'. Thus, for example, the usual position observable does not admit a repeatable measurement. Such a result is connected with our very possibilities of constituting physical objects. The fact that, say, position and momentum cannot be measured together is a basic and well-understood feature of quantum mechanics. However, the idea that the position-momentum uncertainty relations open a way to circumvent this limitation has been more controversial and has been carried out not until rather recently (for an overview, see [Busch, Grabowski, Lahti 1995]). Among the second type of limitations there is the fact that only observables which commute with all conserved observables can be measured at all. For instance, if momentum conservation is a universal conservation law, then position cannot be measured at all. However, conclusions of this type presuppose that the dynamical problem of the previous section has been solved. There are also the fundamental limitations which concern the very nature of the measuring apparatus and the measurement coupling. As became evident in Sect. 6, no proper quantum mechanical object can serve as a measuring apparatus. Moreover, if the pointer observable is to be classical, then the unitary measurement coupling cannot represent an observable in the sense of an interaction Hamiltonian of the object-apparatus system. Or conversely, if a unitary measurement coupling is generated by an observable interaction Hamiltonian, then the pointer observable cannot be classical. These fundamental limitations indicate the directions for searching the possible resolutions of the objectification problem. Finally there is also a limitation on the determination of the past and the future of an object system: complete (statistical) determination of the state (before the measurement) cannot be achieved by means of repeatable measurements. This phenomenon will be explained in some detail in Sect. 9. In the present section we shall discuss limitations on measurability implied by quantum measurement theory. Before entering into this subject, we ought to remind ourselves of some basic positive results of the measurement theory. First of all, for each observable of the object system there do exist premeasurements. Leaving aside the objectification problem, this fact confirms the idea that physical quantities are, indeed, observables, that is, they can be measured. Furthermore, for single sharp observables no a priori limitations on their measurement accuracies arise.

III.8.1 Repeatable Measurements and Continuous Observables If an observable E of S admits a measurement A4 which is repeatable, then E is discrete (Theorem 4.6.1). This fact has an important (though obvious) corollary. COROLLARY 8.1.1. No continuous observable admits a repeatable measurement. This result causes difficulties in our understanding of the operational definition of continuous observables, among them position, momentum, and energy - observables which are most important for the concept of a particle in quantum physics. We


85

shall illustrate these difficulties by considering the localisation observable of an object residing in three dimensional Euclidean space R 3. In fact the localisation observable of such an object can be decomposed into three similar parts referring to the three component spaces R. If the object has a nonzero rest mass, then its localisation observable in R is simply the spectral measure E Q of the position Q:

(Q~o)(x) = x~o(x) for a.e. x 6 R, ~0 6 dom(Q) c 7-/= £2(R, dx)

(1)

E Q(X)~o = Xx ~0 for any X 6 B(R), where Xx is the characteristic function of the set X 6 B(R). The spectrum of Q is the whole real line R. Q is continuous so that it does not admit a repeatable measurement. This raises the question how to define Q operationally. From the results of Sect. 2 we know that Q admits, in particular, unitary measurements, but none of them can be repeatable. Being unable to provide a canonical answer to the question posed, we shall content ourselves with demonstrating that the most obvious way of defining Q operationally is in fact ruled out. To begin with, we recall that the Borel a-algebra B(R) of R is generated by the closed intervals I of R. Hence also the range of E Q, E Q (B(R)) - {EQ(X) • X 6 B(R)}, is generated by the projections EQ(I) associated with such intervals. This means that the mapping I ~ E Q (I) (on the closed intervals) extends to the spectral measure X ~ E Q ( x ) (on B(R)) [Varadarajan 1985]. Thus to define Q it suffices to define the localisations E Q (I) associated with the intervals I. Each E Q (I) can be defined, for instance, via the state transformation T ~ E Q ( I ) T E Q (I) which corresponds to the yes-outcome of the Liiders measurement of the simple observable X, (Q). A diaphragm with a slit I is a prototype of an experimental arrangement leading to this state transformation. The natural question then is whether the mapping I H Lr(I), with I ( I ) ( T ) = E Q ( I ) T E Q (I), extends to a state transformer of Q, that is, whether by varying slit I in the diaphragm one can define Q. The answer to this question is negative. Indeed, if there were a state transformer I Q of Q such that :[Q(I) = ~(I) for all closed intervals I, then due to the additivity of the state transformer

EQ(I)TEQ(I) = EQ(I1)TEQ(I1) + EQ(I2)TEQ(I2)

(2)

for all states T and for any partition of I into disjoint subintervals 11 and/2. But (2) cannot hold true, for example, for the vector states ~ for which (~IE Q (11)~) # 0 # . In fact, one can show that there is no state transformer iTQ associated with Q for which I Q ( x ) = E Q (X)TEQ (X), T 6 S(7"l), for some X 6 B(R). This result holds true for all continuous observables [Bus 90b]. These considerations show that an operational definition of the localisation observable in terms of ideal or repeatable measurements is impossible. The usual way out of this difficulty consists of restricting oneself to discrete versions of Q

86


[yon Neumann 1932] (see also argument b) in Sect. 6.4). A disadvantage of this approach is that it destroys the translation covariance characteristic of the localisation concept. Another approach is to relax strict repeatability into 8-repeatability [Dav 70]. A Q-compatible state transformer Z is 8-repeatable if for all states T and all X 6 B(R) one has tr[Z(X6)Z(X)(T)]

= tr[Z(X)(T)]

(3)

Here X~ = {x 6 l~ : I x - x ' I < 8 for some x' 6 X} denotes the closed 6neighbourhood of X. There do exist 6-repeatable, covariant, completely positive Q-compatible state transformers; but still the known examples are of a somewhat artificial nature. Therefore an even more general concept of approximate repeatability has been proposed admitting more natural realisations. A Q-compatible state transformer is (e, 8)-repeatable if it satisfies the following for all states T and all Borel sets X: tr[~(X~)Z(X)(T)] >_ (1 - e) tr[Z(X)(T)] (4) Here e and ~ are some fixed ('small') numbers. This notion of approximate repeatability may even be applied to the operational definition of general continuous observables [Busch, Grabowski, Lahti 1995].

III.8.2 Complementary Observables Consider two sharp observables A and B of an object system S. Any of their measurements can be combined into the sequential AB- and BA-measurements (cf. Sect. 4.6). The induced state transformers are the composite state transformers ~-S o ~-A and ~-A o ~-B, respectively. It may happen that for some measurements of A and B the sequential AB- and BA-measurements are equivalent, that is, ~'S o ~-A _. ~-A o ~-B. In that case we say that the sequential measurements are order independent. The existence of order independent sequential measurements for a given pair of observables A and B implies their commutativity, or, in general, their coexistence (when arbitrary observables are considered) [Lah 85]. For sharp discrete observables also the converse holds true, that is, commutativity entails the existence of order independent sequential measurements. This fact is closely related to the coexistence of A and B in the sense discussed in Sect. 5.3. Complementary observables represent an important extreme case of noncoexistent observables. Any two observables are complementary if the experimental arrangements which permit their unambiguous (operational) definitions are mutually exclusive [Pauli 1933]. This conception of complementarity of two observables A and B lends itself readily to a formal representation as a relation between the A- and B-compatible state transformers [Busch, Grabowski, Lahti 1995]. For the present purposes it is unnecessary to go into formal details. Instead we state the obvious result:


87

COROLLARY 8.2.1. Complementary pairs of observables have no order independent sequential measurements.

As another related no-go-theorem for the measurability of complementary observables we state the one resulting from the strong noncoexistence (total noncommutativity) of such observables. COROLLARY 8.2.2. Complementary pairs of observables do not admit any joint measurements.

Canonically conjugate position and momentum observables are complementary. The same holds true, for example, for any two (different) spin components of a spin -1 object. Finally we recall that according to the measurement inaccuracy interpretation of the uncertainty relations complementary observables can be measured together if the involved measurement inaccuracy is sufficiently large. For a systematic measurement theoretical justification of this interpretation, see [Busch, Grabowski, Lahti 1995].

III.8.3 Measurability and Conservation Laws We consider next limitations on measurability implied by the existence of universal conservation laws. Such limitations were discovered by Wigner [Wig 52]. To begin with, we restate Wigner's result as elaborated by Araki and Yanase [Ara 60], using our notations. THEOREM 8.3.1. Let (7-lA, AA, ¢, UL>be a Liiders measurement of a discrete sharp observable A. Let K = K ® 1,4 + I ® K,4 be a bounded self-adjoint operator on 7-ls ® TIA. Assume that [( is a constant of motion of S + ,4 with respect to UL) that is, [K, UL] = O. Then also [K, A] = O. This theorem suggests two divergent interpretations: INTERPRETATION I. Any discrete sharp observable A admits a Liiders measurement with a measurement coupling UL. If [( is any additive (bounded) observable of S + ,4 which is a constant of motion with respect to UL, then K commutes with A. If [K, A] ~ O, then ~[ is not a constant of motion with respect to UL.

INTERPRETATION 2. Assume that [~ represents a universal conservation law, that is, it is a constant of motion with respect to all physicaJly admissible evolutions of S + ,4. If [K,A] ~ O, then the Liiders measurement of A, with UL, is not a physically realisable measurement, that is, UL is not (a part of) a physically admissible evolution of S + ,4.

Interpretation 2 follows Wigner who writes: 'Only quantities which commute with all additive conserved quantities are precisely measurable ([Wheeler and Zurek 1983], p. 298; note that here 'precisely' refers to a Liiders measurement). To accept

88


Wigner's interpretation of the above theorem as a limitation on the measurability of certain observables it is desirable to extend this theorem to a larger class of measurement couplings between S and ,4 than the Liiders measurements. The next theorem provides such a generalisation [Belt 90]. THEOREM 8.3.2. Let M'~ be a minimal unitary premeasurement of a discrete sharp observable A. Let K = K ® In + I ® K n be a bounded self-adjoint operator on ?is @ 7-ln. If K commutes with U, then K commutes with A

(a)

provided that one of the following conditions is satisfied: P(P['Ti], P[¢i], U(9~ ® ¢)) = 1

(b)

for any i = 1 , . . . , N, and for all ~ E 7-l, [[ ~ [[= 1 for which 0 ~ N 2 ~ 1; K n commutes with An.

