Less is more : on the Theory and Application of Weak and Unsharp Measurements in Quantum Mechanics

Less is More

On the Theory and Application

of Weak and Unsharp Measurements in Quantum Mechanics

Dissertation

zur Erlangung des akademischen Grades des

Doktors der Naturwissenschaften an der Universit ¨at Konstanz Fachbereich Physik

vorgelegt von

Thomas Konrad

Tag der m ¨undlichen Pr ¨ufung: 16.06.2003 Referent: Prof. Dr. J ¨urgen Audretsch

Referent: H.D. Dr. Lajos Di ´osi

To my parents and my daughter Merle

Contents

Table of Contents i

List of Figures v

Zusammenfassung vii

Introduction ix

1 Generalised Measurements and the Axioms of Q.M. 1

1.1 Introduction . . . 1

1.2 General Representation of Observables and States . . . 4

1.3 The First Axiom: the State Space . . . 10

1.4 Example: Observables and States of a Qubit . . . 11

1.5 Classical Physics . . . 16

1.6 Observables and States in Higher Dimensions . . . 18

1.7 Density Operators and Effects . . . 23

1.8 Generalised Measurements . . . 24

1.9 Dynamics in Quantum Mechanics . . . 25

1.10 Operations . . . 26

1.11 The Second Axiom: Evolution of Closed Systems . . . 29

1.12 Kraus Representation of the State Evolution . . . 31

1.13 The Third Axiom: Measurements . . . 32

1.14 The Forth Axiom: Composed Systems . . . 34

1.15 Conclusion . . . 35

ii CONTENTS

2 Unsharp Measurements 37

2.1 Introduction . . . 37

2.2 What are Unsharp Measurements? . . . 38

2.2.1 Example: Stern-Gerlach Measurement of Spin . . . 38

2.2.2 Projection Measurements . . . 45

2.2.3 Sharp Measurements . . . 47

2.2.4 Definition of Unsharp Measurements . . . 47

2.2.5 Relation to “Unsharp Observables” . . . 49

2.2.6 The Case of a “Loose Pointer” . . . 50

2.3 Unsharp Measurements of Ordinary Observables . . . 51

2.3.1 Stern-Gerlach Measurement of Spin . . . 51

2.3.2 Unsharp measurement of . . . 53

2.3.3 Series of Consecutive Unsharp Measurements of . . . 54

2.3.4 Unsharp measurement of position . . . 59

2.4 The Weak Influence of Unsharp Measurements . . . 61

2.5 Realisation of Minimal Unsharp Measurements . . . 64

2.5.1 Realisation of Generalised Measurements . . . 65

2.5.2 Unitary Premeasurement of a Minimal Unsharp Measurement and Pro- jection of the Pointer . . . 66

2.5.3 Non-Orthogonal Pointer States . . . 67

2.6 Weak Measurements . . . 68

2.6.1 Weak Measurements and the Weak Value . . . 68

2.6.2 The Standard Model of Measurement . . . 69

2.7 Information Gain and Disturbance . . . 71

2.7.1 Reduction of a ”Gemenge”by a Sequence of Minimal Unsharp Mea- surements . . . 72

2.7.2 Reduction of a Superposition by a Sequence of Minimal Unsharp Mea- surements . . . 73

2.7.3 Operations and Bayes Theorem Revisited . . . 74

2.7.4 Meaning of in the Context of Unsharp Measurements . . . 75

2.7.5 Summary and Discussion of Results for Minimal Unsharp Measurements 77 2.7.6 Generalised Measurements and the Relation between Infogain and Dis- turbance . . . 79

2.7.7 Quantum State Estimation and the Balance between Information Gain and Disturbance . . . 81

CONTENTS iii 3 Unsharp Measurements and the Detection of Dynamics 87

3.1 Introduction . . . 87

3.2 Single Unsharp Measurements . . . 91

3.2.1 A Particular Class of Single Unsharp Measurements . . . 91

3.2.2 Realization of Unsharp Measurements . . . 92

3.3 N-series of Successive Unsharp Measurements . . . 93

3.3.1 One N-series . . . 93

3.3.2 Best Guess for the Outcome of One N-series . . . 96

3.4 Measurement of a Dynamically Driven State by a Sequence of N-series . . . . 97

3.4.1 Best Guess as Measurement Readout . . . 97

3.4.2 Level Resolution Time and the Different Regimes of Measurement . . . 98

3.5 Results of the Numerical Analysis . . . 100

3.5.1 Quantum Jump Regime . . . 100

3.5.2 Rabi Regime . . . 102

3.5.3 Intermediate Regime . . . 102

3.6 Statistics of the Sequential Measurement . . . 105

3.7 Conclusion . . . 109

3.8 Appendix A: Why Minimal Unsharp Measurements? . . . 112

3.9 Appendix B: Upper Limit for . . . 113

3.10 Appendix C: Simulation Procedure and Noise Reduction . . . 114

4 Realisation of Sequential Unsharp Measurements 117 4.1 Quantum Optical Weak Measurements Can Visualise Photon Dynamics in Real Time . . . 117

4.1.1 Introduction . . . 117

4.1.2 Set Up of the Experiment . . . 119

4.1.3 The Feedback Mechanism . . . 123

4.1.4 N-series and Interpretation of the Measurement Readout . . . 124

4.1.5 Rabi Oscillations versus Influence of the Measurements . . . 124

4.1.6 Results of the Simulations . . . 125

4.1.7 Feasibility of the Suggested Experiment . . . 126

4.2 Simulation of Limited Detector Efficiency in the Selective Regime . . . 128

iv CONTENTS 5 Non-selective Discription of Unsharp Measurements with Unitary Backaction 131

