master thesis 2009 weak values in quantum measurement ...shikano/master_thesis.pdf · fessor...

Master Thesis 2009Weak Values in Quantum Measurement Theory

— Concepts and Applications —

Yutaka Shikano07M01099

Department of Physics,Tokyo Institute of Technology

Supervisor: Professor Akio Hosoya

February 19, 2009

Version history:

• First version (2.9.2009.) submitted to Tokyo Institute of Technology.

• Second version (2.19.2009) Correcting typo.

Prologue

The virtue of physics is very simple and highly consistent. Also, physics is a powerfultool to understand the nature. In order to understand the nature, we need measure theobject. The fundamental concept of physics is included in measurement. I have studiedthe theory of physics since I faced the phrase;

”Measure the Nature.”

Isamu Sakama.

Isamu Sakama is my first teacher on physics and died at August 8th, 2008.

In Memory of my teacher,

Isamu Sakma

1935 – 2008

i

Contents

Prologue i

Abstract vii

Acknowledgment ix

1 Introduction 1

2 Review of Quantum Measurement Theory 5

2.1 Axiom of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Projective Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Indirect Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Concepts of Weak Values 13

3.1 Definition of Weak Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Experimental Realization to Measure Weak Values . . . . . . . . . . . . . 15

3.3 Review of Two-State Vector Formalism . . . . . . . . . . . . . . . . . . . . 17

3.4 Definition of Weak Operators . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Stochastic Process in Quantum Mechanics 23

4.1 What is Probability in Quantum Mechanics? . . . . . . . . . . . . . . . . . 23

4.2 Extended Probability Theory via Weak Value . . . . . . . . . . . . . . . . 27

4.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Quantum Operations for Weak Values 31

5.1 Quantum Operations for Density Operators — Review . . . . . . . . . . . 31

5.2 Quantum Operations for Weak Operators . . . . . . . . . . . . . . . . . . . 33

5.3 Weak Measurement with Decoherence . . . . . . . . . . . . . . . . . . . . . 37

iii

iv CONTENTS

6 Applications of Weak Values 41

6.1 Geometric Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Conclusions 45

A Uncertainty Relationships 47

A.1 Uncertainty Principle and Uncertainty Relationship . . . . . . . . . . . . . 47

A.2 Quantum Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.3 Detection Limit of Gravitational Wave . . . . . . . . . . . . . . . . . . . . 50

A.4 Heisenberg’s Uncertainty Principle Revisited . . . . . . . . . . . . . . . . . 51

A.5 Ozawa’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.6 Uncertainty Relationships for Joint Measurement . . . . . . . . . . . . . . 55

B Time-Energy Uncertainty Relationships 59

C Optimal Covariant Measurement 67

C.1 Review of Optimal Covariant Measurement . . . . . . . . . . . . . . . . . . 67

C.2 Optimal Measurement Model on a Whole Line . . . . . . . . . . . . . . . . 72

C.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

D Quantum Mechanics on a Half Line 77

D.1 Momentum Operator on a Half Line . . . . . . . . . . . . . . . . . . . . . 77

D.2 Optimal Measurement Model on a Half Line . . . . . . . . . . . . . . . . . 78

D.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

E Mellin Transform 83

E.1 Definition of Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . 83

E.2 Riemann-Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

E.3 Hyperbolic Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

F Deficiency Theorem 87

G Nitrogen Vacancy Center in Diamond 91

G.1 New Scheme to Measure Spin-Lattice Time . . . . . . . . . . . . . . . . . . 91

G.2 Experimental Realization in a Solid State System . . . . . . . . . . . . . . 92

G.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

List of Publications 97

Bibliography 103

CONTENTS v

Epilogue 111

Abstract

The weak value of an observable is experimentally accessible by weak measurements astheoretically analyzed by Aharonov and his collaborators and recently experimentallydemonstrated. Weak values are characterized by the pre- and post-selected states in thetarget system and are complex values and are corresponded to the expectation valuessumming up the post-selected state with a weight of a probability to obtain the post-selected state and are a consequence of the completeness relationship. Therefore, weshould not take the strange weak values too literally but the remarkable consistency ofthe framework of the weak values. The concept of weak values gives a new perspectivedifferent from the conventional quantum measurement theory.

In this thesis, we develop the framework of weak values and discuss the applicationsof weak values theory. In the first half of this thesis, we show that the framework ofweak values is consistent with the conventional quantum measurement theory and canconstruct the formal theory of the probability in quantum mechanics by taking the weakvalues as the random variable. By introducing the weak operator, we construct thequantum operations for weak operator analogous to the density operator. The weakoperator describes the history from the view of the two-state vector formalism. In thelatter half of this thesis, we analyze the geometric phase for mixed state via our proposedframework as the applications.

To consider the weak values may help understand the fundamental questions in quan-tum mechanics, what is a proper expression of an observable and a probability in quantummechanics.

vii

Acknowledgment

I would like to express my sincerest gratitude to Professor Akio Hosoya for his guidance,useful discussions, and continuous encouragements. I have really learned a lot from him.Especially, he exhaustively taught me how to write the paper everyday when we waswriting the first paper [126]. With him, I have really spent a happy-go-lucky time tostudy and to attend a lot of conferences, workshops, and seminars.

I am grateful to Professor Masanao Ozawa for useful discussions and his encourage-ments. He taught me the indirect quantum measurement theory and the relationshipbetween quantum logic and the reality in quantum mechanics. His useful comments al-ways help develop our works. He also checked the contents of the review part of theuncertainty relationships in this thesis (App. A).

I am also grateful to Professor Seth Lloyd for collaboration and his encouragementsduring his stay at Tokyo Institute of Technology as the visiting professor. I and AkioHosoya held the international mini-workshop ”Theoretical Foundations and Applicationsof Quantum Control” during his stay supported by Interactive Research Center of Science,Tokyo Institute of Technology.

I am also grateful to Dr. Christopher A. Fuchs for collaboration and hosting me atPerimeter Institute for Theoretical Physics.

I would like to also thank Takahiro Sagawa for continuous discussions. His substantialcomments always help develop and think of our works.

I would like to also thank Sota Kagami and Dr. Shu Tanaka for collaboration anduseful discussions. Since our project is based on the experiment about Nitrogen-Vacancycenter (NV center) in a diamond (App. G), the useful discussion with them helps clarifythe physical landscape of the weak values and weak measurement.

My thanks also go to Professor Asao Arai, Professor Koichiro Asahi, Professor Im-manuel Bloch, Professor Masahito Hayashi, Professor Takuya Hirano, Professor MasaoHirokawa, Professor Holger Hofmann, Professor Masahiro Hotta, Professor NobuyukiImoto, Professor Richard Jozsa, Professor Masao Kitano, Professor Masato Koashi, Pro-fessor Tatsuhiko Koike, Professor Norio Konno, Professor Mikio Kozuma, Professor PaulG. Kwiat, Professor Lev Levitin, Professor Ryutaro Matsumoto, Professor Michio Mat-sushita, Professor Shuichi Murakami, Professor Mikio Nakahara, Professor HidetoshiNishimori, Professor Izumi Ojima, Professor Kevin Resch, Professor Akira Shimizu, Pro-

ix

x ACKNOWLEDGMENT

fessor Tetsuya Shiromizu, Professor Masaru Siino, Professor Shogo Tanimura, Profes-sor Hal Tasaki, Professor Izumi Tsutsui, Professor Satoshi Tojo, Professor Takashi Ya-mamoto, Professor Yoshihisa Yamamoto, Dr. Daisuke Akamatsu, Dr. Eva Amsen, Dr.Andy O’Barron, Dr. Jen Dodd, Dr. Steve Flammia, Dr. Daniel Gottesman, Dr. Lu-cien Hardy, Dr. Yuji Hasegawa, Dr. Hosho Katsura, Dr. Gen Kimura, Dr. YoshishigeKobayashi, Dr. Koji Maruyama, Dr. Fumiaki Morikoshi, Dr. Michael Nielsen, Dr. Yuki-hiro Ohta, Dr. Yosuke Okudaira, Dr. Masahide Sasaki, Dr. Yuichiro Sekiguchi, Dr. RobSpekkens, Dr. Akito Suzuki, Dr. Jaw-Shen Tsai, Manabu Arikawa, Koji Azuma, SamuelDeleglise, Yujiro Eto, Keisuke Fujii, Shunsuke Fujii, Sinsuke Fujisawa, Ryo Harada,Takayuki Horiuchi, Koji Inokuchi, Koh Iwasaki, Toru Kawakubo, Shingo Kobayashi, YujiKurotani, Kenichiro Kusudo, Satoshi Maeda, Yoshiki Matsuda, Yasumichi Matsuzawa,Jamie McQuay, Miki Nakajima, Shuta Nakajima, Yasunobu Nakamura, Lumi Negishi,Kazuki Nishio, Yoshio Obi, Tomoyuki Obuchi, Seiju Ohashi, Hitoshi Ohmori, NanaOshima, Keiichi Oyamada, Hayato Saigo, Naoyuki Sakumichi, Etsuo Segawa, SusumuSerita, Takanori Sugiyama, Mayuko Suzuki, Shuhei Tamate, Motoyuki Yoshida, ShingoYoshimura, Tomohiro Yoshino, Yu Watanabe, and all other colleagues and friends foruseful and enjoyable discussions. I have been happy to share interests in physics andscience with them. By Professor Yoshihisa Yamamoto’s initiative, we have held ”KantoQuantum Student Chapter” and poster sessions per two months under the financial sup-port from National Institute of Informatics (NII). I have been happy to take a chance tomeet colleagues with the common interests on a quanta.

My thanks go to Professor Charles H. Bennett, Professor Ken Ito, Professor KiyoshiKurokawa, Professor Masanori Ohya, and Dr. Lee Smolin for their encouragements.Especially, Dr. Lee Smolin gave me his draft on the concept of time in quantum fieldtheory and his autograph with his stand on the time in physics ”Future is Open”.

I benefited very much from a one-month stay at Perimeter Institute for TheoreticalPhysics under the support of ”Training for Physicists with International Leadership” atTokyo Institute of Technology in the Initiative for Attractive Education in GraduateSchools at the Ministry of Education, Culture, Sports, Science and Technology (MEXT).I am very thankful for the kind hospitality there. I was also supported by Japan StudentServices Organization (JASSO) during the master course.

I would express my thanks to our secretaries and other groups’ secretaries for miscel-laneous business to support our researches.

Special thanks to the restaurants ”Peace” and ”Love & Tapas” at Futakotamagawa.I thought of the idea when I was eating the delicious foods and seeing the beautiful sceneof the riverside of Tama river there.

Finally, with deceased Isamu Sakama leading the way, I was fascinated physics andhave been able to come this far. I respect his stand on physics as mentioned in Prologueand am deeply grateful to him.

Chapter 1

Introduction

The axiom of non-relativistic quantum mechanics was strictly constructed by von Neu-mann [93] and further Ozawa [97]. Almost all physicists analyze physical phenomena,e.g., we can calculate the probability of the β decay from some atom, based on the axiomof quantum mechanics. Especially, recent development in quantum information scienceand technology, e.g., quantum key distribution between 148.7 km has been demonstratedusing optic fiber [58], is strongly based on considering the axiom of quantum mechanicsand is summarized in the textbook [94, 53].

However, do we completely understand quantum mechanics? We have not consideredthe fundamental questions on quantum mechanics for a long time, e.g., is the wave func-tion a element of physical reality? [39], what is a proper expression of an observable?,why the probability due to the Born rule is special in quantum mechanics?, and are therea formal theory of the probability in quantum mechanics?. The early state of quantuminformation science is strongly related to the fundamental questions in quantum mechan-ics [147]. For example, the quantum Turing machine initiate by Deutsch [34], which worktriggered the quantum computation, is based on his gedankenexperiment on many worldinterpretation motivated by the discussion with Bennett in 1982 at Texas [147] accordingto the talk [45]. Now, I believe that the time has come to reconsider the fundamentalquestions on quantum mechanics by mobilizing the various results in quantum mechanicsand quantum information (e.g. see [35, 31].).

It is known that the seemingly humble requirement of the complete positivity of thequantum operation E together with the trace preservation and the positive convexity forthe density operator ρ implies an explicit representation of the physical operation in theKraus form: E(ρ) =

∑i EiρE†

i , where Ei is called the Kraus operator and Mi := E†i Ei is

the positive operator valued measure (POVM), with the property of the decompositionof unity,

∑i E

†i Ei =

∑i Mi = 1. The Kraus representation of physical operations is a

powerful tool in quantum information theory [94].

However, the probability distribution is not the only thing that is experimentallyaccessible in quantum mechanics. In quantum mechanics, the phase is also an essential

1

2 CHAPTER 1. INTRODUCTION

ingredient and in particular the geometric phase is a notable example of an experimentallyaccessible quantity [125]. The general experimentally accessible quantity which containscomplete information of the probability and the phase seems to be the weak value advo-cated by Aharonov and his collaborators [2, 8]. The weak values are characterized by thepre- and post-selected states for the target system. The key concept of weak values isexperimentally accessible quantities by weak measurement, which is a weak coupling con-stant between the target and probe systems. Furthermore, we can construct the formaltheory of weak values compared with the standard probability theory. The weak valueshave different perspectives from the conventional description in quantum mechanics.

I believe that the weak values may be powerful tools in order to solve the fundamentalquestions in quantum mechanics. Therefore, the aim is to develop the formal frameworkof weak values and analyze these applications in this thesis.

This thesis is organized as follows.In Chap. 2, we review the conventional quantum measurement theory. We recapitu-

late the axiom of non-relativistic quantum mechanics initiated by von Neumann [93] andfurther developed by Kraus [84], Davies and Lewis [33], Ozawa [97], and other people(e. g., see [25, 22]), and the projective measurement a la standard quantum mechan-ics textbooks (e.g., see [123, 85].). Based on the postulate of the generalized quantummeasurement, which is called an indirect measurement, we review the concept and formaldescription of the indirect measurement included in the example of the optical system. InChap. 3, we define the weak value advocated by Aharonov and his collaborators and theweak operator and show that weak values are experimentally accessible quantities, whichis the most important property of the weak values, by the experimental example of thethree box paradox [114], and show the remarkable consistency with the standard descrip-tion of quantum measurement. Furthermore, reviewing the two-state vector formalismfrom the time-symmetric description of quantum measurement, which is the original mo-tivation, we show the properties of the weak operator. In Chap. 4, we construct theformal theory of weak values compared with the standard probability theory. We reviewthe inconsistency with the standard probability theory and the probability according tothe Born rule in quantum mechanics. By taking the weak values as the complex randomvariable, we can naturally extend the probability theory in quantum mechanics to remainthe consistency with the Born rule. In Chap. 5, we construct the quantum operationof the weak operator analogous to the density operator. First, we review the quantumoperation of the density operator. Measurement changes the quantum state as a positivemap from a density operator ρ(> 0) to another density operator E(ρ)(> 0) because of theprobabilistic interpretation. The map E is called a completely positive map (CP map) ifthe map remains positive when the map is trivially extended to any larger Hilbert space.That is, (E ⊗ I)(ρex) > 0 for any state ρex of an arbitrary extended system, when themap E is completely positive. Physically, this is a very reasonable requirement becausethere should always exist the outside of an experimental set up which is inactive duringthe experiment procedures [97]. Therefore, we can construct the Kraus representation of

3

the density operator in quantum measurement. Analogously, we can construct the Krausrepresentation of the weak operator. Finally, as an example, we analyze the target systemwith the noise during the weak measurement by the shifts of the probe observables. InChap. 6, by analyzing the geometric phase for mixed states as the applications of theweak values theory, we show the efficiencies of this framework. In Chap. 7, we summarizesome results in this thesis.

Chapter 2

Review of Quantum MeasurementTheory

2.1 Axiom of Quantum Mechanics

First, we recapitulate the axiom of non-relativistic quantum mechanics founded by vonNeumann [93] as follows.Postulate 1 (Representations of states and observables). Any quantum system S is asso-ciated with a separable Hilbert space HS, called the state space of S . Any quantum state ofS is the element |ψ〉 of the Hilbert space and is represented in one-to-one correspondenceby a positive operator ρ =

∑i ai|ψi〉〈ψi| with unit trace, called a density operator, where

ai is a coefficient and |ψi〉 is a set of states with 〈ψi|ψj〉 = δij. Any observable of S isrepresented in one-to-one correspondence by a self-adjoint operator A densely defined onHS.

In the case of the two-dimensional Hilbert space, the quantum state is called a qubitand has been realized by the polarization of the light, two proper energy level of someatoms, the quantum dots, the superconductor with the Josephson junction, the NitrogenVacancy center (See App. G), and more. When Tr ρ2 = Tr ρ = 1, it is called a purestate. Otherwise, it is called a mixed state. Furthermore, the observable 1 is defined as aself-adjoint operator since the conventional measurement outcome has a real value.Postulate 2 (Schrodinger equation). If S is isolated in a time interval (t, t′), there is aunitary operator U such that if S is in ρ at t then S is in ρ = UρU † at t′.

Conventionally, many physicists have well known the Schrodinger equation as

i~d

dt|ψ(t)〉 = H|ψ(t)〉, (2.1)

1The ”observable” is a technical term. We use this term as a self-adjoint operator a la von Neumannby the Kato-Rellich theorem while the use of this term might be controversial.

5

6 CHAPTER 2. REVIEW OF QUANTUM MEASUREMENT THEORY

for the state |ψ(t)〉 ∈ H and the Hamiltonian H. Since the density operator is expressedas ρ =

∑i ai|ψi〉〈ψi|, we rewrite the above equation as

i~d

dtρ(t) = [H, ρ(t)], (2.2)

where [A,B] = AB − BA is anti commutator and which is called the von Neumannequation. The formal expression of Eq. (2.2) is given by

ρ(t) = Uρ(0)U †, (2.3)

where U = exp(−iHt/~) is the evolution operator.Postulate 3 (Born formula). Any observable A takes the value in a Borel set ∆ in any ρwith the probability Tr[EA(∆)ρ], where EA(∆) is the spectral projection of A correspondingto ∆.Postulate 4 (Composition rule). The composite system S + S′ is the tensor productHS ⊗HS′ of their state spaces.

The above two postulates have the important roles on quantum measurement as dis-cussed in the next session.

While von Neumann discussed measuring processes, he failed to give the mathematicalpostulate of measurement. Davies and Lewis constructed the framework of generalizedmeasurement including the projective measurement by introducing the concept of theinstrument and the positive operator valued measure (POVM) [33]. Thereafter, Ozawaintroduced the postulate of measurement [97] to consider the measured system and probesystem.Postulate 5 (Representation of generalized measurement). When any observable A ofthe measured system is measured in any state ρs before measurement, we obtain that thestate after measurement is M(∆)ρs = Trp[U(ρs⊗ ρp)U

†] and A takes the value in a Borelset ∆ with the probability Trs[ρsM(∆)], where the time evolution operator is defined onthe composite system Hs ⊗Hp.

M(∆) is called a completely positive trace preserving (CPTP) map of measurementdefined in the following chapter. Ozawa introduced the completely positive (CP) instru-ment [97] and showed that the state change by quantum measuring processes can bedescribed in terms of the Kraus operators [84] and proposed a measuring apparatus, i.e.,a scheme of measurement consisting of a probing process described only by quantum me-chanics and a detection process described by the micro-macro coupling, which is to obtainclassical information from quantum information [97, 99]. We discuss the description ofthe measuring process in the following session.

These postulates are central concepts of foundations of quantum mechanics and quan-tum information science. In the follows of this chapter, we consider the finite-dimensionalHilbert space to avoid the technical difficulties.