(c)

The objectification requirement implies that the pointer observable An is classical. Hence condition (c) of this theorem is always satisfied. Condition (b) is the strong state-correlation condition of the measurement, which is fulfilled, for instance, in the case of a repeatable measurement. Theorem 2 extends the scope of Theorem 1 in several respect. However, the relevance of these theorems as limitations on measurability is somewhat open due to the difficulties of realising the measurement coupling U as part of a unitary evolution 1At of S q- ,4. In particular, the classical nature of the pointer observable An, which ensures condition (c) of Theorem 2, entails that the generator of the group/At, the Hamiltonian H of S + ,4, cannot be an observable of S + ,4. Hence this theorem pertains to a rather strange measurement situation which certainly does not belong to the domain of conventional quantum mechanics. Nevertheless these theorems suggest that conservation laws associated with fundamental symmetries may lead to limitations on the measurability of observables. Consequently it would be desirable to investigate further extensions of Theorem 1, both for general (POV) observables [Hell 71] as well as for generalised dynamics. In particular, it is an open question whether also unbounded operators/~ give rise to limitations. There are some results indicating that this is the case for the important position-momentum pair, that is, the conservation of momentum excludes (unitary) measurements of (discretised) position [Bus 85, Ste 71]. On the other hand, it turns out that the limitations stated in Theorem 1 can be circumvented by means of introducing an arbitrarily small inaccuracy [Ara 60, Ste 71, Wig 52]. It has been shown that such inaccurate measurements can be described in terms of POV measures representing unsharp versions of the observables to be measured [Bus 85,89a].

III.9. Preparation and Determination of States

89

III.9. Preparation and Determination of States III.9.1 State Preparation In the present treatise the possibility of preparing arbitrary states T E S(7-/) was taken for granted without further analysis. Nevertheless problems associated with the concept of state were encountered in several places. In Sects. II.2.4 and II.2.6 it was pointed out that the interpretation of mixed states depends on the method by which these states are prepared. The consideration of compound systems and the omnipresence of interactions (Sect. 1.1) provide evidence for the fact that objects can presumably be prepared at best in an approximate way. Finally the ignorance interpretation of mixed states turns out crucial for the objectification problem (Sect.

6). It seems difficult to conceive of a general theory of the preparation of systems in quantum mechanics. Yet the quantum theory of measurement allows for a modelling of the process of preparation in terms of filters. In fact, a repeatable measurement is preparatory in the sense of producing systems with definite real properties. A tilter is a repeatable measurement applied to a class of similar systems and combined with a selection of the systems having a particular value of the measured observable. Combining a sequence of filters associated to a complete set of commuting discrete (sharp) observables yields a filter preparing a pure state. Equivalently a pure state may be prepared by means of a filter using a Lfiders measurement of a maximal (nondegenerate) discrete observable. Clearly this all presupposes that the measurement outcomes can be objectified. A filter based on von Neumann measurements of a degenerate discrete observable prepares mixed states, the interpretation of which depends on the particular choice of pointer observables. To illustrate this point, consider an observable A = ~-~i aiEi, with the spectral projections Ei = ~'~j P[9~ij]. Let A o - ~ aijP[~ij] be a maximal refinement of A, so that A - f(Ao), with f(a~j) - a~ for each j and for all i. Let A/tumL be a Liiders measurement of A0. Then (7"/,4,A`4, ¢, UL, f) and (7-l`4, f(A`4), ¢, UL) are two equivalent von Neumann measurements of A, leaving the object system S in the component states

Ts(i, 7~) - p~A (ai) - 1 ~jP[7~ij]P[7~]P[7~ij].

(1)

The interpretation of this state depends, however, on the measurement applied. Indeed in the first case, the object-apparatus state after reading a value ak is ~ j (I ® P[¢kj]) U(T ® T`4)U -1 (I ® P[¢kj]),

(2)

whereas in the second case the corresponding state is (3)

90


Here T - P[7~] and T.4 - PIe]. In the first case the final state (2) is a mixture representing the Gemenge Fk = ( (l(7~kjlcpll2, P[7~kj] ® P[¢kj]) " j = 1,2,...,n(i)}; for it is assumed that the maximal pointer observable A~4 is objective though its actual value in the set f-1 ((ak}) is ignored. In the second case the entanglement between the object system and the apparatus is not completely destroyed after the reading. In fact state (3) is generally a pure correlated state. In both cases the reduced state of the object system S is Ts(k, 7~) as given in (1) since the two measurements are equivalent. Their interpretation is, however, different. In the first case the state Ts(k, 7~) represents the Gemenge Fk admitting thus an ignorance interpretation. In the second case the state Ts(k, ¢p) is the reduced state of a pure correlated state and, as such, does not admit a similar interpretation. These model considerations demonstrate the possibilities of preparing states - pure states, quantum mixtures as well as Gemenge states - by means of filters. The realisation of these possibilities requires that the measuring results can be objectified, and thus it depends on the solution of the objectification problem, which has not yet been achieved. III.9.2 State Determination

Versus State Preparation

Among the principal limitations on measurements is the following mentioned in Sect. 8. No measurement is capable of determining an arbitrary initial state T from a single measurement outcome. Even the statistics of a measurement generally do not !cad to a unique state determination. In fact for a given observable E there are in general many states T, T~, ... yielding one and the same probability distribution pT E = pE, = .. ". An observable is called in?ormationally complete if it separates the set of states, that is, if for any two states T, T ~ E S(7-/), pT E = pE, implies that

T = T'.

(4)

It is well known that no sharp observable is informationally complete, whereas there do exist informationally complete unsharp observables. This fact gives rise to a new mode of complementarity in quantum mechanics - the complementarity between the determinations of the past and the future of a system [Bus 89b,95b]. Indeed one can show that an informationally complete observable E does not admit a repeatable measurement. If one aims at an optimal determination of the future (preparation) of a system, then one would choose a repeatable measurement. But then the measured observable is not informationally complete. Conversely, an optimal determination of the past via measurement of an informationaUy complete observable excludes optimal state preparation. It turns out that these two complementary goals can be reconciled in an approximative way, using the notion of (e, ~)-repeatability (cf. Sect. 8).

IV. Objectification and Interpretations of Quantum Mechanics IV.1. Routes Towards Solving the Objectification Problem The heart of the measurement problem in quantum mechanics is the objectification requirement and its implications studied in Chap. III. The theorems established there allow one to give a systematic overview of all logical possibilities left open for searching a resolution. We collect the statements and assumptions on a measurement scheme A/f that are involved in the insolubility theorems of Sect. III.6: (PR)

A/f yields probability reproducibility for an observable [Sect. III.1].

(QS)

S is a proper quantum system [Sect. II.2.3].

(QA)

,4 is a proper quantum system [Sect. 11.2.3].

(U)

The measurement coupling is given by a unitary operator [Sect. III.2.2].

(PVD) A/t satisfies the pointer value-definiteness condition for jt [Eq. (III.2.11)]. (PM)

A/f satisfies the pointer mixture condition for 8 + ,4 [Eq. (III.6.5)].

(o)

M fulfills the objectification requirement (Sects. II.2.4, III.6.1).

The quantum mechanical objectification problem, stated in Theorem III.6.2.1, can then be summarised as follows.

(PR) & ( Q S ) & ( Q A ) & (U)&

(PVD)

',. -~(PM)

(OP)

It should be noted that (PM) is stronger than the pointer mixture condition for ,4 alone. Indeed all the properties (PR), (QS), (QA), (V), (PVD), and the pointer mixture condition for ,4 are fulfilled for instance in a v o n Neumann-Liiders premeasurement of a sharp discrete observable A as introduced in Sect. III.2.5. The sentence (OP), as it stands, expresses the logical incompatibility of six assumptions. The physical meaning and relevance of this formal result depends on the evaluation of the role of each of these assumptions and therefore on the kind of interpretation of quantum mechanics that one intends to pursue. This observation leads us back to the decision tree of Table 1, Chap. I. There it was shown how the various intentions that one might attach to the understanding and purpose of a physical theory lead to different decisions as to which interpretation one will have to be committed to. The first two premises are not likely to be open to dispute: (PR) defines the relation between the given measurement scheme and some observable as the measured one, while (QS) expresses the fact that one is interested in understanding measurements performed on a proper quantum system. We shall explicate next

92

IV. Objectification and Interpretations

the evaluation of the remaining statements (QA), (U), (PVD), and (PM) from the point of view of the interpretations indicated in the decision tree. We begin with the standpoint of the realistic intentions pursued in this book, which can be summarised as follows (cf. Table 1). (A) The referents of quantum mechanics are individual systems, their properties and their time evolution. (B) Quantum mechanics is a complete theory: its state description gives an exhaustive account of the properties of a system that can be real at a given time. (C) Quantum mechanics treats measurements as autonomous physical processes which lead to definite outcomes, that is, objectification. (D) Quantum mechanics is a universally valid theory; it applies to all physical systems, whether microscopic or macroscopic. Turning these intentions into an explicit and consistent interpretation of quantum mechanics requires, first of all, specifying the meaning of the terms 'real property', 'definite outcome', and 'objectification'. To begin with, we adopt the definitions given in Chap. II. A property is real in a given state if that state is an eigenstate associated with the eigenvalue 1 of the corresponding effect. To regard this condition as being sufficient and necessary amounts to one way of adopting (B). The objectification condition was based on the ensuing notion of objectivity (Sect. II.2.4) in such a way that one has the implication (O)

~

( P V D ) & (PM).

Now (C) requires us to insist in (O), that is: objectification is to take place in each single run of an experiment. Thus our realistic attitude forces us to read the implication (OP) as follows. (Here, as well as in the forthcoming implications, the premises are taken to be valid, so that the conclusions show the price to be paid.) (PR) & ( Q S ) &

(O)

;- - ~ ( U ) v - ~ ( Q A ) .

To solve the objectification problem, one must therefore abandon either (U) or (QA). One has to commit oneself to one of the following options. (PR) & ( Q S ) & (PR) & ( Q S ) &

(QA)&

(O)

==~ -~(U);

(U)&

(O)

==,

~(QA).

(MD) (CP)

We refer to these conclusions as the modified dynamics (MD) and classical pointer (CP) option, respectively. Both amount to accepting that quantum mechanics cannot fulfil a/1 of the intentions (A), (B), (C), (D) in the sense formalised here. In fact the unrestricted, universal validity (D) of quantum mechanics in its original form is

IV.1. Solving the Objectification Problem

93

called into question: either one takes into consideration some nonunitary dynamical law, or one starts looking for theories of macrosystems that differ from quantum mechanics at least to the extent that some classical observables (superselection rules) are taken into account. In the latter conclusion we have made use of the fact that (QA) is invalidated in the sense described in Theorem III.6.3.1. The two options outlined here require the objectification to take place in the strong sense formulated in Chaps. II and III. They give rise to a class of approaches towards solving the measurement problem which are ready to go beyond quantum mechanics, thereby giving up (D) while maintaining (A), (B) and (C) for the modified theory (Sect. 4). It may still be possible to preserve all four realistic intentions, including the universality (D). The only way then to cope with (OP) is to modify the notions of reality, objectivity and objectification, (PR) & ( Q S ) &

(QA)&

(U)

==,

--(O).