5.1 Introduction . . . 131

5.2 The Elementary Measurement . . . 134

5.3 An Elementary Cell . . . 136

5.3.1 Commutative “Algebra” with Non-Commuting Operators . . . 138

5.3.2 Difference Equation of the Elementary Cell . . . 139

5.3.3 Decoherence and Dissipation in an Elementary Cell . . . 143

5.3.4 Distilling Out Decoherence . . . 150

5.3.5 Summary of the Dynamics of an Elementary Cell . . . 153

5.4 N-series . . . 154

5.4.1 From an Elementary Cell to an N-series . . . 155

5.4.2 Gaussian Operation Elements . . . 159

5.5 Continuum Limit and Master Equation . . . 165

5.6 Selective Evolution in the Continuum Limit . . . 168

5.7 Appendix A: Reordering of Kraus Operators . . . 169

5.8 Appendix B: Open Questiones . . . 171

Conclusion 173

List of Symbols 175

Bibliography 175

Index 183

Acknowledgement 187

List of Figures

1.1 State space of a qubit represented by the Bloch sphere . . . 12

1.2 The state space of a classical system . . . 17

2.1 Stern-Gerlach experiment . . . 39

2.2 Variance of N-series . . . 56

2.3 Shift of Bloch vector due to minimal unsharp measurement . . . 64

2.4 Angle dependence of change of Bloch vector due to minimal unsharp measure- ment . . . 65

2.5 Measurement of the weak value . . . 69

2.6 Information gain of unsharp measurements on a qubit . . . 79

2.7 Estimation and operation fidelity of a sequence of minimal unsharp measure- ments . . . 85

2.8 The pre-measurement state: Trade-off between estimation and operation fidelity 85 2.9 The post-measurement state: Trade-off between estimation and operation fidelity 86 3.1 Quantum jump regime of sequential measurement . . . 101

3.2 Rabi regime of sequential measurement . . . 103

3.3 Intermediate regime of sequential measurement . . . 104

3.4 Averaged state curve, readout and processed readout in the intermediate regime 105 3.5 Disturbance versus deviation of the readout of a sequential measurement for several values of fuzziness . . . 107

3.6 Disturbance versus deviation of the noise-reduced readout of a sequential mea- surement for several values of fuzziness . . . 108

3.7 Density of -curves in the intermediate regime of measurement . . . 109

3.8 Density of noise-reduced -curves in the intermediate regime . . . 110 3.9 Density of the noise-reduced measurement readout in the intermediate regime . 111

vi LIST OF FIGURES

4.1 The meter: relevant levels of Rydberg atom . . . 120

4.2 Experimental setup of the sequential measurement . . . 121

4.3 Results of the simulation: readout, state evolution and readout after noise re- duction . . . 125

4.4 Power spectra of state evolution and readout . . . 126

4.5 Sequential measurement with Possonian statistics . . . 128

5.1 Scheme of sequential measurement . . . 137

5.2 Contraction of the Bloch sphere due to measurement induced decoherence and translation in x-direction due to generalized friction . . . 147

5.3 Decoherence induced by the unitary part of the measurement back-action . . . 151

5.4 Shift of the Bloch sphere in z-direction by an elementary measurement . . . 151

5.5 Feedback scheme to transport the origin of the Bloch sphere into any state with- out selection . . . 152

Zusammenfassung

In dieser Arbeit wird eine spezielle Klasse verallgemeinerter Messungen in der Quantenmecha- nik untersucht. Es handelt sich um sogenannte unscharfe Messungen. Mathematisch sind sie durch vertauschende Effekte und reine Operationen (d.h. reine Zust¨ande werden durch den Messprozess auf reine Zust¨ande abgebildet) gekennzeichnet. Physikalisch bedeutet das, man kann sie als Messungen herk¨ommlicher Messgr¨oßen wie z.B. Energie, Ort oder Spin auffassen, und sie produzieren kein klassisches Rauschen. In der Praxis treten sie u.a. auf, wenn die Kopplung zwischen Messger¨at und zu vermessendem System sehr schwach ist. Daher sind sie verwandt mit den schwachen Messungen von Aharonov und Vaidman, die ¨uber eine solche schwache Kopplung definiert sind. Die Streuung der Messwerte einer unscharfen Messung ist gr¨oßer als die einer vergleichbaren Projektionsmessung, daher die Bezeichnung ”unscharf”.

Damit verbunden ist aber h¨aufig auch eine geringere St¨orung des zu vermessenden Systems.

Falls die St¨orung gering ist, wird die Messung auch ”schwach” genannt. Die Messung heißt minimal, wenn sie eine Zustands¨anderung ohne unit¨aren Anteil induziert.

F¨ur Zwei-Niveau-Systeme kann die Zustands¨anderung aufgrund einer unscharfen Messung geometrisch auf der Bloch-Kugel beschrieben werden. Bei einer minimalen, unscharfen Mes- sung mit Ausgang ” ”, dem der Effekt zugeordnet ist, bewegt sich der Anfangszustand auf einem Großkreis der Bloch-Kugel in Richtung des Eigenzustandes von mit dem gr¨oßten Eigenwert. Je unsch¨arfer die Messung ist, desto geringer ist die Entfernung, die der Zustands- vektor dabei zur¨ucklegt.