2.2. PROJECTIVE MEASUREMENT 7

2.2 Projective Measurement

We shall discuss an exact measurement of an observable A and assume that the targetstate be represented by linear combinations,

|ψ〉 =∑

i

ci|ai〉, (2.4)

where |ai〉’s are orthonormal basis of the observable A. When we perform the measurementof the observable A, we obtain the measurement outcome ai, which is one of the eigenvaluesai of the observable A, with the probability2,

P (ai) = |〈ai|ψ〉|2, (2.5)

which is corresponded to the Born rule (Postulate 3) and such measurement is called theprojective measurement, von Neumann measurement or strong measurement. After themeasurement getting the measurement outcome ai, the target state changes as follows:

|ψ〉 −→ |ai〉, (2.6)

which is often called the ”wavefunction collapse” or ”state reduction”. This implies thatit is necessary to consider the non-unitary process in quantum measurement, which iscalled the von Neumann projection postulate3. Furthermore, we calculate an expectationvalue of the observable A as

〈A〉 =∑

i

aiP (ai) =∑

i

ai|〈ai|ψ〉|2 =∑

i

ai〈ai|ψ〉〈ψ|ai〉 =∑

i

ai〈ai|ρ|ai〉

= Tr Aρ, (2.7)

where ρ = |ψ〉〈ψ| and the observable A can be decomposed as A =∑

i ai|ai〉〈ai|.

2.3 Indirect Measurement

In the above session, we have shown that measurement is highly non-trivial in quantummechanics. In classical physics, we can ignore the effect of measurement for the measured

2 When we confirm whether the experimentally performed measurement (operation) results in themeasurement outcome ai with the probability P (ai) or not, we have to repeat its measurement manytimes under the same condition. This is because that its verification needs the ensemble of the sametarget state.

3This postulate triggered the measurement problem. This problem is when and where the statereduction occurs. Historically, von Neumann tried to inclusively explain the evolution between target andprobe systems, which are both quantum systems, but did not solve this since this procedure concatenates.the probe systems, which is called the von Neumann chain. Even if we arbitrary insert the cut off of thevon Neumann chain, which is called the von Neumann cut, then the measurement outcome does not beaffected [82].


Figure 2.1: Scheme of measuring processes. We switch on the interaction between themeasured and probe systems in the first step to obtain the measurement outcome of theprobe system in the second step. We infer the observable of the measured system at t = 0from the outcome of the probe system at t = tf in the third step.

object4. However, we have to consider the measurement effect in quantum mechanics.In order to perform measurement for the object, we have to prepare the probe and takethe action for the object via the the probe in all fields. Therefore, we assume the twoquantum systems, which is called a target system Hs and a probe system Hp, in a generalframework for quantum measurement, which is called an indirect measurement. Further-more, any indirect measurement can be decomposed into two distinct parts. The firstpart involves an unitary transformation that transfers information of our desired observ-able from the target system to the probe system. The second part involves non-unitarytransformation that achieves the realization of a particular measurement outcome by theprojection postulate. Note that we do not consider a detection process to obtain observa-tional data corresponding to a macroscopic experimental result5. All the above processesare summarized in the book [25], which is illustrated in Fig. 2.1. To discuss measuringprocesses of the probe, we need to specify the Hilbert space corresponding to the probe

4Of course, there exists the effect of experimental errors in classical physics but there exists thetheoretical effect of measurement in quantum mechanics.

5We often call the detection process a magnification process, which is how to observe a pointed valueof measuring devices, e.g., physical processes in a photomultiplier. This process is discussed by manypeople, e.g., see Ojima [95].

2.3. INDIRECT MEASUREMENT 9

system and an interaction Hamiltonian for the combined system of the measured systemand the probe system to calculate an evolution operator. After the combined system isevolved in the measuring time, we acquire the measurement outcome of the probe systemand obtain the state of the measured system after the measuring process taking the partialtrace over the probe system.

Let us discuss the indirect measurement in detail. We assume that the pre-measuredtarget state is expressed by ρs. Summarized the above discussion, any indirect mea-surement model are characterized by a quadruple (Hp, |φ〉p, U,Ap). Here, |φ〉p is a pre-measurement state of the probe Hp. The operator Ap represents the actually measuredobservable As of the probe after a unitary operator U for the combined system Hs ⊗Hp.The probe observable Ap can be decomposed as

Ap =∑

i

ai|ai〉p〈ai|, (2.8)

where |ai〉p’s are orthonormal complete basis in the probe Hilbert spaceHp. In the indirectmeasurement, we give the measuring probability obtaining the measurement outcome ai

P (ai) = Trs〈ai|U(ρ⊗ |φ〉〈φ|)U †|ai〉= Trs E(ai)ρE†(ai), (2.9)

where E(ai) is defined as

E(ai) = p〈ai|U |φ〉p, (2.10)

and is an operator solely defined in the target system and is called a measurementoperator. The measurement operators E(ai)i satisfy the completeness relationship,∑

i E†(ai)E(ai) = I. The corresponding post-measurement state ρ′s of the target system

is given by

ρ′s =E(ai)ρE†(ai)

Trs E(ai)ρE†(ai)=

E(ai)ρE†(ai)

Trs M(ai)ρ, (2.11)

where M(ai) ≡ E†(ai)E(ai) is called a POVM. Furthermore, we calculate the expectationvalue of the observable As as

〈A〉meas =∑

i

ai · Tr ρsM(ai). (2.12)

By the expectation values, we classify the measurement as the follows. In the case that〈A〉 = 〈A〉meas, we call such measurement an unbiased measurement and the other mea-surement are called a biased measurement.

As an example of the indirect measurement, we consider the measurement on thepolarization of light using the mode measurement. The polarization of light is two-level


quantum system, which is characterized by a horizontal polarization |H〉 and a verticalpolarization |V 〉. Let the states |+〉 and |−〉 denote as

|+〉 =|H〉+ |D〉√

2−√2, |−〉 =

|H〉 − |D〉√2−√2

(2.13)

where |D〉 = (|H〉+ |V 〉)/√2. We prepare the initial state |Ψ0〉;|Ψ0〉 = |ψ〉|0〉, (2.14)

where the first and second registers express the polarization and the mode of light, re-spectively, which are corresponded to the target and probe systems. After rotating thepolarization by π/8 and putting the |0〉 mode into the polarizing beam slitter, the statebecomes

|Ψ1〉 = |+〉〈+|ψ〉|1〉+ |−〉〈−|ψ〉|2〉. (2.15)

After putting the |2〉 mode into the beam splitter with the reflection ratio α, the statebecomes

|Ψ2〉 = |+〉〈+|ψ〉|1〉+ α|−〉〈−|ψ〉|2′〉+√

1− α2|−〉〈−|ψ〉|3〉. (2.16)

Note that the reflection ratio α is set by

α =|||H〉 − |D〉|||||H〉+ |D〉|| =

√2−√2

2 +√

2. (2.17)

After rotating the polarization of |2′〉 mode by π/2, the state becomes

|Ψ3〉 = |+〉〈+|ψ〉|1〉+ α|+〉〈−|ψ〉|2′〉+√

1− α2|−〉〈−|ψ〉|3〉. (2.18)

After putting the |1〉 and |2′〉 modes into the 50 : 50 beam splitter, the state becomes

|Ψ4〉 =√

km|+〉〈H|ψ〉|1′〉+√

km|+〉〈D|ψ〉|2′′〉+√

1− α2|−〉〈−|ψ〉|3〉, (2.19)

where km = (1−α2)/√

2. When we distinguish the |1′〉, |2′′〉 and |3〉 modes, we obtain themeasurement operators as

E1 =√

km|+〉〈H|, E2 =√

km|+〉〈D|, E3 =√

1− α2|−〉〈−|, (2.20)

to calculate the POVMs as

M1 = km|H〉〈H|, M2 = km|D〉〈D|, M3 =√

2km|−〉〈−|. (2.21)

The physical interpretation of the measurement is to distinguish the horizontal and D-directed polarizations, which are corresponded to the POVMs M1 and M2, respectively.The failure to distinguish the polarization is expressed by the POVM M3. This procedureis illustrated in Fig. 2.2.

2.3. INDIRECT MEASUREMENT 11

Figure 2.2: The example of POVM to distinguish the polarization of light using the modemeasurement.

Chapter 3

Concepts of Weak Values

Let us consider the expression of measurement outcomes. The motivations of this chapteris to construct another expression of a measurement outcome from the experimentallyaccessible quantities.

3.1 Definition of Weak Values

For an observable A, the weak value 〈A〉w is defined as

〈A〉w =〈f |U(tf , t)AU(t, ti)|i〉

〈f |U(tf , ti)|i〉 ∈ C, (3.1)

where |i〉 and 〈f | are pre-selected ket and post-selected bra state vectors, respectively.Here, U(t2, t1) is an evolution operator from the time t1 to t2. The weak value 〈A〉wactually depends on the pre- and post-selected states |i〉 and 〈f | but we omit them fornotational simplicity in the case that we fix them. Otherwise, we write them explicitly as

f〈A〉wi instead for 〈A〉w. The denominator is assumed to be non-vanishing. Note also thatthe weak value 〈A〉w is independent of the phases of the pre- and post-selected states sothat it is defined in the ray space.

The physical intuition may be enhanced by looking at the identity for the expectationvalue of an observable A,

〈i|U †(t, ti)AU(t, ti)|i〉=

∑

f

|〈f |U(tf , ti)|i〉|2 〈f |U(tf , t)AU(t, ti)|i〉〈f |U(tf , ti)|i〉

=∑

f

pf ·f 〈A〉wi , (3.2)

13

14 CHAPTER 3. CONCEPTS OF WEAK VALUES

where pf = |〈f |U(tf , ti)|i〉|2 is the probability to obtain the final state 〈f | given the initialstate |i〉 [5]. Comparing with the standard probability theory, one may interpret the weakvalue as a complex random variable with the probability measure pf

1. The statisticalaverage of the weak value coincides with the expectation value in quantum mechanics.Further, if an operator A is a projection operator A = |a〉〈a|, the above identity becomesan analog of the Bayesian formula,

|〈a|U(t, ti)|i〉|2 =∑

f

pf ·f 〈|a〉〈a|〉wi . (3.5)

The left hand side is the probability to obtain the state |a〉 given the initial state |i〉.From this, one may get some intuition by interpreting the weak value f〈|a〉〈a|〉wi as the(complex!) conditional probability for the process |i〉 → |a〉 → |f〉2. We believe that theconcept of a quantum trajectory can be formulated in the framework of weak values [144].This interpretation of the weak values gives many possible examples of strange phenomenalike a negative probability3, a negative kinetic energy [6], a spin 100~ for an electron [2]and a superluminal propagation of light [119]. Of course, we should not take the strangeweak values too literally but the remarkable consistency of the framework of the weakvalues due to Eq. (3.5) and a consequence of the completeness relation,

∑a

〈|a〉〈a|〉w = 1, (3.6)

may give a useful concept to further push theoretical consideration by intuition. Theframework of weak values has been theoretically applied to quantum stochastic pro-cess [140], the tunneling traverse time [130], non-locality and consistent history [139],semi classical weak values [131], counterfactual reasonings [90, 91], and quantum commu-nications [23].

1In the standard probability theory [72], the expectation value of an observable A is given as aprobabilistic average,

〈A〉 =∫

dphA(p), (3.3)

with hA(p) and dp being the random variable associated with A and probability measure, which isindependent of A. The standard expression of the expectation value,

〈A〉 =∑

n

an|〈an|i〉|2 (3.4)

, is given by the Born rule. However, this does not fit the standard probability theory because theprobability measure depends on A. We will discuss this later (Chap. 4).

2The interpretation of the weak value as a complex probability is suggested in the literature [91].3The concept of complex probability in quantum mechanics is not new, e. g., see [41, 60].

3.2. EXPERIMENTAL REALIZATION TO MEASURE WEAK VALUES 15

3.2 Experimental Realization to Measure Weak Val-

ues

The most important fact is that the weak value is experimentally accessible by the weakmeasurement (e. g., see [117, 114, 115, 106, 65, 110, 146])4 so that the intuitive argumentbased on the weak values can be either verified or falsified by experiments.

3.2.1 Weak Measurement and Weak Values

First, we recapitulate the idea of the weak measurement [2, 76]. Consider a target systemand a probe defined in the Hilbert space Hs ⊗Hp. The interaction of the target systemand the probe is assumed to be weak and instantaneous,

Hint(t) = gδ(t− t0)(A⊗ P ), (3.7)

where an observable A is defined in Hs, while P is the momentum operator of the probe.The time evolution operator becomes

e−ig(A⊗P ). (3.8)

Suppose the probe state is initially ξ(q) in the coordinate representation with the probeposition q. For the transition from the pre-selected state |i〉 to the post-selected state |f〉,the probe wave function becomes

〈f |V e−ig(A⊗P )U |i〉ξ(q), (3.9)

which is in the weak coupling case5,

〈f |V [1− ig(A⊗ P )]U |i〉ξ(q)= 〈f |V U |i〉ξ(q)− g〈f |V AU |i〉ξ′(q)

≈ 〈f |V U |i〉ξ(

q − g〈f |V AU |i〉〈f |V U |i〉

). (3.10)

In the previous notation, the argument of the wave function is shifted by

g〈f |V AU |i〉〈f |V U |i〉 = g〈A〉w (3.11)

so that the shift of the expectation value is the real part of the weak value, g·Re[〈A〉w]. Theshift of the momentum distribution can be similarly calculated to give 2g·V ar(p)·Im[〈A〉w],

4Weak measurement has only been realized by the optical system. We consider the application to asolid state system (See App. G).

5 ξ(q − g 〈f |V AU |i〉

〈f |V U |i〉)

stands for ξ(q)|q→q−g

〈f|V AU|i〉〈f|V U|i〉

.


where V ar(p) is the variance of the probe momentum before the interaction. Puttingtogether, we can measure the weak value 〈A〉w by observing the shift of the expectationvalue of the probe both in the coordinate and momentum representations. The shift ofthe probe position contains the future information up to the post-selected state.

3.2.2 Three Box Paradox Experiment

As an example of the experimental realization to measure the weak values, we considerthe three box paradox. First of all, the story of the three box paradox is as follows.

1. A macroscopic number N of particles were all prepared at t1 in a superposition ofbeing in three separated boxes,

|ψ1〉 =1√3(|A〉+ |B〉+ |C〉), (3.12)

where |A〉 is the state of a particle in box A.

2. At a later time t2, all the particles were found in another superposition (this is anextremely rare event):

|ψ1〉 =1√3(|A〉+ |B〉+ |C〉). (3.13)

3. In between, at time t, weak measurement of a number of particles in each box, whichare, essentially, usual measurements of pressure in each box, have been performed.The readings of the measuring devices for the pressure in boxes A,B, and C were

pA = p,

pB = p,

pC = −p, (3.14)

where p is the pressure which is expected to be in a box with N particles.

In the above setup, we calculate the weak value as follows,

〈PA〉w =1

〈PB〉w =1

〈PC〉w =− 1, (3.15)

where PA is a projection operator on the state of the particle in box A. Taking the weakvalue into the probability as the next section, Eq. (3.15) means that there exists a particlefor boxes A and B but there exists a negative particle for the box C. We concentrate on

3.3. REVIEW OF TWO-STATE VECTOR FORMALISM 17

Pre-selected state Post-Selected state 〈PC〉w

1√3(|A〉+ |B〉+ |C〉)

|A〉 0|B〉 0|C〉 1

1√3(|A〉+ |B〉+ |C〉) -1

Table 3.1: Weak values for the box C.

the weak measurement for the box C. Then, changing the post-selection state at time t2,we obtain the weak value of the observable PC as Table 3.1.

This experiment was performed by Resch, Lundeen and Steinberg [114] as follows.The box is corresponded to the mode of the light to prepare the three mode. Weakmeasurement is performed by the vertical shift of the light tilting the slide glass in themeasured mode, that is, the vertical and horizontal directions are corresponded to theprobe and target systems, respectively (See Fig. 3.1). Its shift can be detected by theCCD camera. How to change the post-selection state is based on the Mach-Zehnderinterferometer. By changing the optical length, we can choose the post-selected statefrom the interference pattern. Their experimental setup is illustrated in Fig. 3.2. Theydemonstrated the corresponded amount to Table 3.1 (See Fig. 3.3).

gtFlatθ

Mode C

Figure 3.1: Weak measurement for the mode C. By tilting the slide glass, the light axisis vertically shifted. The vertical direction of the light axis is taken as the probe. Theeffective Hamiltonian is H = gPCpp, where pp is the momentum operator for the verticaldirection. Since the coupling constant g depends on the tilting angle and the thick of theslide glass, we can choose the coupling constant g to apply to the weak measurement.

3.3 Review of Two-State Vector Formalism

Historically speaking, the two-state vector formalism was originally motivated by thetime symmetric description of quantum measurement [3] as a new rule for calculatingprobability, which is called the ABL rule. In the case that a final state |f〉 is specified for


the measured system, in addition to the usual choice of an initial state |i〉, the probabilityfor the measurement outcome am for an observable A is

P (am; |i〉, |f〉) =|〈f |U(tf , t)|am〉〈am|U(t, ti)|i〉|2∑

j |〈f |U(tf , t)|aj〉〈aj|U(t, ti)|i〉|2 , (3.16)

where we assume that an instantaneous measurement occurs at a time t intermediate ofthe boundary conditions. If the only the initial condition is specified, Eq. (3.16) shouldreduce to the Born rule,

P (am; |i〉) = |〈am|U(t, ti)|i〉|2. (3.17)

A conceptual leap has been made in the recognition that boundary conditions may bechosen time-symmetrically in contrast to the conventional asymmetric choice of an initialcondition only.

The two-state vector formalism has also been related to the weak values and weakmeasurement [11] also developed by Aharonov et al. [12, 9] as follows. When only aninitial condition is specified for the quantum system, or when the final conditon is identicalto the initial condition, one gets after measurement interaction the expectation value,

〈A〉 = 〈i|U(tf , t)AU(t, ti)|i〉 =∑m

amP (am; |i〉). (3.18)

In the general case, when both initial and final boundary conditions are specified theoutcome is the weak value 〈A〉w, which may be far from any eigenstate of the observableA,

|〈A〉w| =∣∣∣∣〈f |U(tf , t)AU(t, ti)|i〉

〈f |U(tf , ti)|i〉

∣∣∣∣ =∑m

am

√P (am; |i〉, |f〉). (3.19)

3.4 Definition of Weak Operators

We define a weak operator W (t)6 as

W (t) := U(t, ti)|i〉〈f |U(tf , t). (3.20)

To facilitate the formal development of the weak value, we introduce the ket state |ψ(t)〉and the bra state 〈φ(t)| as

|ψ(t)〉 = U(t, ti)|i〉〈φ(t)| = 〈f |U(tf , t), (3.21)

6The weak operator was originally defined by Reznik and Aharonov, which they call this the ’twostate’ [116].

3.5. DISCUSSIONS 19

so that the expression for the weak operator simplifies to

W (t) = |ψ(t)〉〈φ(t)|. (3.22)

By construction, the two states |ψ(t)〉 and 〈φ(t)| satisfy the Schrodinger equations withthe same Hamiltonian with the initial and final conditions |ψ(ti)〉 = |i〉 and 〈φ(tf )| = 〈f |.In a sense, |ψ(t)〉 evolves forward in time while 〈φ(t)| evolves backward in time. Thetime reverse of the weak operator (3.22) is W † = |φ(t)〉〈ψ(t)|. Thus, we can say the weakoperator is based on the two-state vector formalism [10]. The weak operator gives theweak value of the observable A as

〈A〉w =Tr(WA)

Tr W, (3.23)

in parallel with the expectation value of the observable A by Tr(ρA)/ Tr ρ from Born’sprobabilistic interpretation. Furthermore, the weak operator (3.20) can be regarded as aspecial case of a standard purification of the density operator [135]. In our opinion, theweak operator should be considered on the same footing of the density operator. For aclosed system, both satisfy the Schrodinger equation. In a sense, the weak operator W isthe square root of the density operator since

W (t)W †(t) = |ψ(t)〉〈ψ(t)|, (3.24)

which describes a state evolving forward in time for a given initial state |ψ(ti)〉〈ψ(ti)| =|i〉〈i|, while

W †(t)W (t) = |φ(t)〉〈φ(t)|, (3.25)

which describes a state evolving backward in time for a given final state |φ(tf )〉〈φ(tf )| =|f〉〈f |. The weak operator describes the entire history of the state evolution from ti to tfvia t.