(MO)

This line of approach imposes on any interpretation the task of explaining why and in which sense objectification appears to take place 'in the eyes (or minds)' of conscious observers. More generally, one sets out to meet the challenge of understanding how quantum mechanics can account for the emergence of a classical world as it is perceived by human observers. This task has been pursued in the many-worlds interpretation and some more recent variations of it (Sect. 5). Most of these approaches are based on the conception of definite, sharp pointer positions, so that (PVD) is maintained. Also, the realisation of the pointer mixture condition for A is still being pursued, while the corresponding condition for ,S + ,4 (PM), and thereby (O), is explicitly given up. There are two opposing attitudes, an epistemologically optimistic and a skeptical one, which call into question the whole project of a realistic interpretation of quantum mechanics. The optimistic attitude entertains the hope that there is a more comprehensive, hidden variable theory which renders quantum mechanics incomplete; this amounts to denying (B). As a consequence one would view quantum mechanics as a mere ensemble theory, reflecting the statistical aspects of the underlying theory. In this way the formal result (OP) would become irrelevant: the phenomenon of nonobjectivity, which gives rise to (OP), would have to be reinterpreted in terms of some kind of 'cryptodeterminism' (Sect. 3). The skeptical attitude would reject the realistic point of view altogether and stick to the empiricist-instrumentalist wisdom that quantum mechanics, as any physical theory, is just a tool for computing and predicting measurement outcomes. In this view (A) is given up because one does not want to get involved with interpretational problems that may have no bearing on the empirical content of the theory. Instead one adheres to the minimal interpretation and nothing more. A similarly cautious attitude was adopted by the pioneers of quantum mechanics who announced some or another variant of Copenhagen interpretations (Sect. 2).

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With the last-mentioned option we have reached the top of the decision tree (Table 1) and exhausted the list of possible ways of dealing with (OP). There is yet one residual potential loophole to the derivation of (OP): the measurement theory developed in Chap. III makes extensive use of the assumption that initially the object system and apparatus are independent of each other, and that the apparatus can be prepared in one and the same initial state T~ for several runs of the measurement. Since the apparatus is a macroscopic system, this hypothesis of the reproducibility of Tx may be regarded as being of a counterfactual nature. There are various attempts to exploit the possible microscopic nonreproducibility of the apparatus preparation [Mach 80, Schu 91]. In one approach [Schu 91] it is argued that in each single run of an experiment the apparatus is in some 'special initial state' such that after the interaction a definite pointer state is reached; and that some 'pre-established randomness' in the object system's environment is capable of recognising the state of S and thus of conspiring with the apparatus to make it show the right statistics. It does not seem obvious to us how such a view should provide (a natural explanation) for the probability reproducibility condition (PR); at the least, without the assumption of the initial independence of S and ,4, the whole formulation of quantum measurement theory would need a fundamental revision. Therefore we do not pursue this particular option here. Each of the options listed here has been taken up in one or another approach to the interpretation of quantum mechanics, and is still the subject of intensive efforts. A commonly accepted resolution of the measurement problem does not seem to be in reach, and it may still be too early for any conclusive assessments. This seems true even in view of the current appearance of various comprehensive accounts of the different approaches. Therefore we feel free in our subsequent survey to just indicate the systematic position of the different approaches in the logical scheme developed here, with some comments on their merits and open questions. In order to explain the spirit of our modest critical remarks, we first describe our general views of the purposes that an interpretation of quantum mechanics should serve and of the criteria that the formulation of an interpretation should meet. To establish an interpretation of quantum mechanics (or of any physical theory) first of all means to formulate a set of rules that connects symbols and relations of the mathematical formalism with a certain domain of the 'physical world', that is, a domain of experience constituted by a collection of experimental procedures. Thus an interpretation is that part of a physical theory that determines the physical meaning of concepts and laws and thereby makes the formalism a mathematical picture of the given domain of physical experience. In general there is not a unique interpretation associated with a given formalism; and strictly speaking, each interpretation will fix another theory. Therefore it may sound paradoxical to speak of different interpretations of quantum mechanics. We resolve this puzzle by understanding 'quantum mechanics' or 'quantum theory' as that physical theory that is

IV.2. Copenhagen Interpretations

95

based on the usual Hilbert space formalism and equipped with the minimal interpretation introduced in Chap. II; any other 'interpretation of quantum mechanics' then augments this theory with further statements attaching meaning to some symbols and relations. It is important to realise that there is no complete freedom and arbitrariness in choosing an interpretation of quantum mechanics. There is always a tension between the desiderata that one might wish an interpretation to fulfil and the empirical and formal constraints that have to be taken into account. On the one hand, an interpretation should be sufficiently rich so as to be an instructive guide in the consideration of physical problems and the development of new ideas. We maintain that in the last instance it is only with a well-elaborated interpretation that has been checked in as many of its facets as possible that one can ascertain the correctness of the intuitive pictures that one always uses in the formulation of models and the conception of experiments. On the other hand, any interpretation must obey the constraints of empirical adequacy [i.e., there must not be a contradiction with experimental results] and self-consistency [i.e., the interpretation should not entail relations that are in logical conflict with the formal structures of the theory]. The empiricist skepticism points to the danger that too rich an interpretation may produce inconsistencies or paradoxa, which may in fact be quite subtle and hard to discover. The objectification problem itself is an example of the kind of problems that one may run into when it comes to formalising certain ideas: here it is the notion of reality or objectivity that one tries to implement into the quantum mechanical formalism. However, against this skepticism we hold that all the intuition and creativity invested into the development of physics always went far beyond what would be allowed by a purely empiricist or instrumentalist position. Therefore it would be the genuine task of a realistic interpretation, the intentions and expectations of which are expressed in statements such as (A), (B), (C), (D) above, to provide the conceptual tools allowing one to control and hopefully reconfirm the intuitive ideas about what picture of physical reality the given theory yields. The subsequent sections will illustrate further some of the formal problems that need to be addressed in the elaboration of any interpretation of quantum mechanics that sets out to solve the objectification problem. We start with a more or less historical review of the extreme epistemological positions mentioned above which try to evade the objectification problem in different ways.

IV.2. Historical Prelude - Copenhagen

Interpretations

The problem of measurement, that is, the clash between the phenomenon of nonobjectivity and the need for a classical description of the apparatus, was clearly envisaged by the pioneers of quantum mechanics. They tried to deal with it by emphasising the methodological necessity of placing a 'cut' between the object to be

96


described by the theory and the means of observation. In order to evaluate the current attempts at solving the measurement problem in their proper historical context, we find it useful to review briefly some of the representative views of the Copenhagen school on the measurement problem. We shall start by presenting the intuitive ideas developed by Bohr and Heisenberg and then go on with the more systematic studies by von Neumann and Pauli. We also recall the related views of London and Bauer and of Wigner. In speaking of the Copenhagen interpretation of quantum mechanics one is usually referring to the interpretation of quantum mechanics which resulted from the discussions between Niels Bohr, Werner Heisenberg, and Wolfgang Pauli with contributions from Max Born and Johann von Neumann. In very broad terms one may say that these discussions were the first attempts to solve the interpretational problems of quantum mechanics considered as a fundamental theory of individual atomic objects. The classical papers by Born [Born 26], Heisenberg [Heis 27] and Bohr [Bohr 28] are important landmarks in this early development, which led to the systematic treatises by von Neumann [yon Neumann 1932] and Panli [Faun 1933] on the mathematical and conceptual foundations of quantum mechanics. The acceptance of the probability interpretation for the Schrbdinger wave function and the acknowledgement of some fundamental limitations on the applicability of the concepts of classical physics in the description of atomic phenomena were important elements in the Copenhagen interpretation, which, however, never developed into a coherent systematic interpretation of quantum mechanics. Still Bohr's Como lecture may be considered to some extent as a codification of the Copenhagen viewpoint. In fact, as is now well known, the views of Bohr, Heisenberg and Pauli on the interpretation of quantum mechanics were divergent and diffuse to the extent that no unique interpretation could have been built on them and that almost any of the present day interpretations of quantum mechanics can be argued as being a systematic development of some of the 'Copenhagen views'. B o h r . According to Bohr the key to the understanding of the quantum theory was the viewpoint of complementarity which he advocated and developed in a series of articles published during the years 1927- 1962. Rather than attempting to give a systematic exposition of Bohr's views and to argue which of the isms Bohr favoured we provide a few citations from the writings of Bohr which, we think, are revealing with respect to Bohr's account of the measurement process. [Bohr 28] The quantum theory is characterised by the acknowledgment of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests essentially upon the classical physical concepts.


97

• .. Quantum postulate implies a renunciation as regards the causal spacetime co-ordination of atomic processes. [But] the idea of observation belongs to the causal space-time way of description. • .. the complementary character of the description of atomic phenomena • .. appears as an inevitable consequence of the contrast between the quantum postulate and the distinction between object and agency of measurement, inherent in our very idea of observation. [Bohr 39] A clarification of the situation as regards the observation problem in quantum theory ... was first achieved after the establishment of a rational quantum mechanical formalism. • .. In the first place, we must recognise that a measurement can mean nothing else than the unambiguous comparison of some property of the object under investigation with a corresponding property of another system, serving as a measuring instrument, and for which this property is directly determinable according to its definition in everyday language or in the terminology of classical physics. [Bohr 48] An adequate tool for the complementary mode of description is offered by the quantum-mechanical formalism. The idea of the complementarity of observables first conceived by Bohr is expressed in the present approach to quantum measurements by the mutual exclusiveness of state transformers according to Sects. III.2.1 and III.8.2. For Bohr, the referents of quantum mechanics are observed phenomena, and the notion of an individual microsystem is meaningful only within the context of the whole macroscopic experimental setup defining the phenomenon under observation. The classical description of measuring devices is reproduced here by the requirement of the pointer objectification and of the classical properties of the pointer observable (Sect. III.6.3). H e i s e n b e r g . The breakthrough for the proper understanding of the formalism of quantum mechanics lay, according to Heisenberg [Heis 27], in the discovery of the uncertainty relations. These relations showed, in a quantitative way, the limitations on the applicability of classical concepts to the description of atomic phenomena. As mentioned at the end of Sect. III.8.2, the uncertainty relations at the same time may be interpreted as a relaxation of the complementarity. Though Heisenberg's starting point was more formal than Bohr's, Heisenberg later acknowledged Bohr's viewpoint of complementarity as the fundamental basis of quantum mechanics. With respect to an analysis of the measuring process this appears in the fact that, like Bohr, Heisenberg emphasised the methodological necessity for clearly distinguishing the object under investigation and the applied measuring apparatus. Whereas the object was to be described in terms of quantum mechanics, the measuring apparatus

98


had to be described in classical terms. The following extracts from Heisenberg's writings illustrate these remarks.

[Heisenberg 1930] It must also be emphasised that the statistical character of the relation [between the values of two quantum mechanical observables - authors' note] depends on the fact, that the influence of the measuring devices on the system to be measured, is treated in another way than the mutual influence of the parts of the s y s t e m . . . . If one were to treat the measuring device as a part of the system ... then the changes of the states of the system considered above as indeterminate would become determinate. But no use could be made of this determinateness unless our observation of the measuring device were free of undeterminateness. For these observations, however, the same considerations are valid, and we should be forced, for example, to include our own eyes as part of the system, and so on. Finally, the whole chain of cause and effect could be quantitatively verified only if the whole universe were incorporated into the s y s t e m - but then physics has vanished and only a mathematical scheme remains. The partition of the world into observing and observed system prevents a sharp formulation of the law of cause and effect. [authors' emphasis] (The observing system need not always be a human being; it may also be an apparatus, such as a photographic plate, etc.)