Das Verh¨altnis zwischen Informationsgewinn und St¨orung (Zustands¨anderung) durch die Messung nimmt f¨ur minimale, unscharfe Messungen eine einfache Form an: die Zustands-

¨anderung entspricht der ¨Anderung der Wahrscheinlichkeitverteilung der Messergebnisse. Da letztere wiederum dem Informationsgewinn durch die Messung entspricht¹, sind Informations- gewinn und St¨orung ¨aquivalent.

Unscharfe Messungen sind ein probates Mittel um Information ¨uber eine Messgr¨oße zu erhalten, falls die Zustands¨anderung begrenzt sein soll. Wenn es z.B. um die Messung der Dy- namik einer Observablen an einem einzelnen Quantensystem in Echtzeit geht, sind die herk¨omm- lichen Projektionsmessungen praktisch unbrauchbar. Sie beeinflussen die Dynamik des Sys- tems zu stark. Im Grenzwert einer kontinuierliche Projektionsmessung kann es zum Zenon- Effekt (Quantum Zeno Effect) kommen: der Zustand des Systems ver¨andert sich nicht mehr.

1Der Informationsgewinn kann durch die negative ¨Anderung der Shannon-Entropie gemessen werden.

viii Zusammenfassung Es wird gezeigt, dass nichtzerst¨orende Quantenmessungen (QND-Messungen) einer nichttri- vialen Dynamik nur f¨ur Systeme mit unendlich-dimensionalem Hilbertraum existieren. Ein kontinuierlicher Wechsel der Observablen (Mitdrehen der Basis), sodass das System sich zu jedem Zeitpunkt in einem Eigenzustand der gerade zu messenden Messgr¨oße befindet, liefert keine Information ¨uber die Dynamik.

Die Messung von Dynamik an einem einzelnen Quantensystem in Echtzeit ist ein Anwen- dungsgebiet f¨ur unscharfe Messungen. Ein Folge von unscharfen Messungen wurde entwor- fen, die die Rabioszillationen eines einzelnen Zwei-Niveau-Systems in Echtzeit gew¨ahrleisten soll. Als Maß f¨ur die Aufl¨osung einer kurzen Sequenz von Messungen (N-Serie) wurde die sogenannte Niveau-Aufl¨osungszeit aus der Streuung der Messresultate einer N-Serie gewon- nen. Es stellte sich heraus, dass diese proportional zur Dekoh¨arenzzeit ist, die ein Maß f¨ur die St¨orung des Zwei-Niveau-Systems darstellt. Monte-Carlo-Simulationen der sequentiellen Mes- sung zeigen, dass eine Visualisierung der Rabioszillationen m¨oglich ist. Bedingung daf¨ur ist, dass die Niveau-Aufl¨osungszeit ungef¨ahr gleich der Rabi-Periode ist.

Eine Realisierung einer Folge von unscharfen Messungen wird anhand eines quantenop- tischen Systems beschrieben. Der vorgeschlagene experimentelle Aufbau basiert auf einem Ex- periment von Brune Haroche et al.. Ein Photon oszilliert zwischen zwei gekoppelten Mikrowel- lenkavit¨aten. Durch eine der Kavit¨aten werden verstimmte Rydbergatome geschickt, die dort in Abh¨angigkeit von der Photonenzahl eine Phasenverschiebung erleiden. Diese Verschiebung wird mit Hilfe von Ramsey-Interferometrie gemessen. Jede Messung an einem einzelnen Ryd- bergatom l¨asst sich dabei als eine unscharfe Messung der Photonenzahl deuten. In den Betrach- tungen wurden die Lebensdauer der Kavit¨aten, die Streuung der Geschwindigkeiten der Ryd- bergatome und die Detektoreffizienz ber¨ucksichtigt. Monte-Carlo-Simulationen zeigen, dass ein Mitverfolgen der Oszillationen des Photons mit Hilfe der unscharfen Messungen m¨oglich sein sollte.

Wie entwickelt sich ein Qubit mit Hamiltonischer Dynamik unter dem Einfluss einer Folge von unscharfen Messungen mit unit¨arem Anteil in der “Back-action” des Messapparates? Dif- ferenzengleichungen wurden hergeleitet um die zeitliche Entwicklung des Zwei-Niveau-Systems darzustellen. Verschiedene Effekte treten auf: u.a. eine zus¨atzliche unit¨are Dynamik, verursacht durch den unit¨aren Anteil der “Back-action” und Dekoh¨arenz aufgrund des unit¨aren Anteils sowie aufgrund des Informationsgewinnes. Es kommt auch zu einer Art Reibung (Dissipation).

Im Grenzwert kontinuierlicher Messungen verschwinden die meisten dieser Effekte. F¨ur diesen Grenzwert wird eine Mastergleichung angegeben, die die Zustandsdynamik bei einer sequen- tiellen Messung n¨aherungsweise beschreibt.

Das erste Kapitel dieser Arbeit enth¨alt eine Diskussion der Auswirkungen der Stochastik auf die Struktur der Quantenmechanik. Ausgehend von der Zuf¨alligkeit des Ausganges von Quantenmessungen und der nicht-eindeutigen Zerlegbarkeit von gemischten Zust¨anden in reine Zust¨ande ergibt sich eine Motivation f¨ur drei der vier Axiome der Quantenmechanik.

Introduction

Less is more. This paradoxical aphorism which has been applied to architecture² can also be employed in quantum physics: quantum measurements show two competing properties. The more accurate they are the greater is their disturbing influence on the measured system. Less accuracy can thus be preferable if disturbance is expensive and must be kept low.