3.5 Discussions

First, extending our proposed definition of the weak operators, we may consider a super-position of weak operators,

W :=∑

i,f

αifU(t, ti)|i〉〈f |U(tf , t), (3.26)

in analogy to the mixed state which is a convex linear combination of pure states. Al-though this indicates a time-like correlations, the physical implication is not yet clear.This operator may be related to the concept of the multi-time states [116, 7]. In fact, it isshown how the weak value corresponding to the weak operator (3.26) can be constructed


via a protocol by introducing auxiliary states which are space-likely entangled with thetarget states.

Second, the weak operator (3.20) cannot be directly applied to the sequential weakmeasurement [91] since the time evolution among the measurements cannot be dealt. Wehave not clarified the way to mend the framework included in the sequential measurement.In the sequential weak measurement, we obtain the sequential weak value of the observableA and B as

〈B, A〉w :=〈f |U(tf , t2)BU(t2, t1)AU(t1, ti)|i〉

〈f |U(tf , ti)|i〉 . (3.27)

Finally, we apply to the weak measurement for the bounded symmetric operator A,which is a non-observable7. We can define the weak value of the operator A. This meansthat we can extend the postulate of the observable (Postulate 1). There are not clarifiedits classification.

7This was first discussed by Reznik and Aharonov [116].

3.5. DISCUSSIONS 21

Diode Laser

CCD Camera

MS, φΑ

MS, φC

Spatial Filter: 25um PH, a 5cm and a 1” lens

BS1, PBS

BS2, PBS BS3, 50/50

BS4, 50/50

Screen

GP C

GP B

GP A λ/2

λ/2

λ/2

PD

Figure 3.2: The 3-mode Mach Zehnder-style interferometer. The initial mode of a diodelaser is filtered spatially using a pinhole (PH), two lenses and an iris. Light is placed inthe proper coherence superposition of each of the modes labeled A,B, and C using half-wave plates (λ/2) and polarizing beam-splitters (PBSs). Glass plates (GPs) in each armare used to displace each beam transverse to its direction of propagation, and microscopeslides (MS) are used to finely adjust the phases in arms A and C. The three modes arerecombined at two 50/50 beam splitters (BS3 and BS4). The beam shines on a screenand a CMOS camera behind that screen captures its image. A photo diode (PD) in thesecond port of BS4 is used to set the relative phase light in the different modes [114].


0.4

0.6

0.8

1

1.2

1.4

100120140160180200220

Inte

nsity

(ar

bitr

ary

units

)

Pixel Number

Figure 3.3: An experimental data when only the mode C is displaced. We show theexperimental data for individual horizontal beam profiles from mode A (thin solid line),mode B (thin dashed line), mode C (thick dashed line) when a transverse, horizontaldisplacement of beam C is applied. The beam profile of the post-selected beam is also

shown (thick solid line). In fact, the pre-selected state is |i〉 =√

25|A〉+

√25|B〉+

√25|C〉

and the post-selected state is approximately |f〉 = 12|A〉+ 1

2|B〉 − 1√

2|C〉. In this case, the

weak value of the mode C is 1√10

[114].

Chapter 4

Stochastic Process in QuantumMechanics

4.1 What is Probability in Quantum Mechanics?

Definition 4.1 (Expectation Value in Conventional Probabilistic Theory). An expecta-tion value Ex[X] of a random variable X(ω) defined on a probabilistic space Ω,F , P is

Ex[X] =

∫X(ω)dP, (4.1)

where dP is a probability measure.In the conventional probability theory, we give the probabilistic space Ω,F , P. This

does not depend on the measured subject, especially speaking, the probabilistic measureis independent of the observable. Of course, the random variable X(ω) is a function fromΩ to R. On the other hand, in quantum mechanics, according to the Born rule, theexpectation value of an observable A is given by

Ex[A,ψ] = 〈ψ|A|ψ〉, (4.2)

where |ψ〉 ∈ H. In order to remain the consistency to Definition 4.1, we give the followingdefinition;Definition 4.2 (Expectation Value in Quantum Mechanics (Mathematics)). Let F bea subset of the self-adjoint operators (observables) on a Hilbert space H and ψ ∈ H bea state with ||ψ|| = 1. If there exist the probabilistic space Ω,F , P and the randomvariable hA(ω) on Ω,F , P for any observable A ∈ F such that

〈ψ|A|ψ〉 =

∫hA(ω)dP, (4.3)

then the random variable hA(ω) is called a functional representation of the observableA ∈ F .

23

24 CHAPTER 4. STOCHASTIC PROCESS IN QUANTUM MECHANICS

We show a simple example as the above definition. Let ψ(t, x) = 〈x|ψ(t)〉 ∈ H :=L2(R) be a state as

ψ(x, t) = 〈x|ψ(t)〉 = eR(t,x)+iS(t,x), (4.4)

where R(t, x) and S(t, x) are real functions and p be a momentum operator, which is aSchrodinger operator, as

p = −i~∇, (4.5)

and F = x, p, p2 be a set of the observables. In the following propositions, we obtainthe functional representations of p and p2.Propositon 4.1. The functional representation of the observable p is

hp(t, x) = ~∇(R(t, x) + S(t, x)). (4.6)

Proof. The expectation value of the momentum p is

〈ψ(t)|p|ψ(t)〉 = −i~∫

ψ∗(t, x)∇ψ(t, x)dx

= ~∫

ψ∗(x, t) (−i∇R(t, x) +∇S(t, x)) ψ(t, x)dx. (4.7)

Since the momentum operator p is self-adjoint, 〈ψ(t)|p|ψ(t)〉 must be real. Then, weobtain ∫

∇R(t, x)ψ∗(t, x)ψ(t, x)dx =

∫∇R(t, x)µ(t, x)dx = 0. (4.8)

where the probability density µ(t, x) is given by

µ(t, x) := ψ∗(t, x)ψ(t, x) = e2R(t,x). (4.9)

Then, we transform the expectation value of the momentum p to

〈ψ(t)|p|ψ(t)〉 =

∫∇S(t, x)µ(t, x)dx, (4.10)

Because of Eq. (4.8), the functional representation of the momentum p is given by

hp(t, x) = ~∇(R(t, x) + S(t, x)), (4.11)

which is the desired equation.

Propositon 4.2. The functional representation of the observable p2 is

hp2(t, x) = ~2((∇R(t, x))2 + (∇S(t, x))2

). (4.12)

4.1. WHAT IS PROBABILITY IN QUANTUM MECHANICS? 25

Proof. Since p2 = −~2∆, we obtain

〈ψ(t)|p2|ψ(t)〉 = −~2

∫ψ∗(t, x)∆ψ(t, x)dx

= ~2

∫∇ψ∗(t, x)∇ψ(t, x)

= ~2

∫ ((∇R(t, x))2 + (∇S(t, x))2

)µ(t, x)dx. (4.13)

Therefore, we obtain the desired equation.

Lemma 4.1. Except for ∇R∇S = 0, the relation between the functional representationsof p and p2 is

(hp)2 6= hp2 . (4.14)

Proof. Since the functional representation of the momentum p is Eq. (4.6), we obtain

(hp)2 = ~2

((∇R)2 + (∇S)2

)+ 2~2∇R∇S, (4.15)

which is compared to Eq. (4.12).In this simple example, we obtain the strange result on the standard derivations of x

and p as the follows. The variance of the momentum p is given by

V ar(p) = V ar(hp) =

∫~2

(∂R

∂x+

∂S

∂x

)2

µdx−(∫

~(

∂R

∂x+

∂S

∂x

)µdx

)2

, (4.16)

where µ := µ(t, x) is a probability density. The standard derivation of the momentum p isdefined as σ(p) :=

√V ar(hp) =

√V ar(p). From this definition, we obtain the following

theorem.

Theorem 4.1 (Heisenberg Uncertainty Relationship (Mathematics)). There does notexist a lower bound of σ(x) · σ(p).

Proof. First of all, fix a real function R. Since the real function S is independent of R,we set S = εR for arbitrary ε ∈ R. Then, we obtain

V ar(p) = (1− ε)2~2

∫dxe2R

(∂R

∂x

)2

. (4.17)

Since we fix the probability density µ = e2R, the variance of the position x does notchange. For ε → 1, we can arbitrary lower

√V ar(x) ·

√V ar(p).

From the view of physics, this result is very strange since many physicists believe theRobertson inequality (A.5),

σ(x) · σ(p) ≥ ~2, (4.18)


as the principle of quantum mechanics. Of course, the definition of the variance differsfrom Eq. (4.16) and is

σ(A) :=

√〈ψ|A2|ψ〉, (4.19)

where A := A− 〈ψ|A|ψ〉I for any state |ψ〉 ∈ H and any observable A.Note that the product of the standard deviations of x and p has a lower bound in the

case of∫ ∇R∇Sµdx = 0, which is the exceptional case of Lemma 4.1. In this case, we

obtain

σ(x) · σ(p) ≥ ~2. (4.20)

As a simple example [108, Section 53], in the case that the state ψ is given by

ψ(x) = A exp

(−(x− x0)

2

4(δ)2+ i

p0(x− x0)

~

), (4.21)

where x0, p0 and A are constants. From the definition of the wave function, the realfunctions are

R = −(x− x0)2

4δ2, S =

p0(x− x0)

~. (4.22)

We calculate

σ(x) =√

V ar(x) = δ, (4.23)

and the functional representation of p as

hp = − ~2δ2

(x− x0) + p0, (4.24)

to obtain

σ(p) =

√∫(hp)2µdx =

~2δ

. (4.25)

Therefore, we obtain the following relation,

σ(x) · σ(p) =~2. (4.26)

This is because we calculate ∫∂R

∂x

∂S

∂xµdx = 0, (4.27)

which is satisfied with the above condition.This difference originally results from the definition of the random variables. Now,

what is a proper expression of random variables in quantum mechanics?

4.2. EXTENDED PROBABILITY THEORY VIA WEAK VALUE 27

4.2 Extended Probability Theory via Weak Value

First of all, we discuss the strangeness of the Born rule. In conventional quantum me-chanics, the expectation value of the position x is given by

Ex(x, ψ) := 〈ψ|x|ψ〉 =

∫x|ψ(x)|2dx, (4.28)

where ψ(x) ∈ H = L2(R) and |ψ(x)|2 is a probability distribution and |ψ(x)|2dx is aprobability measure. Furthermore, the expectation value of the momentum p is given by

Ex(p, ψ) := 〈ψ|p|ψ〉 =

∫p|ψ(p)|2dp. (4.29)

The probability measure is |ψ(p)|2dp and differs from |ψ(x)|2dx.

Remark 4.1 (Contextual Dependence of Probability Measure). The Born rule leadsthat the probability measure depends on the observables. That is, we cannot define theprobability space a prior and decide this space on deciding the measured observable1.

In order to solve the inconsistency between physics and mathematics, we discuss whatthe random variable. As you know, the examples of the random variable are the obtainedresult of the dice, the prize price of the drawing lots, and so on. The common conceptof the random variable is the accessible quantities. In other words in physics, therandom variable must be an experimentally accessible quantity. In conventional quantummechanics, especially, quantum measurement theory, the experimentally quantity is realsince the observable is self-adjoint operator and only has the real spectrum. In quantummechanics, we, however, can define the complex experimentally accessible quantity, thatis, a weak value.

We calculate the expectation value in quantum mechanics as

Ex(A) = 〈ψ|A|ψ〉=

∫dφ 〈ψ|φ〉〈φ|A|ψ〉 =

∫dφ 〈ψ|φ〉 · 〈φ|ψ〉〈φ|A|ψ〉〈φ|ψ〉 ,

=

∫dφ |〈ψ|φ〉|2 φ〈A〉wψ (4.30)

where hA = φ〈A〉wφ is complex random variable and dP = |〈φ|ψ〉|2dφ is the probabilitymeasure and is independent of the observable A. This formula means that the extended

1As you may know the Kochen-Specker theorem, this theorem means that the spectral representationmay depend on the context, that is, the order of taking the spectral representation of observables (Seethe book [68]). While this remark may be related to this theorem, we have not shown the direct relation.


probability theory is corresponded to the Born rule, that is, the conventional interpreta-tion. From the conventional definition of the variance in quantum mechanics, we obtainthe variance as

V ar(A) = 〈ψ|(A− 〈ψ|A|ψ〉)2|ψ〉= 〈ψ|A2|ψ〉 − (〈ψ|A|ψ〉)2 =

∫dφ〈ψ|A|φ〉〈φ|A|ψ〉 − (Ex(A))2

=

∫dφ|〈φ|ψ〉|2 · 〈ψ|A|φ〉〈ψ|φ〉 · 〈φ|A|ψ〉〈φ|ψ〉 − (Ex(A))2

=

∫| φ〈A〉wψ |2dP −

(∫φ〈A〉wψdP

)2

. (4.31)

This shows the consistent framework in quantum mechanics in the case of putting thecomplex random variable hA = φ〈A〉wφ and the probability measure dP = |〈φ|ψ〉|2dφ.

Summing up this chapter, we have shown the consistency between the extended prob-ability theory and Born rule by considering the meaning of the random variable and haveprepared the fundamental concepts in order to discuss the stochastic process in quantummechanics.

4.3 Discussions

There remains the following problems. First, there are some extensions of the conventionalprobability theory. Especially, quantum probability theory 2 is familiar with quantummechanics from the algebraic view. Furthermore, there are many studies of quantumwalk analogous to the conventional random walk (See the next subsection.). These havenot shown the relation each other. Since every concept is included in quantum mechanics,there may be strong relationships. Second, we consider the application to some physicalsystem. We have not explained the fluctuation of photon numbers by quantum non-demolition photon counting in a cavity during quantum feedback control [24, 48]. Thissystem may have some stochastic process of photon numbers. We hope that this exampleshows a new physical interpretation of quantum stochastic process.

2 We refer the reader to the review paper [111]. There are many formulations on quantum probabilitytheory, which is often called a non-commutative probability theory. All formulations are based on thealgebraic view of quantum mechanics. In this paper, we explain the formulation using the von-Neumannalgebra as follows. As analogous to the conventional probability theory, we define the Hilbert space H,the set of the projection operators in the von-Neumann algebra R on H, P(R), and the state φ associatedwith the event space Ω, the set of the measurable function F , the probability measure P , respectively.

4.3. DISCUSSIONS 29

4.3.1 Quantum Walk

We refer the reader to the book [81]. In analogous to the classical random walk, we define aquantum walk as the follows. Let a particle state denote a tensored state |x〉⊗|c〉 = |x, c〉,a position state |x〉 ∈ Hp := |x〉 : x ∈ Z ' C∞ and a chirality state |ψ〉 ∈ Hc :=α|L〉+ β|R〉 : α, β ∈ C, |α|2 + |β|2 = 1, where

|L〉 =

[01

], |R〉 =

[10

]. (4.32)

The time evolution of the one-dimensional quantum walk is given by the two steps:

Ucoin =

[a bc d

]∈ U(2), (4.33)

where a, b, c, d ∈ C and this is corresponded to a quantum coin toss. The time evolutionC = Ip ⊗ Ucoin solely acts on the chirality states as

C|x, R〉 = a|x,R〉+ c|x, L〉, C|x, L〉 = b|x,R〉+ d|x, L〉. (4.34)

Furthermore, a position shift is subject to the chirality states defined as

S|x,R〉 = |x + 1, R〉, S|x, L〉 = |x− 1, L〉. (4.35)

The time evolution of the quantum walk is given by the unitary operator U = SC per astep. We assume that the initial position and chirality state be origin and φ0, respectively.After t steps, we obtain the probability at the position x as

P (Xt = x) = ||ψ(φ0)t (x)||2. (4.36)

We obtain the weak convergence theorem as the follows.

Theorem 4.2 ( [79, 80]).Xt

t⇒ W (t →∞), (4.37)

where ” ⇒ ” denotes the weak convergence. The density of W is given by

f(x) =I(−1/

√2,1/

√2)(x)

π(1− x2)√

1− 2x2, (4.38)

where IA(x) is the defined function of A in the case of the Hadamard coin (a = b = c =−d = 1/

√2) and the initial chirality state φ0 = (|R〉+ i|L〉)/√2.

Chapter 5

Quantum Operations for WeakValues

Our aim is to find the most general map for the weak operator W in this chapter. Theresult terms out to be of the form E(W ) =

∑i EiWF †

i1. We would like to emphasize

that this form is deduced only by the complete positivity besides the linearity [127].

5.1 Quantum Operations for Density Operators —

Review

Let us recapitulate the general theory of quantum operations of a finite dimensional quan-tum system [94]. All physically realizable quantum operations can be generally describedby a CP map [97, 99], since the isolated system of a target system and an auxiliary systemalways undergoes the unitary evolution according to the axiom of quantum mechanics [93].One of the important properties of the CP map is that all physically realizable quantumoperations can be described only by operators defined in the target system.

Let E be a positive map from L(Hs), a set of linear operations on the Hilbert space Hs,to L(Hs). If E is completely positive, its trivial extension σ from L(Hs) to L(Hs⊗He) isalso positive such that

σ(|α〉) := (E ⊗ I)(|α〉〈α|) > 0, (5.1)

for an arbitrary state |α〉 ∈ Hs ⊗ Hp. We assume without loss of generality dimHs =dimHe < ∞. Throughout this paper, we concentrate on the case that the target state

1Of course, if we introduce a probe Hilbert space and then consider the probability distribution ofthe measurement outcome, we can extract information of the phase by an interference pattern (e. g.,see [132, 16]). The virtue of the CP map is that it is defined solely by operations of the target Hilbertspace. We would like to find out a representation analogous to the Kraus representation for the phaserelated object in a way defined only in the target system.

31

32 CHAPTER 5. QUANTUM OPERATIONS FOR WEAK VALUES

is pure though the generalization to mixed states is straightforward. From the completepositivity, we obtain the following theorem for quantum state changes.

Theorem 5.1. Let E be a CP map from Hs to Hs. For any quantum state |ψ〉s ∈ Hs,there exist a map σ and a pure state |α〉 ∈ Hs ⊗Hp such that

E(|ψ〉s〈ψ|) = e〈ψ|σ(|α〉)|ψ〉e, (5.2)

where

|ψ〉s =∑

k

ψk|k〉s,

|ψ〉e =∑

k

ψ∗k|k〉e, (5.3)

which represents the state change for the density operator.

Proof. We can write in the Schmidt form as

|α〉 =∑m

|m〉s|m〉e. (5.4)

We rewrite the right hand sides of Eq. (5.2) as

σ(|α〉) = (E ⊗ I)

(∑m,n

|m〉s|m〉e s〈n|e〈n|)

=∑m,n

|m〉e〈n|E(|m〉s〈n|), (5.5)

to obtain

e〈m|σ(|α〉)|n〉e = E(|m〉s〈n|). (5.6)

By linearity, we arrive at the desired equation (5.2).From the complete positivity, σ(|α〉) > 0 for all |α〉 ∈ Hs ⊗He, we can express σ(|α〉)

asσ(|α〉) =

∑m

sm|sm〉〈sm| =∑m

|sm〉〈sm|, (5.7)

where sm’s are positive and |sm〉 is a complete orthonormal set with |sm〉 :=√

sm|sm〉.We define the Kraus operator Em [84] as

Em|ψ〉s := e〈ψ|sm〉. (5.8)

Then, the quantum state change becomes the Kraus form,∑m

Em|ψ〉s〈ψ|E†m =

∑m

e〈ψ|sm〉〈sm|ψ〉e = e〈ψ|σ|ψ〉e

= E(|ψ〉s〈ψ|). (5.9)

We emphasize that the quantum state change is described solely in terms of the quantitiesof the target system.