[Heisenberg 1958] In natural science we are not interested in the universe as a whole, including ourselves, but we direct our attention to some part of the universe and make that the object of our s t u d i e s . . . , it is important that a large part of the universe, including ourselves, does not belong to the object. • .. [Before or at least] at the moment of observation our object has to be in contact with the other part of the world, namely the experimental arrangement [which is to be described in terms of classical physics]. • .. After this interaction has taken place, the probability function contains the objective element of tendency and the subjective element of incomplete knowledge, even if it has been a 'pure' case b e f o r e . . . The observation itself changes the probability function discontinuously; it selects of all possible events the actual one that has taken p l a c e . . . . • .. the transition from the 'possible' to the 'actual' ... takes place as soon as the interaction of the object with the measuring device, and thereby with the rest of the world, has come into play; it is not connected with the act of registration of the result by the mind of the observer. The discontinuous change in the probability function, however, takes place with the act of registration, because it is the discontinuous change of our knowledge in


99

the instant of registration that has its image in the discontinuous change of the probability function. • .. quantum theory corresponds to the ideal of objective description of the world as far as possible. Certainly quantum theory does not contain genuine subjective features, it does not introduce the mind of the physicist as a part of the atomic event. But it starts with the division of the world into the 'object' and the rest of the world, and from the fact that at least for the rest of the world we use the classical concepts in our description. The partition of the world into an observed system S and a measuring apparatus ,4 which is inevitable for an objective description of the physical system is reformulated in the present report by the reduction of the state of the compound system S + j t into the reduced states of 8 and j[, respectively (Sect. II.1.2). Within the framework of our approach the apparatus ,4 should be described by means of classical concepts, since only in this way can the objectification of the measuring results be achieved (Sect. III. 6). von N e u m a n n a n d Pauli. von Neumann did not accept Bohr's view, shared by Heisenberg, of the necessity of classical language in the description of atomic phenomena. On the contrary, according to von Neumann, quantum mechanics is a universally valid theory which applies equally well to the description of macroscopic measuring devices as to microscopic atomic objects. It may well be that a detailed description of a measuring apparatus is highly complicated but, according to von Neumann, no limitation in principle is known for a description of a measuring apparatus and thus of the whole measuring process in quantum mechanics. As is well known, and already noted in Chap. III, von Neumann developed the quantum mechanical theory of the measuring process in a way that still meets today's standards of rigour. Within his approach von Neumann clearly faced the problem that the object system and the measuring apparatus had to be separated after the premeasurement and that this problem could not be solved within quantum mechanics. To solve the dilemma, von Neumann introduced what is known as the projection postulate, and he argued that the final termination of any measuring process is in the conscious observer, in his becoming aware of the measurement result. Von Neumann also argued that - apart from the fact that in the course of the development of physical theories the borderline between a physical object, like the measuring apparatus, and the conscious observer is as if shifted in the direction of the latter - the conscious observer cannot be included in the domain of any physical theory. We let von Neumann speak (in the 1955 translation), using, however, the technical notations of the present text.

[von Neumann 1932] We therefore have two fundamentally different types of interventions which can occur in a system $ . . . . First, the arbitrary changes by measurements

100

IV. Objectification and Interpretations (1)

T ~ Zu(R)(T).

Second, the automatic changes which occur with the passage of time (2)

T ~ U~TUt.

• .. In the measurement we cannot observe the system S by itself, but must rather investigate the system S + ,4, in order to obtain (numerically) its interaction with the measuring apparatus A. The theory of measurement is a statement concerning S + ,4, and should describe how the state of S is related to certain properties of the state of ,4 (namely, the positions of a certain pointer, since the observer reads these). • .. the measurement or the related process of the subjective perception is a new entity relative to the physical environment and is not reducible to the latter. • -. But in any c a s e , . - . , at some time we must say: and this is perceived by the observer. • .. Now quantum mechanics describes the events which occur in the observed portions of the world, so long as they do not interact with the observing portion, with the aid of process 2, but as soon as such an interaction occurs, i.e., a measurement, it requires the application of process 1. The dual form is therefore justified. Another member of the Copenhagen school who developed measurement theory in a systematic way was Wolfgang Pauli. His analysis of the measurement problem is almost literally the same as the one given by von Neumann. Indeed Pauli writes as follows (we here give the 1980 translation):

[Pauli 1933] The measurement . . . generates in general ... out of a pure case ... a mixture -.. [cf. T ~-~ Zu(R)(T)]. This r e s u l t . . , is of decisive importance for the consistent interpretation of the concept of measurement in quantum mechanics. For this result shows that we arrive at consistent results concerning the system, whatever be the way in which the division between the system to be observed (which is described by wave functions) and the measuring apparatus is made. (Cf. J.v. Neumann (1932), where in Chap. VI this question is discussed in detail.) • -. It is possible to express the fact that a definite measuring apparatus will be used in the mathematical formalism of quantum mechanics directly. On the contrary, this is not possible with the stipulation that the measurement should give a definite result ... Any statement about a physical fact made with the help of a measuring device (observer or the registration apparatus) which is not counted as part of the system cannot (from the standpoint of mathematical formalism which describes directly


101

only probabilities) present a particular, scientifically not pre-determined act which is to be taken into account by a reduction of the wave-packets... [cf. T ~ Zu(R)(T) ~ [Zu(X)(T)]-1Zu(X)T]. We need not be surprised at the necessity for such a special procedure if we realise that during each measurement an interaction with the measuring apparatus ensues which is in many respects intrinsically uncontrollable. The theory presented in Chap. III can be seen as a further elaboration of the treatment by von Neumann and Pauli, with the important difference that here the measurement problem is not 'solved' with a reference to a conscious observer. S o m e e l a b o r a t i o n s . The analysis of the measuring process as given by von Neumann remained more or less unknown for a long time, perhaps due to its then advanced mathematical presentation. Some authors, like London and Bauer [Lon 39], appreciated this work of von Neumann but they felt the need for a 'concise and simple' treatment of the problem. London and Bauer developed the theory of the quantum mechanical measurement process just in accordance with von Neumann. However, they introduced one significant change into von Neumann's description, namely, by including the conscious observer as a part of the quantum mechanical description of the measurement process. Indeed they considered quantum mechanically a system consisting of the object system S, the measuring apparatus ,4, and the observer (9. To solve the objectification problem, London and Bauer went on to assume that the observer can, by 'introspection' and with his 'immanent knowledge', always rightly create his own objectivity, and thus identify his own pure state. 'I am in the state P[pk]' so that, due to correlations, the measuring apparatus ,4 is in the state P[¢k] and thus the object system $ is in the state P[Tk]. The London-Bauer interpretation of the measurement according to which the objectification of the measuring result is provided only by the activities of the observer's consciousness was criticised by Wigner [Wig 61]. A philosophical analysis of the London-Bauer approach in the light of Wigner's consideration was given by Shimony [Shi 63]. With his 'friend paradox' Wigner gave an illustration that after the measurement interaction between S, ,4 and O the observer (.9 (Wigner's friend) will always be in a state which indicates a definite measuring result. On the other hand, if the total compound system S + A + (.9 were treated by means of ordinary quantum mechanics in Hilbert space, then the observer £9 would be in a superposition of possible final states, that is, in a state of 'suspended animation'. For Wigner [Wig 63] this result appears absurd and he concludes that the quantum mechanical equations of motion cannot be linear, if macroscopic objects like ,4 and (.9 are described. However, no explicit proposal for a nonlinear modification of the SchrSdinger equation was made. Systematic investigations in this direction started only around 1976 (Sect. 4.3).

102


Later Wigner changed his mind and adopted the point of view expressed by Zeh [Zeh 70] according to which the interaction of the macroscopic system (,4 or (9) with the environment destroys the nondiagonal elements in the density matrix of this system (cf. Sect. 5.3). Wigner proposed a modified von Neumann-Liouville equation which describes the (exponential) decrease in time of the nondiagonal elements in the density matrix of macroscopic systems and thus should lead automatically to the desired objectification of the measuring result. The consciousness of the observer, which in the London-Bauer approach was assumed to solve this problem is no longer needed for the objectification. Moreover, according to Wigner [Wig 83], the consciousness of the human observer is beyond the scope of quantum physics and classical physics.

IV.3. E n s e m b l e and H i d d e n Variable Interpretations According to the optimistic epistemological position mentioned in Sect. 1, physical objects and their properties are just as they are observed in the macroscopic world; the classical deterministic realism thus maintained leads one to conclude that the objectification problem is merely an illustration of the incompleteness of quantum mechanics. This theory should therefore be regarded as a statistical framework allowing one to describe the outcomes of measurements performed on an ensemble of equally prepared objects. The rise of the so-called ensemble and hidden variable interpretations of quanturn mechanics is much due to the critique of Albert Einstein and Erwin Schr5dinger on quantum mechanics as a fundamental theory of individual atomic objects. The probabilistic nature of quantum mechanics and the incompleteness argument of Einstein, Podolsky, and Rosen [Eins 35] led, on the one hand, to a consideration of quantum mechanics only as a statistical theory of atomic objects, and, on the other hand, to the development of proper completions of the theory. The critique of the projection postulate, as put forward by Margenau [Marg 36,63], was a further impulse for developing the ensemble interpretations of quantum mechanics. Though much of this critique was later found to be unjustified, these interpretations still have their advocates. The statistical interpretations of quantum mechanics can be divided into two groups, the measurement statistics and the statistical ensemble interpretations (Sects. III.3.2-3). These interpretations rely explicitly on the relative frequency interpretation of probability, and in them the meaning of probability is often wrongly identified with the common method of testing probability assertions. In the measurement statistics interpretation the quantum mechanical probability distributions, such as pA, are considered only epistemically as the distributions for measurement outcomes. The concept of state is taken to characterise conceptual infinite sequences FT A of measurement outcomes at,, a l 2 , ' " such that