Projection measurements were the paradigm of quantum measurement theory ever since its first rigorous formulation by John von Neumann in 1932 [vN96]. They are the most accurate measurements in the sense that they resolve the values which the measured observable can assume. If for example the energy of an atom is measured in a projection measurement the result will be one of the quantised energies which are characteristic for the atom. A superposition of eigenstates of the observable is projected in the course of the measurement process onto the eigenstate which corresponds to the eigenvalue identical with the measurement outcome. How is it possible to carry out a measurement with less disturbance, i.e. a measurement inducing a smaller state change? One possibility are indirect projection measurements: An ancilla system interacts³with the system in question, followed by a projection measurement on the ancilla. The resulting state change will in general be smaller then in a direct projection measurement but at the same time the measurement result will contain less information about the system. Here the question arises how the result of the indirect projection measurement can be interpreted with respect to the system. Is there an interpretation in terms of an ordinary observable of the system such as its energy, angular momentum or position?

In fact the set of all indirect measurements corresponds to the class of so-called gener- alised measurements or POVM measurements⁴. One can show that the class of generalised measurements contains all possible measurements in Quantum Mechanics. In order to add new measurements it would be necessary to change the notion of the state of a physical sys- tem considerably, cf. section 1.7. Generalised measurements have become the new paradigm of quantum measurement theory [Dav76, Kra83, Lud83, BLM91, BGL95, Hol01]. Especially problems in quantum information theory are often stated and solved in the language of “Effects”

and “Operations” which characterise generalised measurements.

2“Less is more” became the guiding principle of the mid-twenty-century architects due to its promotion by Ludwig Mies van der Rohe who used it to characterise his philosophy of minimalism and simplicity in design.

3via a unitary interaction.

4these measurements are also called POVM measurements, because they generate a positive operator-valued measure (POVM) which determines the probability distribution of the measurement results.

x Introduction The interpretation of generalised measurements in terms of the usual observables is in gen- eral difficult. However, for a special class of generalised measurements — the unsharp mea- surements⁵— it is comparatively easy to associate them with ordinary observables. An unsharp measurement conveys information about the associated ordinary observable, only less informa- tion then a projection measurement does.

If appropriately chosen, unsharp measurements lead also to a smaller state change than com- parable projection measurements. The magnitude of the disturbance can be chosen continuously from no disturbance at all (corresponding to no information gain) to the disturbance caused by a projection measurement. If the influence of the measurement on the system is weak enough one also speaks of weak measurements⁶. Operationally one can think of weak and unsharp measure- ments in terms of indirect measurements again: For example in order to measure the position of a particle unsharply one can probe it by a second particle and detect the position of the second particle. The disturbance of the first particle can be reduced by increasing the distance at which the probing particle passes. At the same time the correlation between the detected position of the probe particle and the position of the first particle decreases with increasing distance.

Since the disturbance can be adopted by choosing an appropriate unsharp measurement, this kind of measurement is advantageous if information about a certain ordinary observable is wanted but the disturbance of the system must not exceed a certain level. This is for example the case if the Rabi oscillations of a single atom in a resonant driving field are to be monitored in real time. Then a sequence of projection measurements of the atomic energy would disturb too much. In the limit of a continuous measurement they would even lead to the Quantum Zeno effect, i.e. the Rabi oscillations would be stopped by the influence of the continuous measurement. A sequence of unsharp measurements of energy on the other hand is a promising candidate to track the Rabi oscillations.

As in the case of the interpretation of the results of unsharp measurements the relationship between information gain and disturbance is simple compared to other generalised measure- ments. Unsharp measurements thus grant access to one of the most puzzling problems of gen- eralised measurements: How can we understand the relation between information gain and disturbance? The insights which can be gained for unsharp measurements shed some light on the problem for the general case.

In practice unsharp measurements occur often in experimental realisations of measurements of observables such as energy, position and spin. In many set-ups projection measurements of these quantities are an idealisation. The actual measurements are unsharp, cp. for example the Stern Gerlach measurement of spin (section 2.2.1).

This work is structured as follows:

☞ Chapter 1 introduces the notions which are required for further investigations: Effects⁷, operations, states etc.. A general description of the statistics of measurements is designed

5unsharp measurements can be mathematically characterised by commuting Effects and pure operations, i.e.

pure states are transformed in the measurement process into pure states.

6Sometimes the notion weak measurement refers also to a weak coupling between system and meter.

7An “Effect” is an operator which serves to compute the probability of a corresponding measurement result. In

xi which is broad enough to comprise measurements in classical and quantum physics. Once the state space is determined the structure of generalised measurements emerge from the rules of probability theory. There are static rules (Kolmogorov’s axioms) which lead to positive operator-valued measures and a dynamic law (Bayes theorem) which determines the state change up to a unitary development.

☞ In chapter 2 unsharp measurements are studied. The questions of what are unsharp mea- surements and what quantity do they measure are addressed. The relation to Busch’s un- sharp observables and Aharonov’s and Vaidman’s weak measurement is described. The statistics of a sequence of unsharp measurements carried out on a single qubit is com- pared to the statistics of a repetition of measurements on an ensemble of qubits. The state change of unsharp measurements on a qubit is interpreted geometrically on the Bloch sphere. The realisation of unsharp measurements within the standard model of measure- ment is studied. Eventually the relation between information gain and disturbance is investigated.

☞ Chapter 3 contains the design of a scheme to track the Rabi oscillations of a two-level system by a sequence of unsharp measurements. In chapter 4 a quantum optical experi- ment is proposed to realize this measurement scheme.

☞ Chapter 5 eventually shows how a sequence of unsharp measurement with unitary back- action can influence the dynamics of qubits. Also the limit of a continuous measurement is discussed.