5.2. QUANTUM OPERATIONS FOR WEAK OPERATORS 33

5.2 Quantum Operations for Weak Operators

Let us now define a weak operator as

W (t) := |ψ(t)〉〈φ(t)|, (5.10)

based on the two-state vector formalism by Aharonov and Vaidman [10] and define

〈A〉W :=Tr(AW )

Tr(W ), (5.11)

for an observable A corresponding to the weak value of the observable A [2] as the above2.The weak value is an analog of a probability, and so is the weak operator that of the densityoperators. We discuss a state change by the weak operator and define a map X as

X(|α〉, |β〉) := (E ⊗ I) (|α〉〈β|) , (5.12)

for an arbitrary |α〉, |β〉 ∈ Hs ⊗He. We consider the following states;

|ψ(t)〉s =∑

k

ψk|αk〉s,

|φ(t)〉s =∑

k

φk|βk〉s,

|ψ(t)〉e =∑

k

ψ∗k|αk〉e,

|φ(t)〉e =∑

k

φ∗k|βk〉e, (5.13)

where |αk〉s, |βk〉s, |αk〉e, and |βk〉e are complete orthonormal sets of Hs and He.Then, we obtain the following theorem on the state change of the weak operator such asTheorem 5.1.

Theorem 5.2. Let E be a CP map from Hs to Hs. For any weak operator W =|ψ(t)〉s〈φ(t)|, there exist a map X and pure states |α〉, |β〉 ∈ Hs ⊗Hp such that

E (|ψ(t)〉s〈φ(t)|) = e〈ψ(t)|X(|α〉, |β〉)|φ(t)〉e, (5.14)

which represents the state change for the weak operator.

2 While the original notation of the weak values is 〈A〉w indicating the ”w”eak value of an observableA, our notation is motivated by one of which the pre- and post-selected states are explicitly shown asf 〈A〉wi .

5.2. QUANTUM OPERATIONS FOR WEAK OPERATORS 35

where

〈tm| = 〈sm|U. (5.23)

Similarly to the Kraus operator, we define the two operators, Ei and F †i , as

Em|ψ(t)〉s := e〈ψ(t)|sm〉 (5.24)

s〈φ(t)|F †m := 〈tm|φ(t)〉e. (5.25)

Therefore, we obtain the change of the weak operator as

∑m

Em|ψ(t)〉s〈φ(t)|F †m =

∑m

e〈ψ|sm〉〈tm|φ〉e

= e〈ψ|X|φ〉e= E (|ψ(t)〉s〈φ(t)|) , (5.26)

using Theorem 5.2 in the last line. By linearity, we conclude

E(W ) =∑m

EmWF †m. (5.27)

Note that, in general, E(W )E(W †) 6= E(ρ) although ρ = WW †.Summing up, we have introduced the weak operator (5.10) and obtained the gen-

eral form of the quantum operation of the weak operator (5.27) in an analogous way tothe quantum operation of the density operator assuming the complete positivity of thephysical operation.

It is well established that the trace preservation, Tr(E(ρ)) = Tr ρ = 1 for all ρ, impliesthat

∑m E†

mEm = 1. The proof is simple [94] and goes through as

1 = Tr(E(ρ)) = Tr

(∑m

EmρE†m

)

= Tr

(∑m

E†mEmρ

)(∀ρ). (5.28)

This argument for the density operator ρ = WW † applies also for W †W to obtain∑m F †

mFm = 1 because this is the density operator in the time reversed world in thetwo-state vector formulation as reviewed in the introduction. Therefore, we can expressthe Kraus operators,

Em = e〈em|U |ei〉e,F †

m = e〈ef |V |em〉e, (5.29)


Figure 5.1: The quantum operations for the weak operators. The weak operator W (t) =|ψ(t)〉s〈φ(t)| carries the entire history from the pre-selected state to the post-selectedstate. The quantum operations for the weak operator is described by the two operatorsEm and F †

m. These operators correspond to the Kraus operators for the density operators,WW † and W †W , related to the two-state vector formalism and affect the history.

for some unitary operators U and V , which act on Hs ⊗He. |ei〉 and |ef〉 are some basisvectors and |em〉 is a complete set of basis vectors with

∑m |em〉〈em| = 1. We can compute

∑m

F †mEm =

∑m

e〈ef |V |em〉e〈em|U |ei〉e

= e〈ef |V U |ei〉e = e〈ef |S|ei〉e, (5.30)

where S = V U = U(tf , ti) is the S-matrix. The above equality (5.30) may be interpretedas a decomposition of the history in analogy to the decomposition of unity. The meaningof the basis |ei〉 and |ef〉 will be clear in the following section.

As is well known [94], the physical operation for the density operator can also bedescribed by introducing an environment which is tensored by the target system. Weperform a unitary transformation for a combined state and then take a partial trace overthe environmental states. We can also apply this method to the weak operator. Namely,

E(W ) = Trenv[U(W ⊗ e)V ], (5.31)

where e = |ei〉〈ef | is the environmental weak operator before the physical process. It isstraightforward to formally carry out the partial trace to reproduce the Kraus represen-tation for the weak operator W as Eq. (5.27). Any interaction model of this type willgive the same Kraus representation for the weak operator. The procedure of quantumoperations for the weak operator is illustrated in Fig. 5.1.

A representation of an ensemble of quantum states is usually described by a densityoperator since we only obtain the probability distribution of an observable in the conven-tional quantum measurement theory. The density operator does not contain the phase

5.3. WEAK MEASUREMENT WITH DECOHERENCE 37

information of a quantum state. However, the weak operator gives a weak value of anobservable and retains the information of the phase of the quantum state. We will showa typical example of a geometric phase in Chap. 6.

5.3 Weak Measurement with Decoherence

So far we have formally discussed the quantum operations of the weak operators. Inthis section, we would like to study the effect of environment in the course of the weakmeasurement [2] and see how the shift of the probe position is affected by the environment.As we shall see, the shift is related to the quantum operation of the weak operator E(W )(5.27) which we have investigated in the previous section.

5.3.1 Weak Measurement and Environment

Let us consider a target system coupled with an environment and a general weak measure-ment for the compound of the target system and the environment. We assume that thereis no interaction between the probe and the environment. This situation is illustrated inFig. 5.2. The Hamiltonian for the target system and the environment is given by

H = H0 ⊗ Ie + H1, (5.32)

where H0 acts on the target system Hs and the identity operator Ie is for the environmentHe, while H1 acts on Hs ⊗He. The evolution operators U and V can be expressed by

U = U0K(t0, ti),

V = V0K(tf , t0), (5.33)

where U0 and V0 are the evolution operators forward in time and backward in time, respec-tively, by the target Hamiltonian H0. K’s are the evolution operators in the interactionpicture,

K(t, ti) = T e−iR t

tiU†0H1U0 ,

K(tf , t) = T e−iR tf

t V0H1V †0 , (5.34)

where T and T stand for the time-ordering and anti time-ordering products.Let the initial and final environmental states be |ei〉 and |ef〉, respectively. The probe

state now becomes

〈f |〈ef |V U |ei〉|i〉ξ(

q − g〈f |〈ef |V AU |ei〉|i〉

〈f |V U |i〉)

. (5.35)


Plugging the expressions for U and V into the above, we obtain the probe state as

Nξ

(q − g

〈f |V0〈ef |K(tf , t0)AK(t0, ti)|ei〉U0|i〉N

), (5.36)

where N = 〈f |V0〈ef |K(tf , t0)K(t0, ti)|ei〉U0|i〉 is the normalization factor. We define thedual quantum operation as

E∗(A) := 〈ef |K(tf , t0)AK(t0, ti)|ei〉=

∑m

F †mAEm, (5.37)

where

F †m := 〈ef |K(tf , t0)|em〉,

Em := 〈em|K(t0, ti)|ei〉 (5.38)

are the Kraus operators introduced in the previous section (5.29). Here, we have insertedthe completeness relation

∑m |em〉〈em| = 1 with |em〉 being not necessarily orthogonal.

The meaning of the basis |ei〉 and |ef〉 is now clear as remarked before. Thus, we obtainthe wave function of the probe as

ξ

(q − g

〈f |V0E∗(A)U0 |i〉N

)= ξ

(q − g

∑m〈f |V0F

†mAEmU0|i〉∑

m〈f |V0F†mEmU0|i〉

)

= ξ

(q − g

Tr[A

∑m EmU0|i〉〈f |V0F

†m

]

Tr [∑

m U0|i〉〈f |V0]

)

= ξ

(q − g

Tr[E(W )A]

Tr[E(W )]

)= ξ(q − g〈A〉E(W )), (5.39)

with N = 〈f |V0E∗(I)U0|i〉 up to the overall normalization factor. This is the main resultof this subsection. The shift of the expectation value of the position operator on the probeis

δq = g · Re[〈A〉E(W )]. (5.40)

From an analogous discussion, we obtain the shift of the expectation value of the momen-tum operator on the probe as

δp = 2g · V ar(p) · Im[〈A〉E(W )]. (5.41)

Thus, we have shown that the weak value given by the probe shift is affected by theenvironment during the weak measurement.

5.3. WEAK MEASUREMENT WITH DECOHERENCE 39

Figure 5.2: A weak measurement model with the environment. The environment affectsthe target system as a noise but does not affect the probe. The weak measurement forthe target system and the probe brings about the shift of the probe position at t0. Theamount of the shift depends whether the environmental state is controllable (Chap. 5.3.1)or uncontrollable (Chap. 5.3.2).

5.3.2 Weak Measurement—Decoherence

In many cases, the initial and final states, ei and ef , of the environment, on which thequantum operation E depends, are not controllable so that they have to be statisticallytreated. Let the statistical weight be w(ef , ei) and consider the average,

Ave(g) :=∑ef ,ei

w(ef , ei)g(ef , ei), (5.42)

for a function g of the random variables ef and ei. Note that

Ave

(Nξ

(q − g

Tr[E(W )A]

N

))

≈ Ave

(N

ξ (q)− g

Tr[E(W )A]

Nξ′ (q)

)

≈ Ave(N)ξ

(q − g

Ave(Tr[E(W )A])

Ave(Tr[E(W )])

). (5.43)

We see that the shift of the expectation value of the probe position is on average,

δq = Re

[gAve(Tr[E(W )A])

Ave(Tr[E(W )])

], (5.44)


in the weak coupling case. To obtain a significant shift, one needs some prior knowledgeof the environment. For the case of a detector as the environment, ei and ef are specifiedby the measurement outcome and are definite if the environment is at zero temperature,for example. In general, the shift decreases by the statistical average as one expects.

Chapter 6

Applications of Weak Values

In this chapter, we discuss the efficiency of the weak values by an example.

6.1 Geometric Phase

We present a simple application of our framework of the physical operation of the weakoperators to the geometric phase. There have been many works on the ”geometric phaseof mixed states” but the very definition seems under controversy [129]. We would like tostart with the geometric phase γ in a pure state [128] which is well-defined and can beexpressed in terms of the weak value,

γ := arg〈i|f〉〈f |P 〉〈P |i〉 = arg

[〈f |P 〉〈P |i〉〈f |i〉

]

= arg

[Tr(WP )

Tr(W )

], (6.1)

where P = |P 〉〈P | is a projector to a pure state and W = |i〉〈f |. Here, we simplify thenotations |i〉 := U(t, ti)|i〉 and 〈f | := 〈f |U(tf , t) only in this section. The geometric phaseγ corresponds to the quantum path |i〉 → P → |f〉. By the physical operation E , theweak operator and the density operator are mapped to E(W ) and E(ρ), respectively. Thenew state E(ρ) is in general a mixed state. The new geometric phase γg is correspondinglygiven by

γg = arg

[Tr(E(W )P )

Tr E(W )

], (6.2)

which might be called the geometric phase of the mixed state E(ρ) by a slight abuseof words. This operational definition fits well to experimental situation. That is, anexperimentalist starts with a pure state |i〉 and then makes a trip |i〉 → P → |f〉 by ma-nipulating the external field. If there were no decoherence during that process, one would

41

42 CHAPTER 6. APPLICATIONS OF WEAK VALUES

Figure 6.1: (a) Geometric phase in the case of the pure state for a qubit system. Whenthe path |i〉 → |0〉 → |f〉, where |0〉 is an eigenstate of the eigenvalue +1 of the Pauli zoperator, the half of the solid angle on the Bloch sphere, which is the gray region, corre-sponds to the geometric phase. (b) Geometric phase in the presence of the environmentwith the same path. The quantum operations are represented by Em and F †

m.

get the geometric phase defined above (6.1). Otherwise, one would instead get the value(6.2) for the geometric phase, while one can presume that the state is E(ρ) = E(WW †).Furthermore, we would like to point out that this definition (6.2) coincides with the defi-nition in the Uhlmann approach [135, 129]. In the generalized Kraus representation, thegeometric phase for that path, |i〉 → P → |f〉 can be written as

γg = arg

[∑m

〈f |F †m|P 〉〈P |Em|i〉〈f |i〉

]. (6.3)

This is illustrated in Fig. 6.1.Let us see the decoherence effect on the geometric phase in a simple one qubit system

under a bit flip noise [94]. The Kraus operators are given by

E0 = F0 =√

pI,

E1 =√

1− pσx, F1 =√

1− pσxeiφσz (0 ≤ p ≤ 1). (6.4)

We are going to consider a path, |i〉 → |0〉 → |f〉, where

|i〉 = cos θ|0〉+ sin θ|1〉,|f〉 =

1√2(|0〉 − |1〉), (6.5)

where the angle θ is introduced to make the weak operator well defined but at the veryend of calculation the limit θ → π/4 is taken. A straightforward calculation shows the

6.2. DISCUSSIONS 43

geometric phase γg as

γg = arctan

[p cos θ + (1− p) sin θ

p cos θ − (1− p) sin θtan φ

]− arctan

[cos θ + sin θ

cos θ − sin θtan φ

]. (6.6)

In the no noise, p = 1, and the limit θ → π/4 case, we recover the geometric phaseπ/2, which is the half of the solid angle formed by the three vectors |i〉, |0〉, and |f〉 inthe Bloch sphere. In the case of p = 1/2, the geometric phase vanishes, as we expectbecause the state is completely mixed. It seems the decohered geometric phase has noparticular geometrical meaning while Uhlmann gave an expression for the geometric phasein terms of operators similar to the weak operator [136, 137]. It is curious to point outthat the geometric phase during the measuring process is given by a time evolution ofweak values [107].

We would like to stress that our definition of the geometric phase under the environ-mental noise is operationally defined in the sense that the geometric phase is initiallydefined in a pure state but undergoes a decoherence process while the state becomes amixed state.

6.2 Discussions

There remain the following problems. First, the spin Hall effect have been experimentallydemonstrated by Hosten and Kwiat [65] and theoretically analyzed via the Berry phase byOnoda, Murakami, and Nagaosa [96]. From the inclusive view of the relationship betweenthe Berry phase and the weak values, we reconsider the experiment to verify the quantumHall effect. Second, Giscard recently proposed the operator associated with the geometricphase, especially the Aharonov-Anandan phase [47]. Is there the relationship betweenhis proposed operator and the weak operator? Third, We discuss the non-commutativeproperty of the weak values. From the early stage, quantum mechanics is based on the non-commutative property. This property is most represented by the uncertainty relationship(See App. A on the historical review.). Hall [50, 51] and Johansen [74] analyzed theuncertainty relationships using the weak values. How many relationships are there usingthe weak values? What does the relationship have the physical significance? Finally,motivated by the optimal covariant measurement, our proposed measurement model [126](See App. C.) seems relate to the weak measurement since the measuring time is infinitelylong for the coupling constant to be small. We may analyze our proposed measurementmodel via the weak values and relate to the relationship between the optimal covariantmeasurement and the weak measurement.

Chapter 7

Conclusions

In this thesis, we have developed the framework of weak values, which are experimentallyaccessible quantities and have recently demonstrated in optical systems, and discussedapplications of the weak values. We summarize the master thesis as follows.

We have introduced the weak operator W (3.20) to formally describe the weak valueadvocated by Aharonov et al. which is the more general quantity containing the phaseinformation than the density operator ρ in Chap. 3. We show that we need take theweak values as the random variables in order to remain the consistency with the standardprobability theory in quantum mechanics. Furthermore, we have constructed the formaltheory of the extended probability theory via weak values in quantum mechanics in Chap.4. The general framework is given to describe effects of quantum operation E(W ) (5.27) tothe weak operator W in parallel with the Kraus representation of the completely positivemap for the density operator ρ. We have shown the effect of the environment during theweak measurement as the shift of the expectation value of the probe observables in bothcases of the controllable and uncontrollable environmental states in Chap. 5. We haveshown the advantage of considering the weak values by the following applications of weakvalues theory in Chap. 6. We naturally extend the definition of the geometric phase formixed states by the quantum operations for the weak operator. It is exhibited how thegeometric phase is affected by the bit flip noise.

To conclude, the weak operator is a useful tool in quantum information and quantumfoundations despite its naming.

45

Appendix A

Uncertainty Relationships

In this chapter, we review various uncertainty relationships from the historical view1. Thishistorical review consists of the derivations and the definitions and the physical meanings.

A.1 Uncertainty Principle and Uncertainty Relation-

ship

The uncertainty principle was initiated by Heisenberg [55]. In the paper [55], Heisenbergexplained its physical meaning by the three examples, the gedankenexperiment of gamma-ray microscope gedankenexperiment (position-momentum), the Stern-Gerlach experiment(time-energy) (See App. B.), and the atomic structure (number-phase), and formulatedthese following the Dirac-Jordan theory [36, 75], which is the non-commutative theory. Inthe following, we explain the gamma-ray microscope gedankenexperiment (Fig. A.1) [56].The limits on the accuracy of the location δx of the image is given by

δx ∼ λ

sin ε, (A.1)

where λ denotes the wave length of the scattered radiation2 and ε denotes the half angleof aperture of the object. The direction of the scattered light must then, in principle,be considered as undetermined within this angle ε. Hence, according to the Comptoneffect, the component of the momentum of the material particle in the x-direction isundetermined, after the collision, by an amount

δpx ∼ h

λsin ε. (A.2)

1As far as the author knows, there is no inclusive historical review of the uncertain relationships. Overthe historical review part, we refer to [104, 69]

2λ can be different from the wave length of the incident radiation.

47

48 APPENDIX A. UNCERTAINTY RELATIONSHIPS

Figure A.1: Heisenberg’s gamma-ray gedanken microscope.

From these equations, we obtain

δx · δpx ∼ h. (A.3)

This equation3 was based on classical mechanics but he derived it by the Dirac-Jordantheory.

In the last paragraph of the paper [55], Heisenberg added to the pre-publication proof 4

that Bohr pointed out the direct connection between the uncertain principle and the wave-particle duality. Then, Bohr introduced the complementarity based on the foundationsof quantum mechanics. The concept of complementarity is stated that a single quantummechanical entity can either behave as a particle or as wave, but never simultaneously asboth [19].

In the same year to publish the Heisenberg paper, Kennard derived the followinginequality [78] as

σ(x) · σ(p) ≥ ~2, (A.4)

where σ(x) and σ(p) are the standard deviations of the position and momentum. Heconsidered when measuring some quantity, the amount of this probabilistically changesand we only statistically know the average value as we repeat the measurement. Heconcluded that this inequality was taken as the Heisenberg uncertainty principle (A.3)5

3Almost all physicists interpret the Heisenberg uncertainty principle as Eq. (A.17) or seem to be atcross-purposes with it as Eq. (A.5).