IV.3. Ensemble and Hidden Variable Interpretations

103

n pA(x) = limn--.o~ ~1 ~-']~i=1 Xx (at,). In this pragmatic view quantum mechanics is

only a theory of measurement outcomes providing convenient means for calculating the possible distributions of such outcomes. It may well be that such an interpretation is sufficient for some practical purposes; but it is outside the interest of this treatise to go into any further details, for example, to study the presuppositions of such a minimal interpretation. The measurement problem is simply excluded in such an interpretation. The statistical nature and the alleged incompleteness of quantum mechanics are apparent in the ensemble interpretation of quantum mechanics. In this interpretation a state T characterises a conceptual infinite collection FST of identical, mutually noninteracting systems $1, S2,.... The probability measures pT A defined by a state T describe the distribution of the values of an observable A among the members Si of the ensemble FST. Accordingly, the number pA(x) is the relative abundance of systems $i in FST having the value of A in the set X. The ensemble interpretation of quantum mechanics describes individual objects only statistically as members of ensembles. This interpretation is motivated by the idea that each physical quantity has a definite value at all times. Thus no measurement problem would occur in this interpretation. Some merits of the ensemble interpretation of quantum mechanics are put forward, for example, in [Bal 70,88, d'Espagnat 1971]. But these merits seem to consist only of a more or less trivial avoiding of the conceptual problems, like the measurement problem, arising in a realistic approach. In fact it is only in the hidden variable approaches that one tries to take seriously the idea of the value-definiteness of all quantities. The basic idea of the hidden variable interpretations of quantum mechanics was to 'complete' the probabilistic state description of quantum mechanics with some 'hidden variables' to obtain a dispersion-free state description providing the quantum mechanical probabilities as statistical averages over the 'hidden variables'. Strictly speaking, this type of approach sets out to formulate a more extensive theory than quantum mechanics in that it tries to give a wider mathematical framework than the Hilbert space formalism in order to allow for the hidden variables. Much work has gone into this project, and a number of no-go-theorems emerged restricting the possible forms of such interpretations. It follows from these theorems that only the so-called contextual non-local forms of the hidden variable interpretations are tenable at all. The model of Bohm and de Broglie, also called the causal interpretation of quantum mechanics, is the best known example in this direction [Bohm 52,66,89] It provides a picture of the quantum mechanical reality in which individual particles have definite positions at all times in such a way that they follow trajectories governed by a deterministic Newtonian law of motion. Quantum mechanics and its non-local features are taken into account by the presence of the so-called quantum potential. The underlying deterministic equation of motion generating the trajec-

104


tories can be taken in the form of the Liouville equation and combined with the quantum equilibrium condition, that is, the initial condition that at some fixed (initial) time the distribution function coincides with the squared modulus of some wave function. Then quantum mechanics can be recovered along with its minimal interpretation. This claim, a proof of which was outlined by Bohm, has been substantiated considerably by extensive mathematical investigations in the 'Bohmian mechanics' [Diirr 92,93] approach, which has experienced a remarkable renaissance in the last years. Physical applications of the causal interpretation have been elaborated in numerous examples, providing impressive demonstrations of its intuitive appeal. Holland [Holland 1993] gives a fair and comprehensive account of the merits and difficulties of this approach. The causal interpretation selects the position as a preferred variable to which it attributes value-definiteness in all states. For all other observables it can be shown that due to the definiteness of the positions, measurements always lead to definite outcomes which are causally determined and which occur with the frequency predicted by the quantum mechanical probability. The measurement problem is thus solved in this approach. The nature of this solution becomes apparent if one realises that the Bohm theory is formally equivalent with quantum mechanics: the 'hidden variable' is suggested by the form of the Schrhdinger equation itself, and the quantum equilibrium condition ensures that no statements arise that may be in conflict with quantum mechanical propositions. Moreover, this interpretation maintains the unitary Schrhdinger evolution also in measuring processes, so that it is only the value assignment to the position observable by which it differs from the realistic interpretation of Sect. 1. Thus it turns out that what had been intended to be a hidden variable augmentation of quantum mechanics is a variant of the modal interpretations" the notions of reality and objectivity are modified so as to allow value assignments for a preferred observable in all states [Bub 96] (see Sect. 5.2 below). It should be noted that selecting a particular observable as having definite values stands quite in contrast to the symmetric treatment of all observables in the realistic interpretation fixed in Chap. II. Therefore, in adopting the Bohmian or the modal point of view, one should be able to give good physical reasons as to why there should be such a 'preferred' observable. The Bohm theory is deterministic but non-classical in the sense that the equation of motion for the definite position variable contains the non-local quantum potential. There is a class of hidden variable theories that are genuinely classical in trying to account for the quantum phenomena by introducing local interactions with a fluctuating background medium. Some more recent variants of these approaches are the stochastic mechanics reviewed in [Nelson 1985] or the stochastic electrodynamics (e.g., [Boy 80, Cole 90]). The classical stochastic theories are ruled out in principle as being incompatible with quantum mechanics by the KochenSpeaker-Bell theorems (see, e.g., [Peres 1993]). But as far as experimental testing

IV.4. Modifying Quantum Mechanics

105

is concerned, there is still a debate on whether the apparent agreement between quantum mechanics and the Einstein-Podolsky-Rosen experiments could be due to the low detector efficiencies and whether better experiments might produce results in favour of a classical stochastic theory. As long as the 'efficiency loophole' has not been closed, it is interesting to note that there is completely independent experimental evidence which conclusively rules out a large class of stochastic hidden variables and requires at least that such theories must be endowed with nonlocal features similar to those of the Bohm theory. In fact Baublitz [Baub 93] pointed out that quantum mechanics and classical stochastic theories give different predictions for the average change of energy (AE) per electron in the tunnelling taking place in field emission experiments. Analysing the data of experiments performed since the 1970s, Baublitz [Baub 95] was able to show that the experimental values of (AE) are in good agreement with quantum mechanical calculations but differ significantly from the values required by some local stochastic theories. There are some further recent hidden variable formulations which may not easily be attacked by the known counter arguments. We mention two such approaches, without going into details. The first one, proposed by the late Asim Barut [Barut 90], follows closely the original ideas of SchrSdinger on the matter wave interpretation of the C-function. The other approach, elaborated by Gudder [Gudder 1988], is based on the notion of quantum transition amplitudes and allows for value-assignments at the price of non-standard modifications of the probability theory.

IV.4. Modifying Quantum Mechanics Quantum mechanics does not seem to be easily reconciled with a classical, deterministic conception of reality. Thus, if one accepts the completeness of quantum mechanics and still intends to maintain the goal of objectification (0), then one is forced to question the universal, unrestricted validity of the theory: either one pursues the option ( C P ) in order to ensure the required classical properties of some macroscopic systems, or one searches for a modified description of the dynamics of quantum systems (MD) so that a unification of von Neumann's two types of state changes under one common law is achieved. The first option has gained some general support on the basis of abstract foundational studies (Sect. 4.1) while it was also studied in terms of models (Sect. 4.2). The latter possibility has been dealt with mainly in terms of ad hoc proposals for nonlinear or stochastic modifications of the SchrSdinger equation (Sect. 4.3).

IV.4.1 Operational Approaches and the Quantum-Classical Dichotomy Common to all the variants of Copenhagen interpretations is the emphasis of the operational aspect that quantum machanics makes statements about objects as they appear under the conditions of observation, or measurement. Furthermore,

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the necessity of describing the macroscopic aspects of a measurement in terms of classical physics was always pointed out and seen in contrast to the phenomenon of nonobjectivity, or indeterminacy. This situation forces one to consider the possibility that quantum mechanics needs to be complemented with an essentially classical theory of macroscopic systems in order to account for the classical behaviour of measuring devices and the occurrence of definite pointer readings. A systematic reflection of these points has been undertaken in various approaches aiming at an operational reconstruction of the structure of quantum mechanics. These approaches set out to investigate the pragmatic and physical preconditions for the constitution of physical objects, including the microscopic objects of quantum mechanics. In the quantum logic approach it is found that the formal structures that can be justified in this way are open to the possible existence of (partly) classical systems. Hence according to this approach there are no a priori operational reasons inherent in the general language structure of a physical theory, which would either exclude or require the existence of superselection rules. This is a demonstration of the fact that the measurement problem is indeed specific to the irreducibility of the lattice of propositions of ordinary Hilbert space quantum mechanics. One may thus be led to consider formal extensions of ordinary quantum mechanics by stipulating the presence of superselection rules (Sect. 4.2). Another type of operational approach starts with an analysis of the general statistical structure of physical theories [Ludwig 1983]. This procedure allows one to investigate, in very general terms, the relationship between objective, deterministic theories for macroscopic phenomena and quantum mechanics as a theory for microscopic systems. If it can be shown that the former cannot be derived from a many-body extrapolation of the latter, then the universal validity of quantum mechanics is lost, and there is no reason to expect that measuring devices belong to the domain of quantum mechanics [Lud 83]. Contrary to the usual textbook wisdom it has been argued as a result of careful analysis that macrophysical theories, like thermodynamics or classical mechanics, cannot be derived as approximate limiting cases of quantum mechanics. Such theories should rather be regarded as theories in their own right. Yet as macroscopic systems are aggregates of microscopic systems, it is important to understand the interrelation between quantum mechanics and macroscopic theories. This programme has been carried out to quite a large extent in the form of an embedding of macrotheories into a quantum mechanical many-body theory [Barc 82, Ludwig 1987]. The effect of such an embedding can be described, roughly speaking, as restricting the sets of states and observables of the many-body quantum theory so that the remaining structure allows for the emergence of a classical (deterministic) behaviour of the macroscopic quantities. Some concrete examples of embeddings illustrating the abstract approach can be found in the contributions of Lanz and Melsheimer, and Neumann in [Cologne 1993].


107

The conclusion to be drawn from these investigations is that the reconciliation of quantum mechanics with an objective macroscopic description causes difficulties even to such an epistemologically cautious approach, which abstains from farreaching realistic aspirations and uses only a minimal interpretation. The classical behaviour of macroscopic observables cannot be derived from quantum mechanics; it is only shown that the classical features of macroscopic systems can be consistently described in an approximate way within quantum mechanics [Lud 93]. Even then, the macroscopic quantities must be represented as unsharp observables in order to ensure the consistency of the embedding in question (cf. [Ludwig 1987], Chap. X.2.5). In this way the measurement problem has been dissolved at the expense of giving up the universal validity of quantum mechanics.

IV.4.2 Classical Properties of the Apparatus Sometimes it has been argued that the object system S itself, as well as the apparatus system A, have naturally restricted sets of observables so that not all selfoadjoint operators correspond to observables. Such limitations on measurability may be due to fundamental conservation laws [Ara 60, Wig 52], or due to the limited number of actually existing interactions [Pro 71] (cf. Sect. III.8.3). While in this way one circumvents the implication ( o e ) by exploiting a potential violation of (QS), it is an open question whether objectification can be achieved in this way. Such a solution would not have the status of a theoretically well-founded approach either but would rest upon accidental facts as long as there were no theory of the fundamental interactions. This, admittedly vague, option is mentioned here just for completeness as one logical possibility of trying to deal with (OP). A more effective attempt at tackling the objectification problem is the (more or less ad hoc) consideration of superselection rules for the apparatus. That is, one assumes that the pointer, being an observable of a macroscopic system, should be a classical observable [Beltrametti and Cassinelli 1981, Jau 64,67,68, Wan 80]. Then the objectification (0) is ensured since the pointer observable is objective throughout. However, the dynamical consistency problem described in Sect. III.7 still persists and requires a modification of the dynamics at least to the extent of accepting that the interaction Hamiltonian is no observable [Beltrametti and Cassinelli 1981, Wan 80]. While superselection rules can easily be incorporated into the Hilbert space framework of quantum mechanics, they nevertheless represent an element which does not follow from the standard axiomatics. Thus the problem remains of providing a theoretical explanation for such restrictions on the sets of observables and states. Furthermore, the consistency problem of Theorem 7.2.1 may be interpreted as an indication of the fact that the irreversibility inherent in measurement has not been properly taken into account. In fact, the unitary evolution on a (separable) Hilbert space is quasiperiodic [Perc 61] and generally time inversion invariant, while irreversibility requires a breaking of this fundamental time inversion symme-

108

-


try. Accordingly there are many model considerations aiming at an explanation of the effectively irreversible evolution of the system S ÷ ,4 into a state equivalent to that required by objectification. Usually the equivalence of states is stipulated with respect to a restricted class of macroscopic observables of ,4. The effective irreversibility is achieved by taking account of the macroscopic nature and ergodic properties of ,4 [Cini 79, Dan 62, Haa 68, Ludwig 1961, Ros 65, Weid 67]. In this way one obtains an approximate description of the dynamics in terms of a Markovian master equation, which would be sufficient as far as the macroscopic observables (and their functions) are concerned. Moreover, these approaches allow one to investigate the thermodynamic limit, thus affording a bridge to theories dealing with infinite systems (Sect. 5.4). A more recent attempt along these lines considers irreversibility via dynamical instability properties on the level of the von Neumann-Liouville equation from which an effective restriction of the set of states can be derived; in this way the two important features of the measurement problem irreversibility and classical properties - are shown to be interrelated [Mis 79].