Each chapter can be read without knowing the content of any other chapter. Taking into account a certain redundancy I have repeated special definitions in several chapters in order to save the reader from paging hither and thither.

order to distinguish the notion for operators from the notion “ physical effect” meaning a physical phenomenon, the name of the operator will in the following be written with capital “E”.

xii Introduction

Chapter 1

Generalised Measurements and the Axioms of Q.M.

The purpose of this chapter is to introduce generalised measurements which emerge quite nat- urally from the rules of probability theory and from expressing the probability of measurement results as linear function of the state of the system. A second goal is to demonstrate that the structure of Quantum Mechanics as far as it is given by the first three axioms is intimately related to probability theory.

1.1 Introduction

Imagine a physical experiment. In an experiment there is always a system which is more or less directly measured. This means in most cases the system to be measured (from now on referred to as m-system) interacts with an auxiliary system which in turn interacts with another auxiliary system and so on, until eventually the last of a chain of auxiliary systems is observed. The result of the observation then can be put in statements as, for example, “the red light flashes”

or “the pointer indicates an electric current of strength 0.1 Ampere”. For the observation to be repeatable all influences on its result have to be under control such that the measurement can be started under the same conditions again. Of course the measurement will not be literally the same, because at least time has proceeded or it is carried out at the same time but in another place with different, but equivalent components. One has to assume that the result of the observation is invariant under a translation of the measurement in time or space (assumption i), in order to repeat measurements and observations. If this assumption holds and all other initial conditions of the measurement are the same (assumption ii), one expects the same result. This expectation (assumption iii) is due to our belief in causality, expressed in the statement about the sufficient reason “Nothing happens without sufficient reason” from Leibniz [Lei79]. Here one could say: “The result of the observation in a repeated measurement will not be different without a reason”. In other words, if there is no difference in the initial conditions there cannot occur a different outcome.

2 Generalised Measurements and the Axioms of Q.M.

For small objects, in the realm of Quantum Mechanics, it turns out that certain observations are not repeatable, i.e. measurements are carried out apparently under the same (or at least similar) initial conditions but lead to drastically different results.

For example a silver atom is sent through an inhomogeneous magnetic field and in one execution of the measurement the silver atom is observed to be accelerated in direction of the gradient of the magnetic field and in another execution the silver atom is accelerated in the opposite direction.¹

Now this implies that at least one of the assumptions (i)-(iii) is wrong. A reasonable hierar- chy of doubting the assumptions before knowing anything about Quantum Mechanics may be first (ii), than (iii) and last (i). The reason to reject first assumption (ii) is that a real experiment is a complex endeavour, there may be many influences. And there might be no direct way to prove that all relevant influences are the same from one execution of the measurement to the next, when these executions show different outcomes. In fact, in order to know which influ- ences are relevant for the outcome one has to rely on a theoretical model, which can always be doubted. The results of most measurements scatter and this is why it is necessary to repeat the measurement under at least similar conditions in order to obtain some statistics and estimate the error of the measurement.

To refute the principle of cause and effect (iii) is daring, but it would be even more disastrous for empirical sciences as physics to say that time and space are not homogenous (i), because then measurements cannot be repeated and it becomes difficult if not impossible to estimate the errors of the measurements.² In the extreme case if physics changed dramatically from each moment in time to the next, we never would see a pattern and data from experiments would become an indecipherable code.

Quantum Theory predicts different outcomes to occur for measurements with identical ini- tial conditions (assumptions (i) and (ii) are presupposed). If for example an atom is prepared in a state which consists of a superposition of states with different energies

and , it will show in measurements of energy as result in some executions

and in others . Such a superposi- tion is called a pure state because Quantum Mechanics claims that it is maximally determined.

It has no “loose” parameters, which vary from one execution of the measurement to the other.

It is thus not a statistical mixture of the states with Energy

and . According to Quantum Mechanics all degrees of freedom of the atom are determined and nevertheless the repetitions of the measurement show different outcomes. If assumptions (i) and (ii) are valid and Quantum Mechanics gives a complete description of the degrees of freedom of the atom, i.e. there are no

“hidden variables”, then assumption (iii) is wrong and there is an element of real randomness in the world.³

1The actual results of the observations are black spots on a developed photographic plate, where the atom presumably hit the plate. The experiment is in the literature referred to as Stern-Gerlach experiment.

2In fact, according to general relativity, time and space are not homogenous in the presence of masses or energy but they constitute a curved manifold. For experiments on earth the curvature is approximately constant on the whole planetary surface. Thus the outcome of experiments does not vary much over the surface of earth due to a change of the curvature.

3There is a loophole to escape from this consequence by assuming that all possible measurement results (

1.1 Introduction 3 There is currently no theory known by means of which one could, knowing the initial con- ditions, predict the outcomes of all measurements in principle. Therefore one relies on making statements about the probability of the outcomes. This is reflected in the statistical character of Quantum Theory. Instead of measurement results the theory predicts the probability of the results, depending on the initial physical state of the m-system and the kind of experiment that is conducted.

There have always been different interpretations of the probabilities in Quantum Mechan- ics. Essentially the interpretations can be divided in two groups: The probabilities can be of subjective or of objective nature. If they are interpreted as subjective, then they quantify our knowledge about the outcome of measurements. Supporters of the subjective interpretation of Quantum Mechanics can think that Quantum Mechanics is complete. This then means that there are limits to what we can know about the outcome of measurements. The subjective view also allows to say that Q.M. is not complete, then it is viewed as an effective, statistical theory and there is an underlying more fundamental theory, which disembogues in Q.M. in a statistical limit. If the probabilities are considered to be objective they are seen as properties of the phys- ical objects and not as properties of our knowledge of the physical objects. The probabilities then are not due to ignorance but to real indeterminacies in nature.