4We can get this from the following website.http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr155.1.html

5In the abstract of the original paper: Das Ergebnis kann dahin formuliert werden, dass der Fallen nurin der Hisenbsergschen Unbestimmtheitsrelation zwischen den Werten kanonisch konjugierter Variabelnbesteht.

A.2. QUANTUM MEAN SQUARE ERROR 49

and there occurs the error in classical mechanics on the measurement but the error is thetheoretically inevitable quantity in quantum mechanics.

Weyl derived the same inequality (A.4) following the Cauchy-Bunyakowski-Schwarz in-equality [142, Appendix 1]. Robertson derived the general inequality for non-commutativeobservables [118],

σ(A) · σ(B) ≥ |〈[A,B]〉|2

, (A.5)

which is called Robertson’s uncertainty relationship and where σ(A) and 〈A〉 are thestandard deviation and the average value of the observable A, respectively. Furthermore,Schrodinger derived the following inequality [124];

(σ(A) · σ(B))2 ≥∣∣∣∣〈[A,B]〉

2

∣∣∣∣2

+

(〈A,B〉2

− 〈A〉 · 〈B〉)2

. (A.6)

Summing up Robertson and Schrodinger’s works, we conclude that

σ(A) · σ(B) ≥ |〈AB〉 − 〈A〉〈B〉| ≥ |〈[A,B]〉|2

. (A.7)

The first inequality is due to Schrodinger, and the second to Robertson, which are derivedas follows [86].

Since 〈AB〉 = Tr[√

ρAB√

ρ] is an inner product for A√

ρ and B√

ρ as the Hilbert-Schmidt operators for the density (Hermitian) operator ρ, the Cauchy-Bunyakowski-Schwarz inequality gives

〈A2〉 · 〈B2〉 ≥ |〈AB〉|2. (A.8)

Replacing A by A − 〈A〉 and B by B − 〈B〉, we obtain the first inequality. For theHermitian operators, we obtain

|〈AB〉|2 ≥ 1

4|〈A,B〉+ 〈[A,B]〉|2 =

1

4|〈A,B〉|2 +

1

4|〈[A,B]〉|2 . (A.9)

Deleting the anti-commutator yields the Robertson inequality.

A.2 Quantum Mean Square Error

In classical mechanics and general statistical theory, the definition of the error is almostall taken as the mean square error since Gauss showed its efficiency [46]. Analogously, weintroduce the quantum mean square error as follows.

Definition A.1 (Error operator). For any observable A on the target system Hs, theerror operator for the probe Hp is

NA := U(I ⊗M)U † − A⊗ I := Mout − Ain, (A.10)


where U is some evolution operator for the combined system and M is the meter observablefor the probe Hp. Furthermore,

nA := TrHp NA (A.11)

is called an induced error operator for the observable A.

Definition A.2 (Quantum mean square error). For any state ρ and observable A for thetarget system Hs, the quantum mean square error for the probe Hp is

ε(A) := 〈N2A〉1/2 (A.12)

= Tr |(Mout − Ain)2(ρ⊗ ξ)|1/2, (A.13)

where ξ is the probe initial state.

A.3 Detection Limit of Gravitational Wave

The discussion on the detection limit of gravitational waves substantially contributes tounderstand the Heisenberg uncertainty relationship. A gravitational wave is a fluctuationin the curvature of space-time which propagates as a wave, traveling outward from amoving object or system of objects. A gravitational radiation is the energy transported bythese waves. The existence of the gravitational wave is proven from the Einstein equationby Einstein [38]. As an examples of systems which emit gravitational waves, there arebinary star systems, where the two stars in the binary are white dwarfs, neutron stars, andblack holes. Although gravitational radiation has not yet been directly detected, it hasbeen indirectly shown to exist by slowing the period of revolution of PSR B1913+16 by76 microseconds per a year by Hulse and Taylor [67]. Recently, we try to directly detectgravitational waves by the Mach-Zehnder interferometer, e.g. LIGO6 and TAMA300 7.A passing a gravitational wave then slightly stretches one arm as it shortens the other.Therefore, the interference pattern is changed. The problem is how accurate can we decidethe length of the arms.

To discuss the problem to measure the mirror, the mirror of the interferometer is takenas a free particle with the mass m and a quantum stuff. At some time, t = 0, we measurethe position of the mirror. We measure it again at t = τ . Caves et al. [27] showed

[σ(x(τ))]2 = [σ(x(0))]2 +

(σ(p(0))τ

m

)2

≥ 2σ(x(0)) · σ(p(0)) · τ

m

≥ ~τm

, (A.14)

6Laser Interferometer Gravitational-Wave Observatory.7The 300m Laser Interferometer Gravitational Wave Antenna.

A.4. HEISENBERG’S UNCERTAINTY PRINCIPLE REVISITED 51

which the bound is called the standard quantum limit (SQL). However, Yuen pointed outthat the above inequality was wrong and was proposed as

[σ(x(τ))]2 = [σ(x(0))]2 +

(σ(p(0))τ

m

)2

[σ(x(0)) · σ(p(0)) + σ(p(0)) · σ(x(0))]τ

m, (A.15)

using the contractive state [145] and where the third term is negative. While Caves showedthat we cannot create the contractive state in the von Neumann model [28], Ozawa showedthat the following interaction Hamiltonian gives the contractive state as the ground state;

Hint =πK

3√

3

(2x⊗ P − 2p⊗ X + xp⊗ I − I ⊗ XP

), (A.16)

where K is a coupling constant, which is so large to ignore the individual Hamiltonians,and (x, p) and (X, P ) are the position and momentum operators on the target systemand the probe, respectively [98]. Finally, Maddox judged that Ozawa finally showed thedetection limit of this problem and concluded the paper in the hope to explore a newuncertainty relationship8 in Nature [87].

A.4 Heisenberg’s Uncertainty Principle Revisited

In the previous subsections, we have shown the difference between the error on the mea-surement the standard derivation of wave packets. Then, Ozawa discussed the Heisenberguncertainty relationship [101] as

ε(A)η(B) ≥ |〈[A,B]〉|2

, (A.17)

where η(B) is defined as follows.As analogy to the error (Sec. A.2), we define the quantum mean square disturbance

as follows.

Definition A.3 (Disturbance operator). For any observable B on the target system Hs,the disturbance operator for the probe Hs is

DB := U(B ⊗ I)U † −B ⊗ I := Bout −Bin, (A.18)

where U is some evolution operator for the combined system Hs ⊗Hp. Furthermore,

dB := TrHp DB (A.19)

is called an induced disturbance operator for the observable B.8In the original paper: It is also far from obvious how the particular example on which the conclusion

rests can be turned into a realistic measuring equipment that would allow those who design equipmentto exploit this recipe for beating SQL. But none of this will damp enthusiasm for overcoming what oftenseems an intolerable constraint on the freedom to design accurate measuring equipment.


Definition A.4 (Quantum mean square disturbance). For any state ρ and observable Bfor the target system Hs, the quantum mean square disturbance for the probe Hp is

η(A) := 〈D2B〉1/2 (A.20)

= Tr |(Bout −Bin)2(ρ⊗ ξ)|1/2, (A.21)

where ξ is the probe initial state.

A.5 Ozawa’s Inequality

Ozawa derived the uncertainty relationship included in the error and disturbance on themeasurement and the standard derivation of wave packets as

ε(A)η(B) + ε(A)σ(B) + σ(A)η(B) ≥ |〈[A,B]〉|2

. (A.22)

This inequality means to violate the Heisenberg uncertainty relationship (A.17) and iscalled the Ozawa inequality [101].

On deriving the Ozawa inequality, it is useful the following lemma.

Lemma A.1. Let A and ρ be a target observable and an initial target state on Hs and(Hp, σ, U,M) be a quadruplet about the measurement, the probe Hilbert space, the initialprobe state, the unitary operator on the combined system, and the meter observable. Weobtain

ε(A)η(B) +|〈[NA, Bin]〉|

2+|〈[Ain, DB]〉|

2≥ |〈[A,B]〉|

2. (A.23)

Proof. From the definitions of the error and disturbance operators (A.10, A.18), we obtain

0 = [Aout, Bout]

= [Ain + NA, Bin + DB] (A.24)

to transform[NA, DB] + [NA, Bin] + [Ain, DB] = −[Ain, Bin]. (A.25)

Taking the average value for the density operator ρ⊗ σ on the both sides, we follow thetriangle inequality to obtain

|〈[NA, DB]〉|+ |〈[NA, Bin]〉|+ |〈[Ain, DB]〉| ≥ |〈[Ain, Bin]〉|= 〈[A,B]〉. (A.26)

Using the relationship as the above,

ε(A) ≥ σ(NA), (A.27)

η(B) ≥ σ(DB), (A.28)

A.5. OZAWA’S INEQUALITY 53

and the Robertson uncertainty relationship (A.5), we obtain

ε(A) · η(B) ≥ σ(NA) · σ(DB)

≥ 〈[NA, DB]〉2

. (A.29)

Then, we obtain

ε(A)η(B) +|〈[NA, Bin]〉|

2+|〈[Ain, DB]〉|

2≥ 〈[NA, DB]〉

2+|〈[NA, Bin]〉|

2+|〈[Ain, DB]〉|

2

≥ |〈[A,B]〉|2

, (A.30)

following the second inequality (A.23), which is the desired inequality.From the left hand side of Eq. (A.22), we obtain

ε(A)η(B) + ε(A)σ(B) + σ(A)η(B)

≥ ε(A)η(B) + σ(NA)σ(B) + σ(A)σ(DB)

≥ ε(A)η(B) +

∣∣∣∣〈[NA, Bin]〉

2+〈[Ain, DB]〉

2

∣∣∣∣

≥ ε(A)η(B) +|〈[NA, Bin]〉|

2+|〈[Ain, DB]〉|

2

≥ |〈[A,B]〉|2

, (A.31)

following Eqs. (A.10) and (A.18) in the first inequality, the Robertson uncertainty rela-tionship (A.5) in the second inequality, the triangle inequality in the third inequality, andEq. (A.26) in the forth inequality.

The case of holding the Heisenberg uncertainty relationship (A.17) is .

|〈[NA, Bin]〉|+ |〈[Ain, DB]〉| = 0. (A.32)

In order to characterize a class of measurements satisfying Eq. (A.17), we define that themeasurement interaction is said to be independent intervention for the pair (A,B) if thenoise and the disturbance are independent of the target system; or precisely if there isobservables N and D of the probe such that NA = I ⊗ N and DB = I × B. Under thiscondition, we obtain

〈[NA, Bin]〉 = 〈[Ain, DB]〉 = 0. (A.33)

Therefore, the Ozawa inequality (A.22) reduces the Heisenberg uncertainty relation-ship (A.17). The above conclusion was previously suggested in part by Braginsky andKhalili [22, p. 65] with a limited justification and now fully justified9.

9Generally speaking, the condition that the Ozawa inequality reduces the Heisenberg uncertaintyrelationship is given by [103, Theorems 6.1 and 6.3].


We consider the case to violate the Heisenberg uncertainty relationship (A.17) tomeasure the position. We assume that the interaction Hamiltonian as the above (Sec.A.3) be

Hint =πK

3√

3

(2x⊗ P − 2p⊗ X + xp⊗ I − I ⊗ XP

), (A.34)

where K is a coupling constant, which is so large to ignore the individual Hamiltonians,and (x, p) and (X, P ) are the position and momentum operators on the target systemand the probe, respectively. Then, the evolution operator is given by

U(t) = exp

(−i

Hintt

~

). (A.35)

We denote the initial probe state as |ξ〉 and the time to end the measurement interactionas t = ∆t. In the follows, we consider the Heisenberg picture. From the Heisenbergequation, we obtain

d

dtx(t) =

πK

3√

3[x(t)− 2X(t)],

d

dtp(t) = − πK

3√

3[p(t) + 2P (t)],

d

dtX(t) =

πK

3√

3[2x(t)− X(t)],

d

dtP (t) =

πK

3√

3[2p(t)− P (t)], (A.36)

to calculate x(t) and X(t) as

d

dt(x(t) + X(t)) =

πK√3

[x(t)− X(t)],

d

dt(x(t) + X(t)) = − πK

3√

3[x(t) + X(t)]. (A.37)

We calculated2

dt2(x(t) + X(t)) = −

(πK

3

)2

(x(t) + X(t)) (A.38)

to obtain

x(t) + X(t) = A exp

(iπK

3t

)+ B exp

(−i

πK

3t

). (A.39)

From the initial condition, t = 0, A and B are determined. By the analogous discussion,

A.6. UNCERTAINTY RELATIONSHIPS FOR JOINT MEASUREMENT 55

we obtain

x(t) =2√3

(sin

(π

3(1 + Kt)

)x(0)− sin

(π

3Kt

)X(0)

)

p(t) =2√3

(sin

(π

3(1−Kt)

)p(0)− sin

(π

3Kt

)P (0)

)

X(t) =2√3

(sin

(π

3Kt

)x(0) + sin

(π

3(1−Kt)

)X(0)

)

P (t) =2√3

(sin

(π

3Kt

)p(0) + sin

(π

3(1 + Kt)

)P (0)

)(A.40)

Setting ∆t = 1/K, we calculate

x(∆t)p(∆t)

X(∆t)

P (∆t)

=

x(0)− X(0)

−P (0)x(0)

p(0) + P (0)

(A.41)

to obtain the error and disturbance operators as

Nx = Q(∆t)− x(0) = 0, (A.42)

Dp = p(∆t)− p(0) = −(p⊗ I + I ⊗ P

)(A.43)

. The quantum mean square error and disturbance are given by

ε(x) = 〈N2x〉 = 0 (A.44)

η(p) = 〈D2p〉 = σ(p) + σ(P ) + |〈p〉+ 〈P 〉|2 < ∞. (A.45)

Therefore, the Heisenberg uncertainty relations (A.17) can be transformed as

ε(x) · η(p) = 0. (A.46)

This means that the Heisenberg uncertainty relations (A.17) can be violated.

A.6 Uncertainty Relationships for Joint Measurement

We consider joint measurement of non-commutative observables A and B under the un-biased condition, that is,

〈A〉 = 〈A〉meas, 〈B〉 = 〈B〉meas, (A.47)


hold for the all target states ρ. Ishikawa and Ozawa [70, 71, 100] independently derivedthe uncertainty relationship between quantum mean square errors as

ε(A) · ε(B) ≥ |〈[A,B]〉|2

, (A.48)

which is called the Ishikawa-Ozawa inequality10. They assumed that the probe consistsof macroscopic stuffs to satisfy

[U(I ⊗MA)U †, U(I ⊗MB)U †] = 0, (A.49)

where MA and MB are meter observables to measure the target observables A and B,respectively. From Eq. (A.49), we obtain that

[NA, NB] + [NA, B ⊗ I] + [A⊗ I, NB] + [A,B]⊗ I = 0, (A.50)

where the error operator NA and NB are defined in Eq. (A.10). From the unbiasedcondition, we obtain

Tr(nAρ) = 0, (A.51)

for all ρ to restrict the induced error operator as

nA = 0. (A.52)

Then, we obtainTr([NA, B ⊗ I](ρ⊗ ξ)) = Tr([nA, B]ρ) = 0, (A.53)

and similarly Tr([NB, A ⊗ I](ρ ⊗ ξ)) = 0, where ξ is the probe initial state. Taking theaverage of both sides of Eq. (A.50) in the state ρ⊗ ξ, we obtain

Tr([NA, NB](ρ⊗ ξ)) = −Tr([A,B]ρ). (A.54)

Noting that(σ(NA))2 = (ε(A))2 − |Tr(NA(ρ⊗ ξ))|2 ≤ (ε(A))2, (A.55)

we obtain

ε(A) · ε(B) ≥ σ(NA) · σ(NB)

≥ |Tr([NA, NB](ρ⊗ ξ))|2

=|Tr([A,B]ρ)|

2, (A.56)

10Strictly speaking, the inequality derived by Ishikawa [70] is different by the meaning of the erroroperator.

A.6. UNCERTAINTY RELATIONSHIPS FOR JOINT MEASUREMENT 57

following the Robertson uncertainty relationship (A.5) in the second inequality, whichis desired inequality (A.48). We have applied this to various measurement model (e.g.,see [121]).

From the view of the joint measurement, the Ozawa inequality [102] can be analogouslyderived as

ε(A)ε(B) + ε(A)σ(B) + σ(A)ε(B) ≥ |〈[A,B]〉|2

. (A.57)

Under the unbiased condition, we have proven that this inequality is reduced to theIshikawa-Ozawa inequality. Therefore, the Ozawa inequality is substantial under the casewithout the unbiased condition.

Appendix B

Time-Energy UncertaintyRelationships

In this chapter, we review the papers on the time-energy uncertainty relationships.

Paper 1 (Heisenberg Uncertainty Principle [55]).

ε(E)η(T ) ∼ h, (B.1)

where ε(E) is defined the error to measure the energy of the system and η(T ) is definedthe time to measure the energy. This relation always keeps when we measure the energyof a specified system.

Einstein posed a paradox during the sixth Solvay conference in 1930 to violate thetime-energy uncertainty relationship,

∆E∆t ≥ h, (B.2)

where ∆E and ∆t were defined as uncertainties of the measurement using a box thatemits a photon (See Fig. B.1). This is because the box hangs from a spring scale whichmeasures its weight. Its weight is proportional to its rest mass m, hence to its energy E,according to E = mc2. Einstein supposed that we wait for the box to settle down andaccurately measure the initial scale. This reading can be as slow and accurate as we like.After the photon leaves the box, we measure the final scale, again as accurate as we like,and from the difference between the two positions we get an accurate measurement of theenergy of the emitted photon, that is, ∆E < ∞. On the other hand, the clock in the boxtells exactly when the photon was released, that is, ∆t = 0, so we violate Eq. (B.2)1.

1 Rosenfeld described the Bohr reaction to this argument. ”It was quite a shock for Bohr to be facedwith this problem; he did not see the solution at once. During the whole evening, he was extremelyunhappy, going from one to the other and trying to persuade them that it could not be true, that itwould be the end of physics if Einstein were right; but he could not produce any refutation... The nextmorning came Bohr’s triumph and the salvation of physics...” [105, P.238].

59

60 APPENDIX B. TIME-ENERGY UNCERTAINTY RELATIONSHIPS

Figure B.1: The clock model proposed by Bohr [20] when Bohr triumphed against Ein-stein’s criticism.

Bohr triumphed as follows. Let x denote the position of the pointer on the scale andp its momentum, with ∆x and ∆p the corresponding uncertainties. Once we choose ∆x,∆p is restricted from the Heisenberg uncertainty relationship, ∆x ·∆p ≥ h, as

h

∆x≤ ∆p. (B.3)

Bohr assumed the pointer moves after the photon emission. By hanging little weights onthe box, we lower it to its original position. When it has returned to its original height,the total wight hanging from it equals the weight of the emitted photon. However, theaccurate of this weighting is no better than the smallest added weight g∆m that has anobservable effect. If we add a mass ∆m and wait a time t, the impulsive delivered to thebox cannot be greater than (g∆m)t, which must be greater than ∆p to be observable toobtain

∆p ≤ gt∆m. (B.4)

From Eqs. (B.3) and (B.4), we obtain

h ≤ gt∆x∆m =gt∆x∆E

c2. (B.5)

Einstein assumed that to measure the pointer position could take unlimited time. ButBohr applied a result from general relativity. According to the time-dilation formula of

61

general relativity, a clock in gravitational field ticks more slowly than a clock in free fall.Two clocks at different heights above the Earth will run at different rates, because oftheir gravitational potential difference. If the difference in height is ∆x, the fractionaldifference ∆t/t in their measured times will be

∆t

t=

g∆x

c2. (B.6)

If ∆x is the uncertainty in the vertical position of a clock, then ∆t is the uncertainty in theclock time due to the uncertain gravitational potential. Over a period t, the uncertaintyin the time of the clock amounts to

∆t =tg∆x

c2. (B.7)

Combining this result with Eq. (B.5), we obtain

∆t∆E ≥ h, (B.8)

as required by quantum theory.