As mentioned above, the incorporation of superselection rules needs to be supplied with some physically convincing motivation. The only known physical mechanism for producing strict superselection rules is by means of spontaneous symmetry breaking, which may occur in systems having infinitely many degrees of freedom. There are further arguments pointing in the same direction. In fact approaches based on discrete superselection rules are somewhat artificial as they do not allow for continuous trajectories of the classical (pointer) observables involved. Concrete models of partly classical systems inevitably involve some continuous observables such as position, which are to be considered as classical observables. An early well-elaborated example is that by Sherry and Sudarshan [Sher 78,79] in which the measurement process is described as an interaction between the quantum mechanical object system S and the classical measuring device .A. The latter system is given a quantum mechanical description by means of embedding its observables into the set of self-adjoint operators of some Hilbert space. The classical nature of ,4 is preserved by stipulating that its trajectory variable corresponds to a (continuous) classical (hence superselection) observable. In this theory the dynamical problem of the Hamiltonion not being an observable is taken into account and leads to a requirement of classical integrity of the localisation variable. This approach does not aim at justifying the classical nature of the measuring device, but it provides a very instructive example of how to reconcile classical and quantum mechanical descriptions in the context of measurement. Interesting new contributions in this direction are found in [Bla 93,95, Jad 95]. The 'many-Hilbert spaces' theory by Machida and Namiki [Mach 80] is an attempt to explain the emergence of classical properties of measuring devices due to their macroscopic nature. It is argued that the energy and the number of constituents of an apparatus ,4, being a macroscopic and therefore open system, are


109

not well-defined. Hence the state of A should be a mixture of states with different definite particle numbers, these numbers being distributed in a relatively narrow range around the macroscopically large mean number no. In the limit no ~ c~ one obtains a state operator which can be represented as an average with respect to a continuous size parameter. In this way one effectively performs a transition to a direct integral of Hilbert spaces in which macroscopic observables are defined as continuous averages of microscopic observables. As pointed out by Araki [Ara 80], this procedure leads to macroscopic observables inducing continuous superselection rules. Measurement models involving such continuous superselection rules have been subsequently developed further by other authors (see, e.g., [Fuk 90]). The many-Hilbert spaces theory and its current elaborations are reviewed in [Nam 93]. In this work interesting similarities and differences between this theory and the decoherence approach (Sect. 5.3) become apparent; in particular it seems that the Machida-Namiki theory tries to explain only a rather weak, statistical form of decoherence. Apart from its ad hoc nature, the introduction of classical properties in the description of physical systems faces some other obstacles. In some cases (though not always), the implementation of continuous superselection rules in realistic models forces one into the unfortunate situation of having to deal with nonseparable Hilbert spaces [Ara 80, Gue 66, Piron 1976]. In addition the dynamical consistency problem (Theorem 7.2.1) still persists. The fact that in the case of continuous superselection rules the Hamiltonian generator of the dynamical group cannot be an observable was noticed already by Piron [Piron 1976]. A theoretical framework for systems which are partially classical is offered by the C*-algebraic approach to quantum mechanics. In this theory it is made clear that the best available justification for a restriction of the set of observables has to be based on the limited possibilities of physical observers. This amounts to admitting that objectification cannot be achieved in the strong realistic sense but that only a contextual or relational conception of definite values is tenable (Sect. 5.4).

IV.4.3 Modified Dynamics The goal of objectification may be maintained within a theory of proper quantum systems by considering some modification of the dynamical axiom of quantum mechanics which would lead to dynamically induced superselection rules. For example, one may assume a spontaneous stochastic process to supersede the ordinary unitary evolution. The corresponding generalisation of the Schrhdinger equation is interpreted as representing the autonomous dynamics of isolated systems. The stochastic process in question should become noticeable only for large systems so as to ensure that certain macroscopic observables, like pointers, are practically always found to have (nearly) sharp values. The need for giving up the linearity of dynamics in order to reach objectification was recognised by Wigner [Wig 63] but no explicit proposal for a nonlinear

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SchrSdinger equation was made at that time. Systematic investigations into modified quantum dynamics started only around 1976 (see, e.g., [Bia 76, Dio 89, Fre 90, Gis 84, Ghi 86, Pea 76]). One of the first elaborated examples, and perhaps the best known, of this type of approach is the unitied dynamics theory of Ghirardi, Rimini and Weber [Ben 87, Ghi 86,90a]. In this theory the von Neumann-Liouville equation for a quantum mechanical n-particle system is modified by adding a linear term which models a spontaneous localisation process taking place at random times: n

ih~T

= [H,T]_ ÷ ZAk(/RAqkTAq+d q - T).

(1)

k=l

The integral term is known in the context of the theory of unsharp measurements as the nonselective operation representing the reduced state of the object system after an unsharp position measurement [¢f. nq. (III.2.43)]. The operators Aqk are of the form Aqk = X / - ~ e x p ( - a(Qk- qk)2), where Qk represents (a component of) the position operator of the k-th particle. Equation (1) represents a quantum dynamical semigroup with a non-self-adjoint generator, so that the irreversibility needed for measurement is built in from the outset. The mean rates ()~k) of the localisation processes can be chosen small enough so that systems having only a few number of constituents practically evolve according to the SchrSdinger equation, while the localisation events become noticeable only for systems with a macroscopically large number of constituents. In this way the localisation observable of a macroscopic system is dynamically objectified practically at any instant of time. In the measurement context, the pointer of an apparatus ¢4 becomes thus an effectively classical observable without any ad hoc restriction of the set of states of ,4. The spontaneous localisation theory and its succeeding continuous dynamical reduction variants provide an interesting step towards a unified quantum mechanical description of microscopic and macroscopic systems. Still they are facing a number of problems. First it would be desirable toestablish a theoretical basis for its new dynamical principle, in analogy to and generalising the axiomatic characterisation of the unitary SchrSdinger dynamics [Sim 76]. Some results in this spirit have been obtained for the case of possibly nonlinear or irreversible evolution equations [Dani 89, Gis 83,86]. Second, it should be observed that the spontaneous localisation process relates to an unsharp position observable, so that the induced Gemenge is a continuous family of nonorthogonal and only unsharply localised states (wave functions). This 'tail problem' [Albert 1992, Pea 94] seems to indicate that also in the modified dynamics approach only some form of unsharp objectification may be reached. Another problem consists in the possibility that the spontaneous localisation theory may overshoot its aim: the very stochastic reduction process needed for explaining the classical behaviour of pointer observables could also erase any macroscopic

IV.5. Changing the Concept of Objectification

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quantum effects. That this need not be the case has been indicated for the phenomenon of superconductivity in [Buf 95]. Third, the question still has been raised whether the theory can be formulated in a Lorentz covariant way [Bell 87,89]. The recourse to nonlinearity in quantum mechanics bears with itself some dangerous consequences, such as the possible existence of superluminal signals [Gis 90, Pea 86]. However, it does not seem impossible to formulate such theories in accordance with special relativity and causality [Gis 89]. Recent work provides steps towards an incorporation of stochastic reduction processes into the framework of relativistic quantum theories [Ghi 90b, Pea 94]. Finally we note that the experimental testing of dynamical objectification and of nonlinear evolution equations in general is a difficult and still largely open problem [Pea 84, Shi 79, Wein 89], although it seems that recent optical realisations of quantum jumps may provide a good case in favour of stochastic dynamics (see below). In the last years, systematic and more general investigation of stochastic modifications of the unitary dynamics have been taken up (e.g., [Gis 89,90, Pea 89,93,94]. In particular, it has been realised that equations of the type (1) can be equivalently written as stochastic evolution equations for vector states. A theory of primary state diffusion has been developed as a unified framework of various approaches and models discussed in the literature (for concise accounts, see [Gis 93a, Perc 94]). While these results surely provide valuable and practically useful insights, they still do not satisfy the desire for a principal justification of generalising the quantum dynamics. By contrast, those familiar with the quantum stochastic calculus will point out that state diffusion and stochastic reduction equations can be derived as subdynamics from the unitary evolution of the system of interest combined with some environment system [Bela 95a]. In this way there emerges a technical link between the present approaches based on autonomous stochastic processes and the environmental decoherence approaches to be reviewed in Sect. 6.3. In the present open situation it seems wise to let the two approaches coexist and interact with each other and suspend, as long as necessary, the decision whether the stochastic element is fundamental or derived. In fact this attitude has already led to fruitful exchanges of techniques between the pragmatic modelling of quantum jump equations and the state diffusion approach [Gar 94].

IV.5. Changing the Concept of Objectification There is an interpretation of quantum mechanics which radically takes the objectification problem (OP) as implying that no objectification takes place at all in a measurement process. This extreme form of (MO) is adopted by the many-worlds interpretation (Sect. 5.1). By contrast, the modal interpretations try to maintain a one-world view on the basis of certain modifications of the concepts of reality

112


and objectivity in the sense of essentially contextual, or relational notions of value attributions (Sect. 5.2). The many-worlds interpretation shares a common formal feature with some modal interpretations. They are concerned with unitary premeasurements M ~ of discrete sharp observables A, for which the final state U(~o ® ¢) of the compound system ,~ + ,4 assumes a biorthogonal decomposition with respect to the final component states "~i and ¢i of S and ,4 for any initial state ~ of S (cf. Sect. III.2.5): ® ,) =

®

=

®,,

(1)

ij According to Theorem III.4.4.1, this situation occurs exactly when the premeasurement M ~ produces strong correlations between the component states "yi and ¢i. We shall review several modal interpretations which consider in different ways the state U(~ ® ¢) in its biorthogonal decomposition (1) and ascribe different roles to the reduced states

Ts(~) := ns(P[U(~o ® ¢)1) = Z p ~ (Aa i ) P['Yi] , A

T~(~) := TC~(P[U(~ ® ¢)]) = ~ p ~ ( a ~ ) P[¢,].