In both kinds of interpretations the probabilities obey the same mathematical rules. It is interesting to look at the consequences of these rules for the structure of Quantum Mechanics.

There are “static” and “dynamic” conditions for probabilities. The first are the axioms of Kolmogorov. They state the mathematical properties of probability distributions. From them we can derive the properties of mathematical objects called Positive Operator Valued Measures (POVM) which can be used to determine the probabilities for the measurement results given the initial state of the m-system. The dynamic condition for probabilities (Bayes Theorem) states how probabilities change if additional information is gained. Bayes theorem leads to statements about the way the state of a measured system changes due to the information gain by the measurement⁴, cp. section 2.7. It is important to note that these statistical rules only lead to a frame in which a Quantum Theory can be formulated (and any statistical theory for that matter). The single POVMs, states and state-transformations for particular experiments have to be determined from the physical circumstance, or more precisely, from particular models of the physical circumstances and empirical data. In other words there has to be “physical input”

to get out “physics”, otherwise one just stays in a statistical theory which also could apply for example to economics.

and ) actually occur simultaneously in one execution of the experiment. But since this has never been observed, one has to postulate the emergence of simultaneously existing worlds in each of which a different outcome of the experiment is realized (cp. Everett’s many worlds interpretation [Eve73, BLM91]), or the schizophrenia of the observer (cp. many minds interpretation [Loc96, Deu96]).

4We will see later that there are equivalent ways to describe the change of the probabilities of measurement results: it can be described as change of the state of the system (“Schr¨odinger picture”) or as change of the POVMs associated with the system (“Heisenberg picture”).

4 Generalised Measurements and the Axioms of Q.M.

1.2 General Representation of Observables and States

Let us stipulate some names for the objects we are concerned with. The time immediately before the measurement is called . The states of the system to be measured (m-system) are represented as ,

or . In this context the ’s are just symbols for the states. At this point let us pretend not to know anything about the mathematical representation of quantum mechanical states in calculations. The set of possible measurement results

is called

.

Because the outcomes of a quantum mechanical measurement seem in general to resist an exact prediction, we have to assign probabilities to the possible outcomes. It will also be necessary to assign probabilities to composed events such as “result or result occurs”.

All composed events can be represented by subsets of . For example the event “ or ” is represented by the subset . Elementary events as “ ” are represented by subsets with one element like

. Events can also be composed by “and”, “or” and “not”. If two events are represented by the subsets then “first or second event” translates into “ ”, “first and second event” reads “ ” and “not the first event (all events but the first)” is represented by the complement of : . From now on we identify the events and the subsets of

which represent the respective events. The events form an algebra which is closed under countably many unifications ( ), countably many intersections ( ) and the formation of the complement.

It also contains

itself, which stands for the composed event: “one of the outcomes of the measurement will come out”. This is a tautology and it will always occur, therefore is called the certain event. The technical name for the algebra of all events (composed and elementary) is “sigma algebra”^!#"

$

.

We want to assign to each event a probability. Regardless of the interpretation of probability there are basic properties of such probability assignments (probability distributions).

Definition 1 A function^% which assigns to each element of a sigma algebra ^{!#" $} a real num- ber, such that

%&"'

$)(+*

-,.!#"

$

(1.1)

%&"

$/

(1.2)

01324

1

65 1724

%&"'

1$ for countably many, pairwise disjoint

1

,.!#"

$

(1.3) is called a probability distribution or probability measure.

Properties (1.1-1.3) are referred to as Kolmogorov’s axioms.

Definition 2 The function^{%8"9 $} defined by

%&":

$<;=

%&"'

$

%&">

$ %&">

$@?6*

AB,.!#"

$

(1.4) is called conditional probability.

1.2 General Representation of Observables and States 5

%&": $

is the probability that event occurs, provided that also event occurs. We note, that in the Baysian view of probabilities, the defining properties of probability distributions (1.1- 1.3) and of conditional probabilities (1.4) are not mere axioms but they can be derived from consistency arguments. Bayesianism interprets probabilities as subjective degrees of conviction (degrees of confidence, degrees of credence). Although the degrees of conviction are subjective they are assumed to be reasonable, i.e. all probability assignments have to be coherent. For example, if someone assigns to the event the probability , i.e. he is sure that will occur, he cannot at the same time be sure that the opposite of will happen. When saying%&"'

$/

he is forced to assign the probability

*

to the complement of . Kolmogorov’s axioms are equivalent to a coherent assignment of degrees of conviction, cp. chapter 3 of [HP89]. Bayesianism has been employed lately for a subjective interpretation of Q.M. [CFS01].

Now we have the means to discuss the statistics of an experiment. Surely the probability to obtain a certain measurement outcome (say ) depends on the initial state (denoted by ) of the system to be measured:

%8":

$/

%&": $

Let us consider the case where the system is at the beginning of the measurement either in state

or in state with probabilities

and

, respectively. This is called a mixed state, let us denote it by . The probability to find as measurement outcome is then given by

%8":B

$/

%&":

$

%8":B $

(1.5) The right hand side of (1.5) follows from considering the conditional probabilities that hap- pens under the condition that initial state is

or the initial state is . If we are allowed to represent the mixed state by

for

,

*

and

, Eq. (1.5) shows that the probability to measure has to be a convex-linear function of the initial state. In other words: The conditional probabilities that enter the total probability to obtain a measure- ment result if the initial state is not precisely known, are correctly represented by writing the initial state as a sum of the possible initial states weighted by their probabilities and postulating that the probability of is a convex linear function of the initial state.