Paper 2 (Salecker and Wigner [122]). Let us consider a linear clock model.

∆tT >

√~TE

, (B.9)

where ∆tT = ∆x/〈v〉 means that a clock measures a time after T , i.e., this is an accuracy.We call the first Salecker-Wigner (S-W) inequality.

E >~

2∆tT

√T

∆tT. (B.10)

We call the second S-W inequality.

We derive the first S-W inequality as follows. From the Heisenberg uncertainty rela-tionship (A.5), we have ∆x∆p ≥ ~/2, where ∆x is a variance in the clock position. Weassume that the spread in velocity ∆v = ∆p/M , where M is a mass of the clock, remains.Over time t, the variance of the position grows as

∆x2t = ∆x2

0 + t2∆v2 = ∆x20 +

~2t2

4M2∆x20

. (B.11)

Fixing the overall time T and minimizing over ∆x0 yields a clock accuracy

∆xT ≥√~TM

. (B.12)


Since 〈v〉 < c, Eq. (B.12) can be transformed to the desired inequality.We derive the second S-W inequality as follows. The reading of the clock is connected

with the emission of a light signal of duration ∆tT and this imparts to the clock anindeterminate momentum ~/c∆tT . This momentum wold be even greater if a particle ofnonzero rest mass were used as a signal. As a result of the emission of the light signal, thevelocity of the clock acquires a spread of the amount ~/Mc∆tT , where M is the mass ofthe clock. After a further time interval T2, it may be at a distance ~T2/Mc∆tT from thepoint where it would have been without having been read. Therefore, the actual distancebetween the two points in space time, at the first of which the clock read T1 less than atthe time of the emission of the signal, at the second of which it reads T2 more than at thetime of the emission of the light signal, is

√(T1 + T ′

2)2 −

(~T2

Mc2∆tT

)2

∼ T1 + T ′2 −

~2T 22

2M2c4(∆tT )2(T1 + T ′2)

, (B.13)

where

T ′2 =

√T 2

2 +

(~T2

Mc2∆tT

)2

. (B.14)

Hence, the actual distance (B.13) differs from the time difference T = T1 + T2 shown bythe clock, in the approximation considered, by

− T1T2

2(T1 + T2)

(~T2

Mc2∆tT

)2

. (B.15)

The inaccuracy of the clock will be within the limit ∆tT if Eq. (B.15) is less than ∆tT .If one considers the first order to be of the order of magnitude T , we obtain

M >~

c2∆tT

√T

∆tT, (B.16)

which is corresponded to the desired equation.Summing up the two S-W inequalities, the first inequality is restricted from the con-

stant light speed c and the second one is restricted from the causality.

Paper 3 (Margolus and Levitin [89]).

∆t ≥ π~2E

, (B.17)

where ∆t is defined as the time which can move from one state to an orthogonal statewith a fixed average energy, denoted as E.

63

This relation is derived as the follows. An arbitrary quantum state can be written asa superposition of energy eigenstates

|ψ0〉 =∑

n

cn|En〉. (B.18)

Note that we assume that a system has a discrete spectrum En and choose the groundenergy zero so that E0 = 0. If |ψ0〉 is evolved for a time t, then it becomes

|ψt〉 =∑

n

cne−iEnt/~|En〉. (B.19)

In order to judge the orthogonality for these states, |ψ0〉 and |ψt〉, we let

S(t) ≡ 〈ψ0|ψt〉 =∑

n

|cn|2e−iEnt/~. (B.20)

Since we want to solve the time which can move from one state to an orthogonal state,we want to find the smallest value of t such that S(t) = 0. To do this, we note that

Re(S(t)) =∑

n

|cn|2 cos

(Ent

~

)

≥∑

n

|cn|2(

1− 2

π

(Ent

~+ sin

(Ent

~

)))

= 1− 2E

π~t +

2

πIm(S(t)), (B.21)

where we have used the inequality cos x ≥ 1 − (2/π)(x + sin x) for x ≥ 0. On S(t) = 0,both Re(S(t)) = 0 and Im(S(t)) = 0, and so Eq. (B.21) becomes

0 ≥ 1− 2E

π~t. (B.22)

Then we obtain the desired inequality.

Paper 4 (Anandan and Aharonov [14], Vaidman [138]).

∆t ≥ π~2∆E

, (B.23)

where ∆t is defined as the time which can move from one state to an orthogonal state and∆E is defined as the variance of the energy.


In this thesis, Vaidman’s proof [138] is as follows. In general, for a given observableA, we can decompose

A|ψ〉 = α|ψ〉+ β|ψ⊥〉, (B.24)

where |ψ⊥〉 is orthogonal to |ψ〉 and β is real and non-negative. Then 〈A〉 ≡ 〈ψ|A|ψ〉 =〈ψ|(α|ψ〉 + β|ψ⊥〉) yields α = 〈A〉, and 〈ψ|A†A|ψ〉 = (α∗|ψ〉+ β∗|ψ⊥〉) (α|ψ〉+ β|ψ⊥〉)yields β =

√〈A2〉 − 〈A〉2 ≡ ∆A to obtain

A|ψ〉 = 〈A〉|ψ〉+ ∆A|ψ⊥〉. (B.25)

From this and the Schrodinger equation, we obtain

d

dt|ψ(t)〉 = − i

~H|ψ(t)〉 = − i

~(〈E〉|ψ(t)〉+ ∆E|ψ⊥(t)〉) , (B.26)

where 〈ψ(t)|ψ⊥(t)〉 = 0. Furthermore, we calculate

d

dt|〈ψ(t)|ψ(0)〉|2 = 2Re

(〈ψ(t)|ψ(0)〉〈ψ(0)| d

dt|ψ(t)〉

). (B.27)

From these, we obtain

d

dt|〈ψ(t)|ψ(0)〉|2 = −2∆E

~Re (i〈ψ(t)|ψ(0)〉〈ψ(0)|ψ⊥(t)〉) . (B.28)

Furthermore, we expand the initial state |ψ(0)〉 as

|ψ(0)〉 = 〈ψ(t)|ψ(0)〉|ψ(t)〉+ 〈ψ⊥(t)|ψ(0)〉|ψ⊥(t)〉+ α|ψ⊥⊥(t)〉, (B.29)

where 〈ψ(t)|ψ⊥⊥(t)〉 = 0 and 〈ψ⊥(t)|ψ⊥⊥(t)〉 = 0. The normalization of the quantumstates, then, requires that

|〈ψ(0)|ψ⊥(t)〉|2 = 1− |〈ψ(t)|ψ(0)〉|2 − |α|2. (B.30)

Therefore, the maximum value of |〈ψ(0)|ψ⊥(t)〉 is obtained for α = 0, and it is equal to√1− |〈ψ(0)|ψ(t)〉|2. Thus, the maximum possible absolute value of the rate of change of

the square of the overlap is

2∆E

~|〈ψ(0)|ψ(t)〉|

√1− |〈ψ(0)|ψ(t)〉|2. (B.31)

We find that this maximum rate, indeed, depends only on the value of the overlap and onthe energy uncertainty. Therefore, the condition for the fastest evolution to an orthogonalstate is that during the whole period of the evolution the right-hand side of Eq. (B.28) isequal to minus Eq. (B.31):

d

dt|〈ψ(t)|ψ(0)〉|2 = −2∆E

~|〈ψ(0)|ψ(t)〉|

√1− |〈ψ(0)|ψ(t)〉|2, (B.32)

65

After introducing a parameter φ, cos φ = |〈ψ(0)|ψ(t)〉|, Eq. (B.32) becomes

d

dtφ =

∆E

~. (B.33)

Since the orthogonal state corresponds to φ = π/2, the minimum time is, indeed

∆t =π

2φ=

h

4∆E. (B.34)

The relationship is desired.

Paper 5 (Mandelstam and Tamm [88]).

∆H∆T ≥ ~2, (B.35)

where ∆H is defined as an uncertainty of the energy and ∆T is defined as the shortesttime, during which the average value of a certain quantity is changed by an amount equalto the standard deviation of this quantity.

This equation is derived as the follows. Let R and S denote any two physical quantitiesand at the same time the corresponding symmetric operators. We have shown that usingthe Cauchy-Schwartz inequality,

∆S∆R ≥ 1

2|〈RS − SR〉|, (B.36)

where ∆S and ∆R are the standard deviations of the quantities S and R and 〈·〉 denotesas the average value, and the Heisenberg equation as

~∂〈R〉∂t

= i (〈HR−RH〉) . (B.37)

Putting in (B.36) S ≡ H, we obtain the inequality,

∆H∆R ≥ ~2

∣∣∣∣∂〈R〉∂t

∣∣∣∣ . (B.38)

The absolute value of an integral cannot exceed th integral of the absolute value of theintegrand. Hence, integrating (B.38) from t to t + ∆t and taking into account that ∆His constant one gets

∆H∆t ≥ ~2

|〈Rt+∆t〉 − 〈Rt〉|∆〈R〉 . (B.39)

This relation is desired.


Paper 6 (Aharonov and Bohm [4]).

∆H∆T ≥ ~2, (B.40)

where ∆H is defined as an uncertainty of the energy and ∆T is defined as the shortesttime, during which the average value of a certain quantity is changed by an amount equalto the standard deviation of this quantity. Aharonov and Bohm basically showed thatMandelstam and Tamm bound is universal as follows.

They define the ”clock” operator,

Tc =1

2

x,

1

p

, (B.41)

noting that this operator is singular. When the ”clock” Hamiltonian is given Hc = p2/2M ,we obtain the commutation relations,

[Hc, Tc] = i~. (B.42)

Essentially, there are three criteria on the time-energy relationship but meanings of∆t are different. About Margolus and Levitin’s paper and Anandan and Aharonov’spaper, this situation is that a certain state moves to the orthogonal state, i.e., we cantake two states with perfect distinguishable. Its minimum time is denoted as ∆t. AboutMandelstam and Tamm’s paper, this situation is that a certain state moves to the statesuch that we can two states with distinguishable for a given observable. Note that thesesituations restrict the Hamiltonian each other.

Appendix C

Optimal Covariant Measurement

For measuring processes, we shall consider an optimal measurement initiated by Hel-strom [57]. He defined an optimality of a measuring process to minimize the variancebetween an outcome of a measured system before the interaction and a measurementoutcome of a probe system after the interaction. The optimal measurement sets upperlimits to a POVM. In this chapter, we explicitly construct a model Hamiltonian whichreproduces the optimal POVM in a special case, while a general method is not availableto construct a measurement model from a given POVM.

C.1 Review of Optimal Covariant Measurement

Let us consider a measuring process described by an interaction between a measuredsystem and a probe system, the latter of which is the part of the measuring apparatusas a whole. To establish the relationship between the measured and probe systems, weconsider the momentum space Ω = R and a projective unitary representation of the shiftgroup of Ω. Stone’s theorem tells us that the unitary representation is given by

p → Vp = e−ipx, (C.1)

where x is the position operator.

Definition C.1. A POVM M(dp) is covariant with respect to the representation p → Vp

if

V †p M(∆)Vp = M(∆−p), p ∈ Ω (C.2)

for any ∆ ∈ A(Ω), where

∆p = p′|p′ = p + p′′, p′′ ∈ ∆ (C.3)

is the image of the set ∆ under the transformation p and A(Ω) is the Borel σ-field of Ω.

67

68 APPENDIX C. OPTIMAL COVARIANT MEASUREMENT

The covariant POVM has the property in the following form by using the Born for-mula [93, 63],

Prp ∈ ∆p‖ρp+p′0 = Tr ρp+p′0M(∆p) = Tr V−pρp′0V†−pM(∆p)

= Tr ρp′0V†−pM(∆p)V−p = Tr ρp′0M(∆)

= Prp ∈ ∆‖ρp′0. (C.4)

That is, when the measured system is arbitrarily shifted, the measurement outcome isshifted by the same amount. This idealized measurement is called a covariant measure-ment. Realistic measuring devices, however, satisfy this condition only locally as discussedby Hotta and Ozawa [66].

By von Neumann’s spectral theorem, any Hilbert space H can be formally describedas the direct integral of a Hilbert space Hx,

H =

∫⊕Hxdx, (C.5)

so that any state vector ψ ∈ H is described by the vector-valued function ψ = [ψx] withψx ∈ Hx introducing a convenient notation [ · ] [61, 63]. There, a position operator x actsas multiplication operators

xψ = [xψx] (C.6)

in this notation. The same notation [ · ] is used for an operator-valued function. A kernel[K(x, x′)], where K(x, x′) is a mapping fromHx′ toHx for all x and x′, defines an operatorK on H. We can write

Kψ = [K(x, x′)] [ψx′ ] =

[∫K(x, x′)ψx′dx′

]. (C.7)

Equations (C.6) and (C.7) can be rephrased by the bracket notation as

x|ψ〉 =

∫dx|x〉x〈x|ψ〉, (C.8)

K|ψ〉 =

∫dx

∫dx′|x〉K(x, x′)〈x′|ψ〉, (C.9)

respectively. Also we express the norm in Hx as ‖ · ‖x.We are now in a position to explicitly describe the covariant POVM as follows.

Theorem C.1 (Holevo [61]). Any covariant POVM in H has the form

M(dp) =

[K(x, x′)ei(x−x′)p dp

2π

], (C.10)

where [K(x, x′)] is a positive definite kernel satisfying K(x, x) ≡ Ix, the identity mappingfrom Hx to itself.

C.1. REVIEW OF OPTIMAL COVARIANT MEASUREMENT 69

In the above discussion, we have assumed that system and probe observables areisometric to obtain (C.10) as the POVM. The proof of Theorem C.1 is given as follows.

A measure ν is called invariant on A(Ω) if and only if for any p ∈ Ω there exists ameasure ν such that

ν(∆p) = ν(∆), (C.11)

where ∆ is an element of A(Ω) and ∆p is defined in (C.3).The following lemma is useful.

Lemma C.1. Let M(dp) be a covariant POVM with respect to a projective unitary rep-resentation p → Vp of the parametric group G of transformations of the set Ω. Forany density operator ρ on the Hilbert space of the representation and for any Borel set∆ ∈ A(Ω) ∫

Ω

Tr VpρV †p M(∆)µ(dp) = ν(∆) (C.12)

where µ(dp) is the proper Lebesgue measure and the extent Ω of the integral is a space ofparameters in G and ν is an invariant measure.

Proof. Define

1∆(p) =

1 on p ∈ ∆0 on p /∈ ∆

. (C.13)

Then we can write the left hand side of (C.12) as

∫

Ω

Tr VpρV †p M(∆)µ(dp) =

∫

Ω

Tr ρM(∆−p)µ(dp) =

∫

Ω

∫

Ω

1∆(p+p0)µρ(dp0)µ(dp), (C.14)

where µρ(dp0) ≡ Tr ρM(dp0) is the probability distribution of the momentum for the stateρ noting that the second integral on the rightmost side is over the whole momentum spaceΩ. We see that ∫

Ω

µρ(dp0)

∫

Ω

1∆(p + p0)µ(dp) = ν(∆). (C.15)

The following is the proof of Theorem C.1.

Proof. Let us assume ρ = |ψ〉〈ψ| without loss of generality. Then we see that

∫

Ω

VpρV †p

dp

2π=

[∫

Ω

eipxψx · ψ†x′e−ipx′ dp

2π

]=

[δ(x− x′)ψx · ψ†x

]. (C.16)

Noting that the operator M(∆) is defined by the kernel M∆(x, x′), we obtain from LemmaC.1 and Eq. (C.16) ∫

G

ψ†xM∆(x, x)ψxdx =mes∆

2π, (C.17)


where G is the parametric group and mes∆ denotes the Lebesgue measure of ∆. Since [ψx]is arbitrary, we see that M∆(x, x) = 1

2πmes∆ · Ix noting that Ix is the identity mapping

from Hx to itself. From the positive definiteness of M∆(x, x′), we can derive that

|ψ†xM∆(x, x′)ψx′| ≤√

ψ†xM∆(x, x)ψx

√ψ†x′M∆(x′, x′)ψx′ = ‖ψx‖x‖ψx′‖x′

mes∆

2π(C.18)

using the Cauchy-Schwartz inequality. Therefore, the measure M∆(x, x′) is absolutelycontinuous with respect to the Lebesgue measure, so that we can express

M∆(x, x′) =1

2π

∫

∆

Kp(x, x′)dp, (C.19)

with Kp(x, x′) being some positive definite density satisfying Kp(x, x) = Ix. From thecovariance properties, it follows that

Kp(x, x′) = ei(x−x′)pK0(x, x′). (C.20)

Putting K0(x, x′) = K(x, x′), we get (C.10) in Theorem C.1.Next we turn to a measuring process. First, we couple a measured system to a probe

system. Second, the combined system is evolved in time. Finally, we measure the probeobservable. The sequence of processes enables us to retrospectively evaluate the systemobservable at the starting time by the measurement outcome of the probe observableat the end time (See Fig. 2.1). So we define the optimal covariant measurement as anoptimal evaluation of the system observable by the outcome of the probe observable.

Let us assume that W (p − P ) is a deviation function, which expresses the variancebetween the inferred ”measurement” outcome p of the system momentum before theinteraction and the measurement outcome P of the probe momentum after the interaction,satisfying

W (p) = −∫

eipxW (dx), (C.21)

for an even finite measure W (dx) on R. Let us consider the condition to minimize thevariance

RpM =

∫

Ω

W (p− P )µρ(dp), (C.22)

where µρ(dp) ≡ Tr ρM(dp) is the probability distribution for the pure state ρ = |ψ〉〈ψ|.Because of covariance, we rewrite (C.22) as

R0M =

∫

Ω

W (p)µρ0(dp)

= −∫

Φρ(x)W (dx), (C.23)

C.1. REVIEW OF OPTIMAL COVARIANT MEASUREMENT 71

where

Φρ(x) ≡∫

Ω

eixp〈ψ|M(dp)ψ〉 (C.24)

is a characteristic function of µρ(dp). We get from Eq. (C.10)

Φρ(x) =

∫〈ψµ|K(µ, µ− x)ψµ−x〉dµ. (C.25)

Since the integral converges by the Cauchy-Swartz inequality and the condition K(x, x) =Ix,

ReΦρ(x) ≤ Φ∗(x) ≡∫‖ψµ‖µ‖ψµ−x‖µ−xdµ, (C.26)

so that

R0M ≥ −∫ ∫

‖ψµ‖µ‖ψµ−x‖µ−xdµW (dx)

≡ R0M0, (C.27)

where

M0(dp) =

[ψx · ψ†x′

‖ψx‖x‖ψx′‖x′ei(x−x′)p dp

2π

], (C.28)

by transforming µ− x to x′. Note that Eq. (C.28) does not depend on the choice of thedeviation function W (p − P ) because of the covariance. It is curious to point out thatthis POVM corresponds to the optimal POVM under the unbiased condition [54]. In thecase of the whole line system, the optimal covariant POVM (C.28) in the bracket notationexpresses

M0(dp) =

∫

Rdx

∫

Rdx′|x〉ei(x−x′)p dp

2π〈x′|, (C.29)

noting that the normalized termψx·ψ†x′

‖ψx‖x‖ψx′‖x′is the identity in the bracket notation. By

using the Fourier transformation,

|p〉 =1√2π

∫

Rdxeipx|x〉, (C.30)

Eq. (C.29) is transformed to the following equation:

M0(dp) = |p〉〈p|dp, (C.31)

to obtain the projective measurement of a momentum on a whole line. To summarizethe above discussion, we obtain the optimal covariant POVM (C.28) to minimize theestimated variance between the system and probe observables [61, 62]. We emphasizethat Eq. (C.28) remains valid even when we change the domain of x.