(2)

(3)

In these interpretations the strong correlation premeasurement is taken to be the whole measuring process. The many-worlds interpretation has been the starting point for another line of research, known as the decoherence approach. In this approach one tries to explain in terms of quantum mechanics how and why the macroscopic world shows (perhaps only to conscious observers) a largely deterministic, classical behaviour. Thus, definite properties, such as pointer positions, should arise due to a process of decoherence induced by interactions of the system under consideration with its environment. It turned out that decoherence in this sense is crucial in understanding certain features of the many-worlds and modal interpretations. On the other hand, the decoherence approach has itself initiated the search for new interpretations in the spirit of (MO) (Sect. 5.3). The measurement models studied in the decoherence approach and the modal interpretations all seem to encounter, at some stage, the problem of explaining how the 'preferred pointer basis' (that is, {¢~}) is singled out and made dynamically robust in nature. It seems that the algebraic theory of superselection sectors can provide valuable insights into the necessary formal structures of the systems involved and the processes they undergo (Sect. 5.4). Still it turns out that the preferred pointer basis can at best be determined in an approximate sense for finite systems; that is, both the pointer value-definiteness and pointer mixture property for ,4 may be only approximately realisable. It may therefore turn out that unsharp objectit]cation is all one can ultimately hope to achieve (Sect. 5.5).


113

IV.5.1 M a n y - W o r l d s I n t e r p r e t a t i o n The first attempt to interpret the quantum mechanical formalism without additional assumptions concerning the objectification was made by Everett [Eve 57] and Wheeler [Whe 57]. Everett investigated unitary premeasurements of sharp discrete observables A and studied the decomposition (1) of the state • = U ( ~ ® ¢ ) of the compound system S + ,4 into a sum of products of two states 7i and ¢i, one referring to the object system S and one referring to the apparatus ,4, the 'observer'. The 'measurement' is then nothing else but the correlation between the respective state ~/i of the system and the 'relative state' ¢i of the observer who is aware of the system's state ~/i. The large variety of alternatives which coexist in the state • has later been interpreted by some authors as an ensemble of 'really' existing 'worlds' an idea which has given rise to the name many-worlds interpretation [DeW 70,71, Whe 57]. According to Everett's analysis the state (1) which provides strong statecorrelations between S and ,4, can be considered already as a description of the complete measuring process. Any definite measurement outcome is described by a product state ~i = 7i ® ¢i, where 7i is the state of the system and ¢i represents the observer (that is, the apparatus and some registration devices) as aware that the system S is in the state ~/i. Hence the state vector • - ~ v/pA(ai)~i describes the complete variety of possible final states of a measurement. Assume next that Ad~ is even a strong value-correlation measurement of A, so that the vectors 7i are eigenvectors of A: A~i = aiTi for any i = 1, 2 , . . . and all ~. Then the minimal interpretation and the probability reproducibility condition require that the coefficients pA(ai), which appear in the decomposition (1), are the probabilities for the outcomes ai in the/~na/states of S and ,4. This postulate was justified in the sense of the relative frequency interpretation of probabilities (Sect. III.3). However, in the present case such an interpretation cannot be given. The A (hi) cannot be interpreted here as relative frequencies of outcomes ai numbers p~, in the two Gemenge { (pA(ai), P[-yi]) • i = 1, 2 , . . . } and { (pA(ai), P[¢i]) • i = 1, 2 , . . . } which correspond to the decompositions (2) and (3) of the reduced states Ts(~o) and TA(~), simply because these mixed states play no role in the description of a measurement in the many-worlds interpretation. The whole system ,9 + jI evolves according to the unitary coupling T ® ¢ ~-, U(T ® ¢) and no state reduction or objectification will take place. The numbers p~A (ai) constitute also in the present situation a probability measure in the formal sense. But now the results of Sect. III.3.2 cannot be used for the justification of a relative frequency interpretation. Yet, even in this 'interpretation without objectification' the relative frequency interpretation of the probabilities p~,A (ai) can be justified in the following sense. Let ,S (n) = $1 + . . . + Sn be a compound system of n identical, equally prepared systems with states ~o and denote the compound state by ~(n). If on each system the observable A is measured, that

114


is, the observable A (n) is measured on S (n), the observer will register a sequence -- (/1,-..,In) of n index numbers lk indicating the values ark and store it in her memory. The relative frequency of index numbers i in a 'memory sequence' are denoted f~n)(l). If one defines a relative frequency operator F (n) as in Eq. (III.3.6), it is obvious that T(n) is not an eigenstate of F (n). However, one obtains the following 'large n' result due to Finkelstein [Fin 62] and Hartle [Sar 68]. THEOREM 5.1.1. Let {81,..., ,.qn} be a set of n equally prepared identical systems S} with states ~oand let ~o(n) be the state of the compound system S (~). If F (n) is the relative frequency operator for the value ai appearing in the eigenvalue (al, , . . . , al~)

of A (~), then A

nlirnoo(~0(n)[ (F (n) - p ~ ( a i ) )

2

~o(n)) = 0

for any i = 1, 2,... and for all ~o. It has been pointed out by Ochs [Ochs 77] that the strong and somewhat unrealistic premises of Hartle's proof (pure state ~o, equally prepared and independent systems) can be considerably relaxed and replaced by more realistic assumptions. Furthermore Ochs emphasised that Theorem 5.1.1 is essentially a 'law of large numbers' which shows that quantum probabilities fulfil this important requirement. The theorem means that for a given observer, that is, an apparatus plus a registration device, the relative frequency f~n)(g) of values ai will (for large n) be very close A to the probability p~(ai) in almost every memory sequence ~ - (/1,... ,ln) [DeW 71]. At first glance this result is somewhat surprising since none of the n systems 8i with preparation ~o possesses an A-value ak in an objective sense. However, this argument does not invalidate the relevance of the mentioned result, since within the present interpretation for the measuring process the objectification of the measured observable is not required in any sense. Theorem 5.1.1 has been referred to as 'the meta-theorem' of quantum mechanics, with the reading that quantum mechanics implies its own interpretation. This view has been criticised by many authors (cf., e.g., [Squires 90, Squ 90], and references therein). In fact Theorem 5.1.1 assigns (Hilbert space norm) 'measure zero' to non-random sequences of measurement outcomes with relative frequencies deviating from the quantum mechanical value. This does not entail by itself that the corresponding set of worlds would 'occur with probability zero' ([Des 85], p. 20). Such an interpretation would presuppose the probability interpretation to apply on the ensemble level. Therefore it is necessary to add an extra interpretational axiom to the qantum formalism, such as the one proposed in IDes 85]: 'the world consists of a continuously infinite-measured set of universes', which then ensures the required probability interpretation. On the basis of this short account of the many-worlds interpretation we mention some of its consequences relating to the measuring process and illustrating


115

its advantages and disadvantages. For details we refer to the literature [DeW 71]. First, for the description of the measuring process the consciousness of the observer is not needed. Automata for the registration of memory sequences are completely sufficient. Several automata can be used for the formation of a measurement chain without thereby changing the registered results in the respective memory sequences. The results of the automata persist throughout the whole chain. Moreover two automata are also allowed to communicate with each other since this exchange of information will never lead to a paradoxical situation like the paradox of Wigner's friend. In the many-worlds view, quantum theory in Hilbert space does not describe the reality which we usually have in mind but some reality which is composed of many distinct worlds. The observer is only in one of these worlds. For this reason she is unable to recognise the full determinism of the totality of many worlds which are described by ~, but only one part of it which is governed by a probability law, the justification of which is given by Theorem 5.1.1. It follows from these arguments that quantum mechanics can be applied to an individual system. Every automaton, that is, a measuring apparatus with a memory sequence, describes the system correctly according to the probability laws. For this reason quantum theory can be applied even to the universe, its formation and evolution. The state of the universe corresponds to a decision tree with an enormous number of branches. Every measuring process splits the world again into many alternative components, each representing mutually exclusive alternatives; but the whole description is completely consistent. This interpretation of the branching or splitting occurring in measurement-like processes applies only to situations where the biorthogonal decomposition (1) is unique. But this is the case only if all the initial probabilities p~A (ai) are different from each other. Otherwise, there seems to be no sufficient reason why the relevant 'pointer basis' should be singled out physically. This problem disappears if besides the object and registration system the common environment is taken into account, as is the case in the existential interpretation [Zur 93] formulated in the context of the decoherence approach (Sect. 5.4). Under these circumstances one is dealing with a triorthogonal expansion, which is always unique [Cli 94, Elby 94]. The many-worlds interpretation thus seems to have the status of a proper interpretation of quantum mechanics with its usual axioms. It does not presuppose the pointer objectification and the value objectification, thereby avoiding the problem (OP). Accordingly, it can make use of Theorem 5.1.1 as a legitimation of the probability interpretation, since this theorem need not be understood here in the sense of an ignorance interpretation. The price for this interpretation of quantum measurement theory is a very strange ontology: quantum mechanics does not describe any longer the one world in which we are living but at the same time the totality of all possible worlds.

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Modulo the qualifications mentioned above, the basic merit of the many-worlds interpretation consists in the observation that the quantum mechanical formalism provides an interpretation of the theory, including the relative frequency interpretation of the formal probabilities; this view is valid if the strong correlations between the object state and the memory state of the observer are considered as already representing the 'measurement'. However, the additional statement that the possible 'worlds' really exist seems to be rather irrelevant for the following two reasons: first, the 'many-worlds hypothesis' is not compatible with any physically acceptable conception of reality (cf., for example, [dEsp 89]); second, the existence of the alternative worlds cannot be falsified by any quantum mechanical experiment. Accordingly there have since been a variety of attempts at formulating alternative ontologies that would allow one to maintain the no-objectification spirit of the many-worlds interpretation and to explain the apparent occurrence of definite events to observers. Along with the development of the decoherence theories (Sect. 5.3) there have emerged pictures of one deterministic quantum world in which the appearance of different measurement outcomes is associated with the 'branching' of the observer's consciousness, where stable correlations between pointer states and brain states are established by processes of decoherence [Albert 1992, Alb 88, Lockw 89, Sau 93,95, Squ/res 1990, Stapp 1993]. It does not seem clear yet whether the resulting ontologies are less bizarre in each case than the many-worlds picture. The mathematical structures required for the physical description of observers needed in these 'one-world', or 'many-minds' interpretations of quantum mechanics have been investigated in a series of papers by Donald [Don 90,92,95].