How does this representation effect our notion of a state? Apparently there are states which can be expressed as convex combinations of other states. Whether conversely all convex combi- nations of states again represent states of physical systems is not obvious. Because this cannot be excluded a priori, it is reasonable to suppose that the set of all states, a physical system can assume, is convex, i.e. with two states

and , it also contains the state

"

$

for ^{, *} . States which can be written as non-trivial convex combination ( ^{? *} ) are called mixed states. States which cannot be expressed as non-trivial convex combination are called pure states. The decomposition of a mixed state into a convex combination of pure states is in general not unique if more than two pure states exist. Two convex combinations which correspond to different preparation procedures of the state before the measurement but are rep- resented by the same mixed state cannot be distinguished by any measurement on the m-system alone. This follows from the convex linearity of the probability distribution of the outcomes of

6 Generalised Measurements and the Axioms of Q.M.

any measurement as function of the initial state of the m-system:

5 1 % 1 1 5

%

;

(1.6)

5 1 % 1

%&": 1 $ 5

%

%8":B

$/

%&":

$ (1.7)

where the^%

1

, ^% are statistical weights, i.e.^%

1 % , *

for all , and ^%

1 %

. Mixed states as can be understood as equivalence classes of all preparation procedures of the initial state which cannot be distinguished by any measurement of the m-system, cp. [Kra83], p.6. This theoretically derived structure is confirmed by experiments. For example, a photon can be prepared in a mixed state by sending it through a polariser either in x-direction or in y-direction with probability^{% -*} , respectively. Here x and y are spatial directions which are perpendicular. Such a photon leads to the same probability distribution of results of any measurement as one that has been sent either through a -polariser or a -polariser with prob- ability^{% -*} (state ), where and are any two perpendicular directions different from x and y. Even though these preparation procedures can be clearly distinguished by looking at the experimental set up, the resulting states of the photon are indistinguishable by means of measurements carried out on the photon alone. The state of the photon as a system which is separated from the polariser, does only describe the properties of the photon and not of the po- lariser. Therefore and which show the same properties with respect to all measurements on the photon but are connected to different settings of the polariser are identified. If a certain decomposition of a mixed state is distinguished because the preparation procedure is known, the mixed state is called a Gemenge⁵.

As all convex sets, the set of states of a system can be embedded into a vector space.

This means

1 1 1

can be interpreted as linear combination of vectors

1

with real coefficients

1

. We will thus in the following represent the states of systems as elements of a vector space over the field of real numbers . For this purpose we take the smallest vector space which contains the set of states. This vector space is given by the linear span of the states which consists of all finite linear combinations of states:

5 1 1 1 1

, 1

, (1.8)

We note that the set of states always contains a basis of the linear space in which we embed it. Therefore the convex linear mapping%&":

$

from into the real numbers can be uniquely expanded to a linear mapping from

into the reals (a linear form on

). For brevity let us here only investigate the case of finitely many linearly independent states, thus the dimension of is finite. All linear forms on are then of the form ^"

$

, where ^, and

3 is an inner product on

. This is because

is an isomorphism between V and the set of all linear forms on (dual of ) . Hence, given a measurement and a corresponding probability distribution^%8":B

$

there exists for each event an event vector^"'

$ ,

, such that

%&"'B

$/

9"'

$ ! " ,# (1.9)

5“Gemenge” is the german word for an inhomogeneous mixture.

1.2 General Representation of Observables and States 7 The rules for assigning such vectors to events are determined by Kolmogorov’s axioms (1.1)- (1.3). For each measurement with event algebra ^{!#" $} there must exist a map ^{; !#" $} , such that for all ^,

(

9":

$

!

( *

-,.!#"

$

(1.10)

A"

$

!

(1.11)

01724

1

!

5

1724

9"' 1 $

! for countably many, pairwise disjoint

1

,.!#"

$

(1.12) In a first step we employ these rules to further determine the set of states and the assigning of vectors to events, i.e. the mapping

;

!#"

$

. This will lead to a quite general struc- ture. In order to further specify and the mapping we will study in a second step the change of states due to measurements and postulate the existence of so-called “preparation measure- ments”. Preparation measurements represent a means to produce pure states . The structures we will find among others are that of density operators and positive operator valued measures (POVM). The first represent states in Quantum Mechanics, the latter generalised observables (i.e. measurable quantities). If we only invoke the subclass of generalised observables repre- sented by commuting positive operators we obtain structures reminiscent of classical physics.

It is convenient to look at the implications of rule (1.11) first. Without restriction of gen- erality the vector associated with the certain event

can be said to point in direction of some normalised vector ^,

"

$/

(1.13)

where is a positive, real number. By (1.11) the first component of the state vector ^., is now determined:

5

1

1 1

(1.14) Here the ^{" $}

form an orthonormal basis of . From Eq. (1.10) one obtains the requirement

(

(+*

" ,#." , (1.15)

where represents the set of all event vectors (for all measurements which can be carried out on a particular m-system). In order to obtain the previous equation, both – the event vector and the state – have been expressed w.r.t. the basis ^"

$ of . The components of the event vector in this basis read

. ^;=

1

1 1

is the inner product in the dimensional subspace spanned by

3 3

. Now in order to fulfil Eq. (1.15), constraints on the angle defined via the inner product

!