C.2 Optimal Measurement Model on a Whole Line

In the previous section, we have obtained the optimal covariant POVM. We are now goingto explicitly construct a Hamiltonian for a measurement model to realize the POVM.While it is straightforward to calculate the POVM and the probability distribution ofthe system observable for a given Hamiltonian of a combined system, it is not to find aHamiltonian from a given POVM. In the two dimensional case, there is a way to constructa model Hamiltonian from a given POVM [94]. Once the Hamiltonian for the combinedsystem is found, we can physically realize the given POVM in principle. In the infinitedimensional case, we heuristically explore the optimal covariant POVM for the momentumin measuring processes in the following way. In this section, we preparatively discussmeasurement of the momentum of a particle on a whole line and then apply the resultsto that on a half line in the next section. To make our exposition shorter, we assume thatthe wave functions ψx are normalized and the measure dp

2πis omitted in Eq. (C.28).

Then Eq. (C.28) is simply

M0 =[ψx · ψ†x′ei(x−x′)p

]. (C.32)

Let us consider a model Hamiltonian,

Hcom =1

2mp2 +

1

2MP 2 + gP xδ(t) +

mω2

2x2

≡ H0 + gP xδ(t), (C.33)

where a pair (x, p) are the position and the momentum operators of the measured sys-tem, a pair (X, P ) are those of the probe system and δ(t) is the Dirac δ-function. ThisHamiltonian is modeled from the following consideration. We take the potential of themeasured system as a harmonic oscillator for simplicity and the probe system is assumedto be a free particle system. Furthermore, the interaction is assumed to be instantaneouswith a coupling constant g. The interaction term gxP δ(t) is chosen by the following rea-soning. Because of the covariance, i.e., the measurement value P of the probe observablecorresponds to the ”measurement” value p of the system observable at a certain time, weare led to an interaction of the momentum P of the probe system. Since the exponentsin the optimal covariant POVM (C.32) has a quadratic form, a possible interaction termis either gxP or gpP . The latter is excluded because it does not influence the momentumof the measured system.

Furthermore, we assume that the measured system itself is weakly coupled to a bulksystem at zero temperature. We consider the measuring process from the time t = 0− to

C.2. OPTIMAL MEASUREMENT MODEL ON A WHOLE LINE 73

t = tf . Then the evolution operator U becomes

U = T exp

(−i

∫ tf

0−Hcomdt

)

= T exp

(−i

∫ tf

ε

H0dt

)exp

(−i

∫ ε

−ε

gP xδ(t)dt

)

= T exp

(−i

∫ tf

ε

H0dt

)exp

(−igP x(0)

), (C.34)

where ε is an infinitesimal positive parameter and T stands for the time-ordered product.We construct the Kraus operator[Axx′ ] from the evolution operator as follows. Given

the initial probe state |X = 0〉 =∫ |P 〉dP , an eigen state of the position X of the probe

system1, we see that

Axx′ =

∫〈P |〈x|U |x′〉|P 〉dP

=∑

j

〈x|T exp

(−i

∫ tf

ε

H0dt

)|j〉ψ†x′,j exp

(−igPx(0)

)

→ ψx · ψ†x′ exp(−igPx(0)

)as tf →∞, (C.35)

where |P 〉 is an eigen state of P , ψx,j is the wave function corresponding to the j-th energy

eigen state |j〉 and ψ = [ψx] is the ground state of the free Hamiltonian H0. In the lastline of (C.35), the ground state is picked up in the limit tf → ∞. Physically speaking,we measure the probe observable after sufficient time passes. Recall that the standard iεprescription [1] implicitly assumes that the measured system itself is weakly coupled tothe bulk system at zero temperature. Equation (C.35) is the matrix element of the Krausoperator [Axx′ ].

From the Kraus operator, we calculate the POVM as

M =

[∫A†

x′x′′Axx′′dx′′]

(C.36)

=[ψ†x′ · ψx exp

(−igPx(0)− x′(0)

)]. (C.37)

We identify gP with the measurement outcome P itself of the probe observable to repro-duce the optimal covariant POVM (C.32).

Now, we physically describe how we optimally infer the momentum of the measuredsystem just before the measuring process. First, we instantaneously couple the measured

1In our paper [126], we have taken the mistake about the setting the initial state of the probe system.However, we confirm the later discussion to calculate the POVM is right.


Figure C.1: An optimal covariant measurement model. By the instantaneous interactionbetween the measured and probe systems, the measured system is entangled with theprobe system. On the other hand, the measured system is coupled with the bulk systemat zero temperature to dissipate the energy of the measured system. Thus we optimallyevaluate the system observable at t = 0 inferred from the outcome of the probe systemat t = ∞ by the momentum conservation law.

system to the probe system. Second, we keep the measured system in contact with thebulk system at zero temperature and wait for a sufficiently long time. Since the energy ofthe measured system is dissipated to the bulk system, the state of the measured systemsettles down to the ground state. If we let the energy of the ground state zero, i.e.,ω → 0 of the interaction Hamiltonian (C.33), the momentum of the measured systempsys,∞ becomes zero at tf = ∞. According to the momentum conservation law, we obtain

psys,0 + pp,0 = psys,∞ + pp,∞ = pp,∞, (C.38)

where psys,t and pp,t are the momenta of the measured system and the probe system at atime t. Since we can control the probe system, we can precisely infer the ”measurement”value psys,0 of the momentum of the measured system at the beginning of the measuringprocess from the measurement outcome pp,∞, which we measure in the probe system attf = ∞ (See Fig. C.1). If ω of the Hamiltonian (C.33) were finite, the variance of themomentum of the measured system would remain finite due to the zero point oscillationand Eq. (C.38) would be modified.

Although we have assumed that the potential of the measured system is given bythe harmonic oscillator, the potential could actually be any convex function since the iεprescription picks up the ground state at tf →∞.

C.3. DISCUSSIONS 75

C.3 Discussions

The following point remains to be clarified. We have only discussed the covariant case.Peres and Scudo, however, pointed out that the covariant measurement may not be opti-mal and mentioned counterexamples in quantum phase measurement [109]. We have tocheck whether the optimality for any measurement is the optimal covariant measurementin our setup or not.

Appendix D

Quantum Mechanics on a Half Line

Let us recall that observables are defined as self-adjoint operators (See Postulate 1.). Ina general case of symmetric operators, we cannot apply the projection postulate sincesymmetric operators cannot generally be decomposed to real spectra, so that we have toconsider generalized measurement [33]. As an arch-typical example, we consider a halfline system R+ ≡ [0,∞) in quantum mechanics in this chapter. There have been manyworks concerning this problem since the beginning of quantum mechanics [113, 30, 40],e.g., the singular potential [26, 83, 49, 44]. Recently, Fulop et al. have studied boundaryeffects [42, 133, 43] and Twamley and Milburn have discussed a quantum measurementmodel on a half line by changing the coordinate x ∈ R+ to log x ∈ R [134] (See App. E).

D.1 Momentum Operator on a Half Line

According to the functional analysis, on which the mathematical foundation of quantummechanics [93] is based, an operator A is symmetric if A = A†, where A† is the Hermiteconjugate. Further, a symmetric operator A is self-adjoint if D(A) = D(A†), where D(A)is the domain of the operator A. In quantum mechanics, the observables are defined asself-adjoint operators, which have real spectra [13]. Symmetric operators, however, donot necessarily have a real spectrum. We need to classify symmetric operators into self-adjoint operators, essentially self-adjoint operators, self-adjoint extendable operators andnon-self-adjoint extendable operators1(for the definitions, see the book [13]). A criterionis known as the deficiency theorem (See App. F).

In the following consideration, we characterize the half line system as follows. Let us

1 Many physicists do not classify symmetric operators into self-adjoint operators and often call asymmetric operator Hermitian and identify a Hermitian operator with an observable without checking adomain of a operator (See, e.g., [123, 85]).

77

78 APPENDIX D. QUANTUM MECHANICS ON A HALF LINE

take a Hilbert space H+ ≡ L2(R+) and a momentum operator p+ in H+ defined by

p+ψ(x) =1

i

d

dxψ(x),

D(p+) =

ψ ∈ H+; ψ(0) = 0,

∫ ∞

0

∣∣∣∣d

dxψ(x)

∣∣∣∣2

dx < ∞

(D.1)

in analogy to the standard momentum operator on a whole line. Throughout this chapter,we take the unit ~ = 1.

Then we can see that p+ is symmetric since

〈φ|p+ψ〉 =1

i

∫ ∞

0

φ(x)d

dxψ(x)dx

=

[1

iφ(x)ψ(x)

]∞

0

− 1

i

∫ ∞

0

d

dxφ(x)ψ(x)dx

=

∫ ∞

0

1

i

d

dxφ(x)ψ(x)dx

= 〈p†+φ|ψ〉, (D.2)

ψ ∈ D(p+) φ ∈ D(p†+), (D.3)

where p†+ = 1i

ddx

with

D(p†+) =

ψ ∈ H+;

∫ ∞

0

∣∣∣∣d

dxψ(x)

∣∣∣∣2

dx < ∞

. (D.4)

Therefore we conclude that (p+,D(p+)) (p†+,D(p†+)) since D(p+) 6= D(p†+). So themomentum operator p+ on a half line is symmetric but not self-adjoint, i.e., not anobservable.

D.2 Optimal Measurement Model on a Half Line

Let us apply the optimal measurement model to the half line system. We have alreadyseen that the momentum operator (D.1) is not self-adjoint. First, we extend the domainof p+ a la Naimark so that the extended operator p is self-adjoint. The extended Hilbertspace is

H = H+ ⊗H2, (D.5)

where H ≡ L2(R), H+ ≡ L2(R+) and H2 is the two dimensional Hilbert space of the twolevel system with the orthonormal bases |0〉 and |1〉. We choose the form of the extendedmomentum operator as

p = p+ ⊗ |0〉〈0| − p+ ⊗ |1〉〈1|. (D.6)

D.2. OPTIMAL MEASUREMENT MODEL ON A HALF LINE 79

Figure D.1: A Naimark extension. An auxiliary two dimensional Hilbert space H2 istensored to the Hilbert space H+ to prepare the two (original and copied) Hilbert spaces.Then we spatially invert the copied Hilbert space around the zero point. Finally, wecombine the original and inverted Hilbert spaces to obtain the extended Hilbert space,H = H+ ⊗H2 = H+ ⊕H−.

By the unitary transformation Π1, which is the space inversion around the zero point onlyfor the spin state |1〉, the Hilbert space H is unitarily equivalent to

H = H+ ⊗ |0〉+H− ⊗ |1〉 = H+ ⊕H−, (D.7)

where H− ≡ L2(R−) and R− ≡ (−∞, 0]. Then we transform the extended momentumoperator (D.6) by Π1 as

Π1pΠ†1 = p+ ⊗ |0〉〈0|+ p− ⊗ |1〉〈1|, (D.8)

where p+ and p− are momentum operators, which have the following domains

D(p+) =

ψ ∈ H+ ; ψ(0) = 0,

∫ ∞

0

∣∣∣∣d

dxψ(x)

∣∣∣∣2

< ∞

D(p−) =

ψ ∈ H− ; ψ(0) = 0,

∫ 0

−∞

∣∣∣∣d

dxψ(x)

∣∣∣∣2

< ∞

, (D.9)

respectively. Then the extended operator p is self-adjoint extendable since the domain isthe Hilbert space for the whole line system. For a more precise argument, see App. F,where the choice of a boundary condition ψ(0) = 0 is also justified. These operations areexhibited in Fig. D.1. It is curious to point out that this operator p is PT symmetricnoting that the spin states |0〉 and |1〉 are interchanged by the time reversal T and themomentum operators p+ and p− by the parity inversion and the time reversal PT . Wecan see that the spectrum of p is real also from this reasoning [18].

80 APPENDIX D. QUANTUM MECHANICS ON A HALF LINE

We adopt the form of the model Hamiltonian (C.33) with p being replaced by the righthand side of (D.8) and x ∈ R, so that all the operators in the Hamiltonian (C.33) areself-adjoint to construct the optimal covariant measurement in the same way as describedin Sec. C.2. We, then, calculate the Kraus operator from the model Hamiltonian by usingthe iε prescription. Since we have chosen ψ(0) = 0, we end up with the ground state withodd parity with the energy 3

2ω. The Kraus operator is then

Π1[Axx′ ]Π†1 =

[ψx+ · ψ†x′+ exp (−igP+x+(0))

]⊗ |0〉〈0| (D.10)

+[ψx− · ψ†x′− exp (−igP−x−(0))

]⊗ |1〉〈1|. (D.11)

From Eq. (C.36), the Kraus operator (D.11) gives the following POVM,

Π1M0Π†1 =

[ψx+ · ψ†x′+ei(x+−x′+)p+

]⊗ |0〉〈0|+

[ψx− · ψ†x′−ei(x−−x′−)p−

]⊗ |1〉〈1|. (D.12)

By taking the partial trace over H2, we obtain the reduced POVM

M0 ≡ Tr2 M0

=[ψx+ · ψ†x′+ei(x+−x′+)p+

], (D.13)

up to a normalization constant. Here in Eq. (D.13), we have transformed (D.12) backto M0 by the unitary operator Π1 and reproduced the optimal covariant POVM (C.32)restricted to positive parameters x and x′.

Finally, we calculate the probability distribution of the momentum on a half line in

the optimal case. As an example, let us assume the pure state ρ =[φx+ · φ†x′+

], which is

a plane wave with a momentum ptrue,

φx+ = Aeiptruex+ , (D.14)

for the measured system before the measuring process. We assume that the state (D.14)is properly localized to be an element of the Hilbert space H+. The state (D.14),

[φx+

],

is relaxed by the measuring process to the ground state ψx+ ∈ H+ given by

ψx+ = 2

((mω)3

π

) 14

x+ exp(−mω

2x2

+

). (D.15)

Then we obtain the probability distribution of the momentum as

Tr(ρM0) = Tr([

φx′′+ · φ†x′+] [

ψx+ · ψ†x′′+ei(x+−x′′+)p])

=

∫ ∫φx′′+ · φ†x+

· ψx+ · ψ†x′′+ei(x+−x′′+)pdxdx′′

= 16

√π

(mω)3|A|2 (p− ptrue)

2 exp

(− 1

mω(p− ptrue)

2

), (D.16)

D.3. DISCUSSIONS 81

which has two peaks at p = ptrue±√

mω and vanishes at p = ptrue. If we take ω → 0, i.e.,the measured system is a free particle system, we can precisely evaluate the momentumof the plane wave since we obtain Tr(ρM0) = δ(p − ptrue). Otherwise there remainsuncertainty by quantum zero point oscillation and the momentum with the maximumprobability deviates by

√mω from the precise momentum ptrue. When the potential of

the measured system is a general convex function, the probability distribution for themomentum becomes the modulus square of the Fourier transformation of the odd parityground state wave function.

To summarize this section, we have obtained the optimal covariant POVM on a halfline, which enables us to explicitly construct the measuring process of the momentum ona half line.

D.3 Discussions

The following points remain to be clarified. First, Ozawa have recently constructed anew Heisenberg uncertainty principle [101, 103] (See Sec. A). The inequality expresses aquantum limit of measuring processes. It will be interesting to examine Ozawa’s inequalityin our framework. Second, there is an analogy between a momentum operator on a halfline and a time or time-of-arrival operator since a energy has a lower bound. However,there has been a long debate about mathematical formulations and physical meanings ofa time or time-of-arrival operator (for example [63, 4, 17, 15, 59]). It will be interesting toshow physical meanings of this operator motivated by our framework. Third, we have onlydiscussed one example of the Naimark extension, that is, the boundary condition at theorigin is ψ(0) = 0. Physically, we can construct the generalized boundary conditions at theorigin parameterized by U(1). Furthermore, to experimentally demonstrate and certificatethe generalized boundary conditions, experimental procedures remain to be considered.Finally, we have presented the model Hamiltonian (C.33) to physically realize the optimalcovariant POVM (C.28). We do not know a general method to construct a Hamiltonianfrom an arbitrary POVM. Our analysis may be a clue to the general method to solve theinverse problem. Furthermore, to experimentally demonstrate the measurement model,experimental setups remain to be considered for our proposed model Hamiltonian.

Appendix E

Mellin Transform

E.1 Definition of Mellin Transform

We refer the reader to the book [32]. Let us recall that the Fourier transform pair can bewritten in the form

A(ω) =

∫ ∞

−∞a(t)eiωtdt, α < Im(ω) < β, (E.1)

and

a(t) =1

2π

∫ iγ+∞

iγ−∞A(ω)e−iωtdω, α < γ < β. (E.2)

The Mellin transform and its inverse follow if we introduce the variable changes;

p = iω,

x = et,

f(x) = A(ln x), (E.3)

so that Eqs. (E.1) and (E.2) become

Mf(p) = F (p) =

∫ ∞

0

xp−1f(x)dx, α < Re(p) < β, (E.4)

M−1F(x) = f(x) =1

2πi

∫ c+i∞

c−i∞x−pF (p)dp. (E.5)

Equation (E.4) is the Mellin transform, and Eq. (E.5) is the Mellin inverse formula. Thetransform normally exists only in the strip α < Re(p) < β, and the inversion contourmust lie in this strip.

83

84 APPENDIX E. MELLIN TRANSFORM

As a typical example, we assume the follows,

f(x) = (1 + βx)−γ, γ > 0, | arg β| 6= π (E.6)

F (p) =

∫ ∞

0

xp−1

(1 + βx)γdx = β−p

∫ ∞

0

yp−1

(1 + y)γdy (E.7)

The substitution y = z/(1− z) reduces the integral to the standard form

F (p) = β−p

∫ ∞

0

zp−1(1− z)γ−p−1dz = β−p (p− 1)!(γ − p− 1)!

(γ − 1)!, (E.8)

where for the integral to converge, we must have 0 < Re(p) < γ. The inversion formulathen gives us

(γ − 1)!f(x) =1

2πi

∫ c+i∞

c−i∞(p− 1)!(γ − p− 1)!(βx)−pdp, (E.9)

where the contour separates the two sets of poles.