IV.5.2 Modal Interpretations Rather than abandonning the goal of objectification altogether, one may try to formulate alternative consistent value-attributions and thus a conception of the reality of properties that allows for an understanding of definite measurement outcomes within quantum mechanics. This is the task pursued in the so-called modal interpretations of quantum mechanics, several variants of which have already been put forward [Bub 92,94; Di 89,94; Healey 1989; Koch 85, vFra 81,90, van Fraassen 1991]. They all aim at going beyond the minimal interpretation in order to provide a language for quantum mechanics in accord with the realistic intentions (A), (B), (C), (D), which is weak enough to evade the objectification problem (OP). In this section we discuss some of these interpretations in the context of measurement theory, and we show that they imply particular specifications of the measurement process [Cas 93,94,95]. Consider any physical system S, be it the object system, a measuring apparatus, or their composition. With any state T of the system we associate two sets of


117

(sharp) properties:

~:~1(T) • = { P E P(?-l) • tr [TP] - 1 } ;

(4)

P#o(T) "= {P E P(7-l) • tr[TP] ¢ 0}.

(5)

If P E P l ( T ) , then a yes-no measurement of P is determined to yield the yesresult, and we say that the property P is real in the state T. On the other hand, if P E P#0(T), then a yes-no measurement of P can lead to the yes-result, but it does so only with the probability t r [ T P ] . The objectification problem is now essentially due to the strong formulation of the objectification requirement in terms of the following two postulates: the requirement that a measurement leads to a definite result, understood as the assumption that the pointer observable has the corresponding value after the measurement; and the stipulation that an observable has a value exactly when the system is in an eigenstate of that observable. The very idea of the modal interpretations is to modify this reality criterion so as to give room for the possibility that a system in (a mixed or entangled) state T could 'have' other properties R E P#o(T), be it in addition to the properties P E :P~ (T) or alternative to them. Such properties would pertain to the system With the corresponding measurement outcome probability. Therefore, in the first place, a modal interpretation aims at defining a set of properties Pm.~.(T) that the system may possibly possess in the state T, without T necessarily representing a Gemenge of eigenstates of these properties. There are a variety of proposals for Pm.i.(T) currently under discussion in the literature, with no conclusive agreement in sight; on the contrary, some of the modalists maintain a more or less exploratory attitude as to what algebraic structures such a set Pm.i. (T) should have. It will depend on the decisions made in this respect whether the proposed set :Pm.i. (T) admits a consistent value attribution in the sense of a structure-preserving map (i.e., an appropriate homomorphism) onto the set {0, 1} (cf. Sect. II.2.2). First systematic investigations into these questions are reported in [Bacc 95, Cli 95, Bub 96, Bub 1996]. Instead of trying to cover each of the known proposals, we formulate some natural candidates for Pm.i. (T) as they arise from the decomposability properties of the state T and from its support projection. We consider first some examples which satisfy the constraint P l ( T ) c_ Pm.i. (T) C_ P#o(T),

(6)

so that the ensuing reality and objectivity criteria are relaxations of those formulated in Sect. II.2.4. The support projection of a state T, PT, is the smallest projection to which the state assigns probability one, tr[TPT] -- 1. Thus, for any P e P(7-/), P e Pl (T) if and only if PT _ E T ({ti}) }

(11)

(12)


119

where the abbreviations ran, and sd stand for the range and the spectral decomposition of T, respectively. The set Psd(T) is basic to those variants of the modal interpretations which build on the polar decomposition of the final entangled objectapparatus state [Di 89,94, Healey 1989, Koch 85]. For any state T, one has [Cas

95]: VI(T) c V d(T) C V o (T)c

c V o (T)c V¢0(T).

(13)

There are two types of intuitive ideas behind the above sets of possible possessed properties ~m.i. (T). First, the motivation for considering the sets Reran(T) and ~sd(T) is that in addition to the smallest property which the system has in the state T with certainty, the property PT, it could also have some stronger (smaller) properties. In the case of :PercH(T) one allows for the possibility that any atomic property P[~] contained in PT could also pertain to the system in that state. In the case of Psd(T) this possibility is restricted only to the spectral properties ET({ti}). The motivations for the sets Prsq(T) and Pran(T) are slightly different, referring to the decompositions of the state T. In the first case one argues that when the system is in a mixed state T, it could, in fact, be in one of the vector states occurring as a convex component of T, whereas in the second case this is restricted to the irreducible, or even orthogonal, vector state components of the state. In order to explain the role and implications of the modal interpretations in the framework of measurement theory, we assume now that the state in question is the state of the apparatus after the measurement. To be more specific, we consider a unitary measurement M u of an observable E with respect to a fixed reading scale 7~. For any initial state ~ of the object system, the final apparatus state is the reduced state TA(~) = T~A(P[U(~® ¢)]). The basic stipulation of the modal interpretations then is the following: if the probability is nonzero for an E-measurement to lead to a result in a set Xi, then the pointer observable P ~ could have the corresponding value i after the measurement, and it would have that value with the probability p~E (Xi). The assumption that the pointer observable could have such a value is formalised by the requirement that the corresponding property is in the set Pm.~. (TA(~)). This amounts to posing the following modal conditions on the measurement [Cas 95]: MC. For any unitary measurement .Mu of an observable E, any reading scale T~, all P[~] e S(?-ls), i e I,

tr[TA(~)PA (f-I (Xi))] - p~E (X,);

(MC1)

if p~S(xi) ¢ O, then PA(f-l(Xi)) e 7~m.i.(TA(~)).

(MC2)

The modal condition (MC1) is just the probability reproducibility condition and is thus always satisfied in any measurement. However, in general, the set Pm.i. (TA(~))

120


is a proper subset of the set :P#0(TA(~)). Therefore condition (Me2) is an additional constraint on the measurement process. In the class of the minimal unitary measurements of a discrete sharp observable the modal conditions have exhaustively been characterised for various choices of the set ~m.i.(TA(~P)) [Cas 94]. Let {¢ij} be the generating set of vectors of a measurement ~ 4 ~ of an observable A = ~ aiPi, as described in Theorem III.2.5.1. If ~ is the initial object state, then A Ts(cp) = Zp~,(ai)P["/i]

(14)

TA(cp) = Z V/pA(ai)pA(aj)(~i I"Yj ) lCj )(¢il

(15)

are the final states of 8 and A, respectively (Sect. III.2.5). In order to discuss the solutions of the condition (MC2) for the cases (7), (9), (11), and (12) in this model, we assume that the vectors ~i are such that lin{~l,...,'yi,.. "} - 7-/. This assumption simplifies the discussion without implying any loss of generality. The following points are then obtained [Cas 94]. First, the linear independence of the vectors ~/i is necessary for the condition (MC2) with Pm.i. (TA(~)) = P c r a n (TA(~p)). However, it is not sufficient to ensure (MC2) except when 7-/is finite dimensional. To this end some stronger requirements are to be posed on the set {-yi}. The weakest possible 'topological' strengthening of the linear independence of {~/i} is the gl-linear independence: for each sequence of complex numbers (ci) with ~ Icil < oo, the relation ~ ci~/, = 0 implies that ci = 0 for all i. Then one can prove: P[¢i] e Pcra,~(T~a(~P)) for any i and ~ with pA(ai) ~ 0 if and only if {-yi} is ll-linearly independent. Second, one can show that P[¢i] e 7~rsq(TA(~)) for each i and ~, such that p~A (hi) ~ O, if and only if Ts(~) - ~'~pA(ai)P[~/i] is an irreducible decomposition. This is to say that each ¢i, with pA(ai) ~ O, is a possible convex component of the final apparatus state TA(~) if and only if the final pointer component states Z(ai)(P[~]) /p~(~) A a -- P[~/~] form an irreducible decomposition of the final object state Ts (~). Third, the sequence {~/i} has the/initeness property if it is linearly independent and for each i, ~/i = Oi + ~-~Ml,j#iaj~/j for some Oi E li---n{'yl,'" ,'yi-l,~/i+l," "}±. One then obtains the following: P[¢i] e Pran(TA(~)) for each i and ~, such that p~A (hi) ~ O, if and only if {~/~} has the finiteness property. Finally, if {-y~} is an orthonormal sequence, then the decompositions (11) and (12) are the spectral decompositions, in which case the condition (MC2) is satisfied for 7~m.i.(TA(~)) -- 7~sd(TA(~)). Conversely, if for that choice of 7~m.i. (TA(~)), (MC2) is fumned for each i and ~ for which pA(ai) # O, then {~i} is orthonormal [Lah 90]. From Sect. III.4.4 we recall that the orthogonality of the vectors 7i is equivalent to Az[~ being a strong-state correlation measurement.


121

To summarise, the following characterisations of the modal conditions (MC2) on measurements are given: for any P[~o] e S(7-ls), and for each i = 1, 2,..-, such that p~A (hi) ~ O, P[¢i] e Pcran(T.4(qo)) if and only if {'yi} is/1-independent;

(16)

P[¢i] e 7)rsq(TA(qo)) if and only if {~i} is irreducible;

(17)

P[¢i]

e

~ran (TA(~O))

if and only if {-yi} has the finiteness property;

P[¢i] e :Psd(TA(qo)) if and only if {-yi} is orthonormal.

(18) (19)

There are two further version of the modal interpretations, which we briefly mention here. They arise from the polar and from an orthogonal decomposition of the entangled vector state • - U(qo®¢) of the compound object-apparatus system. Let • = ~ v / - ~ i ® ~i be the polar decomposition of • (which, for simplicity, we assume to be unique). One may then consider the set :=

®

(20)

where pd stands for the polar decomposition. We note that ~i)pd(~) C ~i)y£O(~), whereas P l ( ~ ) is comparable with Ppd(~) if and only if the two sets coincide, which is the case only when • is of the product form • = ~ ® ~/. Ppd(~) is the set of properties which the system could have in the state • according to the modal interpretations which are based on the polar decomposition. Due to the biorthogonality of that decomposition the reduced states obtain the spectral decompositions T~s(~) = ~-]~wiP[~i] and nA(~) = ~,wiP[~l~]. It then follows that {I ® R • R e P~d(T~(~)) } C_ Ppd(~), which shows to what extent the property attributions to a compound system and its subsystems according to Eqs. (20) and (12) are consistent. A similar consistency result is not obtained in the Copenhagen variant of the modal interpretation, where it typically occurs that P[¢i] e : P c ~ (TA(qo)), but I ® P[¢i] ~t : P c ~ (V(~o ® ¢)) = Pl (V(~o ® ¢)). In that interpretation this special feature is ascribed to the holistic nature of a quantum mechanical composite system [van Fraassen 1991]. To introduce the other modal interpretation based on a particular orthogonal decomposition of the entangled state V, let Po(~) := {P e 7)(H) : ( ~ I P V ) = 0}, and recall that the s e t ~i)obj(~ ) =- Pl(~I/) [.J ~O0(~I/) is the set of (sharp) properties which are objective in the state ~. If (P~) is a sequence of mutually orthogonal projection operators such that E Ri = I, we write • = E P ~ - E II Ri~ II ~i, with the agreement that ~i = O, whenever Ri~ = O. We define:

bj po(R,)

:=

'obj(v ) ~(~)

(21)

Clearly, for any P ~ P(?-ls ® 7-l.a), P ~ , obj (~) if and only if P[~i]

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