"

#%$'&

with

;=)(

*

(1.16)

could be postulated. For the sake of simplicity we start by allowing all angles ^,

*

,+.-

$

. This way we shall arrive at a correct description of the simplest quantum mechanical systems. In order to model more complex systems one has to restrict the angle .

8 Generalised Measurements and the Axioms of Q.M.

Proposition 1.2.1 The set of states of the observed system is bounded, i.e. there exists a real number ^* , s.t. for all ^, .

Proof: could only be unbounded in a direction perpendicular to all event vectors (w.r.t.

the inner product on ), otherwise Eq. (1.15) would not hold. Since all probability distribu- tions would then be independent from this direction, it would correspond to a non-measurable property of the m-system and would not enter the state (which represents only measurable prop-

erties).

The component of w.r.t. the basis vector is already determined by condition (1.11):

. Let the remaining components be bounded by

)

5

1

1

& (1.17)

where is a positive, real number. The set of states then reads

5

1

1

1

(1.18)

This particular set of states is called instead of in order to remind us, that we allow to point in any direction. As a consequence the angle in (1.16) is not restricted. In general the set of states can be any convex subset of with appropriate constants and .

Eq. (1.18) leads to a reformulation of requirement (1.15). Now, the inner product assumes its maximum if both vectors point in the same direction and its minimum, if they point in opposite directions:

2

2

(1.19) Because we allowed to point in any direction these values can be realized by preparing the corresponding physical states. This has the following consequences for the event vectors.

(

(1.20)

*

(1.21) These two inequalities are equivalent to the statement:

( (

(1.22) In particular this implies ^{( (+*} . The space of event vectors thus reads

5

1

1

1

( (

(1.23)

1.2 General Representation of Observables and States 9 We note again that was obtained under the assumption that the set of states is given by , i.e.

can point in any direction. In general the set of states will be a subset of , as a consequence there can be more event vectors than contained in , such that (1.15) still holds, thus

. (1.24)

Proposition 1.2.2 The set of probability distributions for measurements on the m-system al- lowed by and is not restricted by any special choice of the positive constants and in and .

In other words we are free to choose ^* as we like.

Proof: Given a probability distribution with%&":

$B

& ! , where

% , is

the vector assigned to the event ^{,!#" $} and

''% , represents the state of the m-system. Then this probability distribution can also be expressed by means of

%&"'B $

&

! where

(1.25) and

(1.26) Both vectors are represented w.r.t. the basis ^"

$ of .

The coordinates of the new state vector and the event vector

satisfy

and

*

and

*

(1.27)

(

(

( (

(1.28) Hence a probability distribution%&": $

, which is expressed by states ^, and event vectors

B, , can also be represented by states ^, and ^B, , where and are obtained from in (1.18) and in (1.23), respectively, by replacing the positive constants and by positive constants and

.

Let us sum up the investigation so far. Due to the rule for conditional probabilities, the probability distribution of a measurement of the m-system can be written as a linear function of the initial state of the m-system, if the space of states is represented by a convex subset of a linear space. Kolmogorov’s axioms lead to a state space which is contained in

-

"

$

,

< (1.29)

and the space of event vectors which contains the set

"

$ ,

(

( * (1.30)

10 Generalised Measurements and the Axioms of Q.M.

Here and are arbitrary positive, real numbers. The mapping which assigns vectors to the events must have the following properties:

;

!#"

$

(1.31)

9"

$/

(1.32)

0 1 1 5 1

9":

1 $ for countably many, pairwise disjoint

1 , !#"

$

(1.33)

1.3 The First Axiom: the State Space

The first axiom of Quantum Mechanics reads (see e.g. [NC00]):

Axiom 1: Associated with any isolated physical system, there is a complex Hilbert space , i.e. a complete inner product space over the complex numbers . The state of the system can be represented by a positive operator⁶ on with tr .

The set of all states of a quantum system thus reads ^{" $ ;=- ; (+*} tr . Let us briefly introduce the notation. ^"

$

denotes the set of linear operators on with finite trace norm tr

;=

tr ⁽ (trace-class operators). The superindex “

” stands for the positivity of the elements of^{" $} whereas the index reminds us that they posses unit trace.

One often refers to the elements of^{" $}

asdensity operators. In the following we will meet another class of operators: the bounded self-adjoint operators on denoted by ^"

$

. The strict classification unfolds its full importance only on infinite dimensional Hilbert spaces – in finite dimensions all linear operators are bounded and of trace class. The extra effort is small and it makes the transition to the general case easier to use the notions which are also correct for dim . This is common practise in Quantum Theory, compare for example the usage of the notion of Hilbert space for finite dimensional vectorspaces.

The first axiom says how states of isolated systems are represented in Quantum Mechanics.

A system is isolated if it is not interacting with any other system. We will use the terms “iso- lated” and “closed” here as synonyms. The axiom sounds quite peculiar to someone who starts to study Quantum Mechanics. Let us see how we can motivate it from the previous discussion of the general representation of states and observables.

We looked at the situation where a system is measured and although the initial conditions are fixed, the outcome of the measurement is not determined. More precisely, the initial conditions only determine a probability distribution for the possible outcomes of the measurement. We understood the state of the system to comprise those properties of the system which determine the probability distributions of all measurements. It does not comprise properties which are not related to the outcomes of any measurement, i.e. properties which are not measurable.

With this notion of a state we derived from the axioms of probability theory a superset for the space of states . While in general , there is a special class of quantum mechanical systems for which

, the so-called qubits.

6positivity of means ^"! .