E.2 Riemann-Zeta Function

The class of transcendental functions we consider the three parameter family of highertranscendental functions sometimes known as the Lerch transcnedents Φ(z, s, u), whichcontain the Riemann-Zeta function ζ(s). The Lerch transcendent is given as the analyticcontinuation of the series,

Φ(z, s, u) =∞∑

n=0

zn

(u + n)s, |z| ≤ 1, u 6= −1,−2,−3, . . . , (E.10)

which converges for u ∈ R+, z, s ∈ C, with either (|z| < 1, Re(s) > 0), or (|z| = 1, Re(s) >1). Special cases include the analytic continuations of the Riemann-Zeta function, forRe(s) > 1,

ζ(s) =∞∑

k=1

1

ks= Φ(1, s, 1). (E.11)

E.3 Hyperbolic Phase Space

In this section, we define a new space via the Mellin transformation as the momentumoperator is self-adjoint [134]. Let us examined the operator which generates dilations of

E.3. HYPERBOLIC PHASE SPACE 85

the x operator,

pη ≡ 1

2(xp+ + p+x) = xp+ − i

~2

(E.12)

= i~(

xd

dx+

1

2

). (E.13)

where the last line is pη evaluated in the x representation. We consider the operatorpη = xp++ p+x, to be a new conjugate variable describing dilation/conformal momentum.From [x, pη] = i~x, the Sack algebra [120], we see that the action of this new momentumon x, is as expected

eiµpη/~xe−iµpη/~ = xeµ. (E.14)

From the short calculation, we obtain the deficiency indices (n+, n−) = (0, 0), indicatingthat pη is an self-adjoint operator. Defining η = ln x, we can recover the standard Heisen-berg algebra and displacement operation, albeit in the ’exponent space’: [η, pη] = i~, and

eiαpη/~xe−iαpη/~ = η + α. (E.15)

Thus by making the unitary transformation of the quantum mechanics on a half linedescribed by the ’conjugate operators’, (x, pη), to the exponential representation on awhole line, (ln x, pη), we can regime the familiar Heisenberg algebra. Dilations of x nowbecome displacements of η ≡ ln x, and one can formulate all of the conventional quan-tum mechanics on R+, in this hyperbolic phase space. The eigenstates of pη by taking〈x|pη|pη〉 = pη〈x|pη〉, and where ψpη(x) = V pη(x)/

√x, is obtained as

〈x|pη〉 =1√2π

xipη/~−1/2. (E.16)

Wee see that the representation of a given wavefunction 〈x|ψ〉, in the |pη〉, basis is,

〈pη|ψ〉 =

∫ ∞

0

〈pη|x〉〈x|ψ〉 (E.17)

=1√2π

∫ ∞

0

dxψ(x)x−1/2−ipη (E.18)

=1√2πMψ(s = 1/2− ipη). (E.19)

This corresponds to the Lerch transcendent, we obtain that

Ξ

(z, s =

1

2− ipη, u

)≡ Γ(s =

1

2− ipη)Φ(z, s =

1

2− ipη, u) (E.20)

=

∫ ∞

0

dte−(u−1)t

et − zt−1/2−ipη (E.21)

= Mχ(z, s = 1/2− ipη, u), (E.22)

86 APPENDIX E. MELLIN TRANSFORM

where

χ =e−(u−1)t

et − z(E.23)

Thus, to simulate the three parameter family of Lerch transcendents, we take the Mellintransform of χ(z, t, u), evaluated on the critical line s = 1

2− ipη.

Appendix F

Deficiency Theorem

We refer the reader to the book [112] and the paper [21] for details. We shall give acriterion for closed symmetric operators to be self-adjoint operators.

Definition F.1 (Deficiency indices). Let (A,D(A)) be a densely defined, symmetric andclosed operator. One defines the deficiency subspaces N± by, for a fixed γ > 0,

N+ = ψ ∈ D(A†) ; A†ψ = iγψ (F.1)

N− = ψ ∈ D(A†) ; A†ψ = −iγψ (F.2)

of respective dimensions n+ and n−, which are called the deficiency indices of the operatorA and denoted by a pair (n+, n−).

The following theorem holds.

Theorem F.1 (Deficiency theorem). For any closed symmetric operator A with deficiencyindices (n+, n−), there are three possibilities:

1. A is self-adjoint if and only if n+ = n− = 0.

2. A has self-adjoint extensions if and only if n+ = n−. There exists one-to-onecorrespondence between self-adjoint extension of A and unitary maps from N+ toN−.

3. If n+ 6= n−, A has no self-adjoint extension.

This theorem is firstly discussed by Weyl [141] and generalized by von Neumann [92].

Proof. The following proposition immediately follows the proof.

Definition F.2. λ ∈ C is called a point of regular type of the operator A if there existsk = k(λ) > 0 such that for all |ψ〉 ∈ D(A),

||(A− λI)|ψ〉|| ≥ k. (F.3)

87

88 APPENDIX F. DEFICIENCY THEOREM

From this definition, it follows that λ is a point regular type of the operator A if andonly if (A− λI)−1 exists and is bounded.

Propositon F.1. If Γ is a connected subset of the field of regularity of a linear operatorA, then the dimension of the subspace H ªD(A−1)(λ) is the same for each λ ∈ Γ.

Proof. Let Pλ denote a projection operator on the subspace

Rλ := H ªD(A−1)(λ). (F.4)

If we show that for arbitrary λ1, λ2 ∈ Γ,

||Pλ1 − Pλ2|| < 1, (F.5)

then it follows that the dimensions of the subspaces Rλ1 and Rλ2 are equal. In order toprove the inequality (F.5) for arbitrary λ1, λ2 ∈ Γ, it is sufficient, by using the Heine-Boreltheorem, to show that for each λ0 ∈ Γ there exists δ = δ(λ0) > 0 such that

||Pλ − Pλ1|| < 1, (F.6)

for |λ − λ1| < δ. Thus, let λ0 be an arbitrary point of the region Γ and let δ = δ(λ0) ≤k(λ0)/3. Since

k(λ0) ≤ ||(A− λ0I)|ψ〉|| ≤ ||(A− λ0I)|ψ〉||+ |λ− λ0|, (F.7)

we have, for |λ− λ0| ≤ δ,

||(A− λI)|ψ〉|| ≥ 2

3k(λ0). (F.8)

Then, for each |φ〉 ∈ Rλ,

||(I − Pλ0)|φ〉|| = sup|ψ〉∈D(A)

|〈φ|(A− λ0I)|ψ〉|||(A− λ0I)|ψ〉||

= sup|ψ〉∈D(A)

|〈φ|(A− λ0I)|ψ〉+ (λ− λ0)〈φ|ψ〉|||(A− λ0I)|ψ〉||

= sup|ψ〉∈D(A)

|λ− λ0| · |〈φ|ψ〉|||(A− λ0I)|ψ〉|| ≤

1

2, (F.9)

and, for each |φ〉 ∈ Rλ0 ,

||(I − Pλ)|φ〉|| ≤ 1

2. (F.10)

From Eqs. (F.9) and (F.10), we obtain

||Pλ − Pλ0|| ≤1

2. (F.11)

89

Thus, the proposition is proved.Therefore, we prove the deficiency theorem.Let us apply this theorem to the momentum operator (D.1) on a half line. First, we

solve the differential equations,

p+ψ±(x) = −id

dxψ±(x) = ±iγψ±(x), (F.12)

where γ is real and positive to obtain

ψ±(x) ∼ e∓γx. (F.13)

Because of ψ ∈ L2(R+), only ψ+(x) is allowed. Therefore, we obtain the deficiency indices(1, 0) and conclude, by the deficiency theorem, p+ has no self-adjoint extension.

As another example, we show that the extended momentum operator (D.6) is self-adjoint extendable. We obtain the deficiency indices (0, 1) of −p+ in the same way. So thedeficiency indices of the extended momentum operator (D.6) are (1, 1) and the operatoris self-adjoint extendable by the deficiency theorem. Since the self-adjoint extension isparametrized by U(1), ψ(0+) = eiθψ(0−) where θ ∈ R, we have a freedom to choose theboundary conditions at the origin by that amount. The boundary condition ψ(0) = 0chosen in the main text, which comes from the physical requirement to the half linesystem, is mathematically legitimate in the extended system because it is a special caseof the U(1) variety.

Appendix G

Nitrogen Vacancy Center inDiamond

A nitrogen vacancy center, NV center, in a diamond (Fig.G.1) is studied as a promisingcandidate for a qubit to realize quantum information processing since its electron spinstate has the long coherence time at room temperature and can be optically initializedand read out (Fig. G.2) [73, 37, 52]. However, there have been only limited methodsand interpretation to understand its properties. Our aim is to apply weak measurementadvocated by Aharonov et al. [2] to this system as a new measurement scheme. Weexpect that weak measurement has possibility to determine properties of the NV centermore precisely (e.g. see [65] in linear optics).

Experimentally, we have constructed the confocal laser scanning microscope (See Fig.G.3) and observed some fluorescent spots from nitrogen vacancy centers in type Ib di-amond (Fig.G.4). Measuring an intensity correlation function to each observed spot,we identified whether its fluorescence came from a single or an ensemble of NV centers.Applying microwave to the single NV center as an oscillating magnetic field by flowingAC current through the thin gold wire, we obtained the electron spin resonance (ESR)signal as change in fluorescence intensity, which is called the optically detected magneticresonance (ODMR) showing that the single electron spin was interacted with the appliedexternal field.

G.1 New Scheme to Measure Spin-Lattice Time

We are preparing to do the pulse ODMR experiment in which the electron spin stateis initialized with laser pulse excitation, manipulated with proper sequence of pulsedmicrowave, and read out by irradiating laser pulse again. By arranging this laser andmicrowave pulse sequence, the idea of weak measurement is adapted. We propose thissequence as follows. First, we measure the electron spin state by applying two microwave

91

92 APPENDIX G. NITROGEN VACANCY CENTER IN DIAMOND

π with the interval of a fixed time τtar after the initializing laser pulse. Second, we repeatthe measurement putting a weak coherent light between two π pulses at a time τ . Thisprocess is corresponding to weak measurement and slightly changes the electron spin statein Bloch sphere. We illustrated them in Fig.G.5. Applying this scheme, we arbitrarilyadd an auxiliary noise by the weak coherent light and infer an electron spin state at τ .For example, we extrapolate the spin relaxation time T1 from the fidelity to the spin stateat τtar without the weak coherent light.

G.2 Experimental Realization of Weak Measurement

in a Solid State System

We are going back to our original aim to apply weak measurement advocated by Aharonovet al. [2] to this system as a new measurement scheme. We would like to first experimen-tally demonstrate the weak value of some observable in a solid state1.

G.2.1 Weak Measurement using NV Electron and 13C NuclearSpins

We assume that the single NV electron and the 13C nuclear spins be target system andprobe, respectively. We define a ”MAP” and ”INT” operations as the follows. The 13Cnuclear spin is located near the single NV electron spin using the Type II-a diamond.The hyperfine energy level of the grand state is almost same between |0〉e|0〉n and |0〉e|1〉nbut different between |1〉e|0〉n and |1〉e|1〉n. First of all, we take the laser to the object topick up the electron state to |0〉e to obtain the state a|0〉e|0〉n + b|0〉e|1〉n. Taking the πpulse of the microwave with the bit-flip between |0〉e|0〉n and |1〉e|0〉n, we obtain the statea|1〉e|0〉n + b|0〉e|1〉n. Waiting the time π/ωn where ωn is the bare 13C Larmor frequency,we obtain the state a|1〉e|0〉n + b|0〉e|0〉n, which is called a ”MAP” operation correspondedto the SWAP operation between the NV electron and nuclear spins. Furthermore, we takethe laser to pick up the electron state |0〉 to finally obtain the state |0〉e|0〉n. First, werepeat the ”INT” operation due the imperfect single operation to initialize the electronand nuclear spin states, |0〉e|0〉n and manipulate the electron spin state |ψ〉 by irradiatingthe microwave. The electron and nuclear spins take free time evolution by the time t0,which is shorter than T2 time of the electron spin. Then, we pick up the post-selectedstate |φ〉 := |0〉e by irradiating the coherent light. The interaction Hamiltonian betweenthe electron and nuclear spins [29, Eqs. (1)-(19) in Supporting Online Material] is given

1As far as the author knows, there has already been the proposal in the quantum dot using thequantum point contact [143].

G.3. DISCUSSIONS 93

by

Hint =∑

µ,ν=x,y,z

αµνSµIν , (G.1)

which is corresponded to the weak measurement since the coupling constant αµν is givenby

αµν = µeµn

[(−8π

3|ψ(r)|2 + 〈 1

|r− r0|3 〉)

δµν − 3〈 nµnν

|r− r0|3 〉]

(G.2)

is very small where ψ(r) is the wave function of the NV electron and r is the positionof the nearest neighborhood nuclear spin and r0 is the position of the NV electron spin.This is included in the decoherence effect due to the term Sx · Iz etc. Furthermore, theHamiltonians for the electron and nuclear spins with the external magnetic field Bz are

HNV = ∆S2z − µeBzSz, (G.3)

HB = µnBz · Iz, (G.4)

where the hyperfine coupling constant ∆ is 2.88 GHz and the external field Bz is experi-mentally 50G. Since the electron and nuclear spins take the dipole-dipole interaction, thenuclear spin state rotates around the electron spin. After that, we take the ”MAP” oper-ation to transfer the nuclear spin state to the electron spin. Furthermore, we irradiate thecoherent light and detect the photon counting to obtain the rotation angle of the nuclearspin state. This angle is corresponded to the weak value of σz for the electron spin withdecoherence.

The total Hamiltonian is given by

Htot = HNV ⊗ IN + Ie ⊗HB + Hint, (G.5)

where we ignore the decoherence effect for the nuclear spin. There remains the problemwhether we can experimentally measure quantities of the tilting nuclear spin or not.

G.3 Discussions

In Sec. G.1, we propose the measurement sequence of measuring the T1 time. However,the evaluation method has been not clear. In Sec. G.2, we have to show whether theamount of the weak value can be experimentally accessible or not.


Figure G.1: Structure of a single NV center in a diamond lattice.

Figure G.2: Scheme of electronic and spin energy levels of NV centers. Arrows indicatespin-selective excitation and fluorescence emission pathways.

G.3. DISCUSSIONS 95

Objective Lens

YAG Laser

Fluorescence

SM fiber

Signal Generator

Cu Wire （25um）SPCM

AOM

Diamond Sample

＋ＰＺＴ Stage

PIN-Switch

AWG

Microwave

Figure G.3: Diagram of the experimental approach. Single NV centers are imaged usingthe confocal microscopy. The position of the focal point is moved to change a piezo-drivenobjective mount [77].

Figure G.4: Spatial fluorescence image of a part of a diamond sample, showing brightspots corresponding to single NV centers [77].


Figure G.5: Measurement sequence for measuring the spin lattice time of a single NVelectron spin.

List of Publications

• Paper

1. Yutaka Shikano and Akio Hosoya,

”Optimal Covariant Measurement of Momentum on a Half Line in QuantumMechanics”

Journal of Mathematical Physics 49, 052104 (2008).

e-Print: arXiv:0710.1724.

• Preprint


”Weak Values with Decoherence”

e-Print: arXiv:0812.4502.

• Oral Presentations


”Quantum Operations for Weak Values”

JPS 2009 Annual Meeting at Rikkyo University (2009.3.30).

2. Yutaka Shikano,

”Stochastic Process in Quantum Mechanics via Two-State Vector Formalism”

MCYR5 at Hokkaido University (2009.3.4).

3. Seth Lloyd, Akio Hosoya, and Yutaka Shikano,

”Optimal Quantum Covariant Clock”(invited)

DEX-SMI Workshop on Quantum Statistical Inference at NII, Tokyo (2009.3.2).


”Theory of ”Weak Measurement””

RIMS Research Meeting ”Micro-Macro Duality in Quantum Analysis” at Ky-oto University (2008.11.6).

97

http://arxiv.org/abs/0710.1724

http://arxiv.org/abs/0812.4502

98 LIST OF PUBLICATIONS



JPS 2008 Annual Meeting at Kinki University (2008.3.26).

6. Yutaka Shikano,


CCYR4 at Hokkaido University (2008.2.12).


”How to Optimally Measure a Momentum on a Half Line in Quantum Me-chanics”


• Oral Presentations (Another Presenter)

1. Sota Kagami, Yutaka Shikano, Shu Tanaka, and Koichiro Asahi,

”Detection and Manipulation of a Single Electron Spin of Nitrogen VacancyCenter in Diamond and its Application of Weak Measurement”

JPS 2009 Annual Meeting at Rikkyo University (2009.3.30).

2. Seth Lloyd, Akio Hosoya, Yutaka Shikano, Vittorio Giovannetti, Lorenzo Mac-cone, and Mankei Tsang,

”Optimal Quantum Clock”


• Poster Presentations


”Framework of Waak Measurement with Noise”(refereed)

nanoPHYS 2009 at International House of Japan (2009.2.23).

2. Yutaka Shikano and Sota Kagami,

”Measurement of Spin-Lattice Relaxation Time of Nitrogen Vacancy Center inDiamond via Weak Measurement”(refereed)

International Symposium on Physics of Quantum Technology at Nara Prefec-tual Hall (2008.11.25).

3. Yutaka Shikano,

”On Weak Measurement”

99

Fourth Quantum Information Future Theme Meeting at Hotel Sunrise Chinen,Okinawa (2008.9.2 and 8).


”Optimal Covariant Measurement of Momentum of a Particle in QuantumMechanics”(refereed)

AQIS2008 at KIAS, South Korea (2008.8.27).



QCMC2008 at University of Calgary, Canada (2008.8.20).

6. Yutaka Shikano,

”Relationship between Optimal Measurement and Weak Measurement”

KEK Research Meeting ”Recent Issues and Prospects on Quantum Theory” atKEK, Tsukuba (2008.3.20).


”How to Optimally Measure a Momentum on a Half Line in Quantum Me-chanics”(refereed)

TQC2008 at Hongo Campus, University of Tokyo (2008.1.30).



Third Open Symposium ”Nanometer-scale Quantum Physics” at Gotanda,Tokyo (2007.12.20).


”How to Optimally Measure a Momentum on a Half Line in Quantum Me-chanics”

QIT17 at Sanyo Shinbun SANTA Hall in Okayama (2007.11.21).

10. Yutaka Shikano,

”Quantum Measurement Theory on a Half Line”

The 52nd Condensed Matter Physics Summer School Poster Session at Kimi-idera Garden Hotel Hayashi (2007.8.7-8).

• Poster Presentation (Another Presenter)

1. Sota Kagami, Yutaka Shikano, and Koichiro Asahi,

”Detection, Manipulation of a Single Electron Spin of Nitrogen Vacancy Centerin Diamond and its Application of Weak Measurement”(refereed)

nanoPHYS 2009 at International House of Japan (2009.2.24).

100 LIST OF PUBLICATIONS

2. Sota Kagami and Yutaka Shikano,

”Detection and Manipulation of a Single Electron Spin of Nitrogen VacancyCenter in Diamond and its Application of Weak Measurement”(refereed)

ISNTT 2009 at NTT Atsugi R & D Center (2009.1.21).

• Seminar Presentations

1. Ozawa Lab. Seminar at Nagoya University (2009.2.24).

”Weak Values Theory — Definitions and Concepts —”.

2. NEC Nano Electronics Lab. Seminar at NEC Nano Electronics Lab. (2009.1.6).

”Consistent Framework of Weak Values with Decoherence” (in English).

3. Theoretical Astrophysics Group Colloquium at Tokyo Institute of Technology(2008.12.2)

”Weak Values in Quantum Measurement Theory — Concepts and Applications—”.

4. Ohba and Nakazato Lab. Seminar at Waseda University (2008.11.13)

”Quantum Operation of Weak Values”.

5. Konno and Kajihara Lab. Seminar at Yokohama National University (2008.10.28).

”Theory of ”Weak Measurement””.

6. Asahi Lab. Seminar at Tokyo Institute of Technology (2008.6.26)

”Introduction to Weak Measurement -Mysterious Weak Values’ World-”.

7. Theoretical Astrophysics Group Colloquium at Tokyo Institute of Technology(2008.4.22)

”Seeking the SIC POVM”.

8. Ozawa Lab. Seminar at Tohoku University (2007.4.20)

”Quantum Measurement Theory on a Half Line”.

• Other Activities

1. Co-organizer: Titech/KEK Workshop: Measurement and Physical Reality inQuantum Mechanics at Tokyo Institute of Technology (2009.3.16-17.) co-organized and supported by KEK.

2. Co-Organizer: Kanto Quantum Student Chapter (Chief Organizer: The 3rdKanto Quantum Student Chapter Poster Presentations at Tokyo Institute ofTechnology (2008.8.8)).

3. Chief Organizer: International Mini-Workshop : Theoretical Foundations andApplications of Quantum Control at Tokyo Institute of Technology (2008.7.11)organized and supported by Interactive Research Center of Science, TokyoInstitute of Technology.

101

4. Chief organizer: Colloquium for Department of Particle-, Nuclear-, and Astro-Physics at Tokyo Institute of Technology (2007.5.25).

Bibliography

[1] E. S. Abers and B. W. Lee, Phys. Rep. 9, 1 (1973).

[2] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).

[3] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964).

[4] Y. Aharonov and D. Bohm, Phys. Rev. 122, 1649 (1961).

[5] Y. Aharonov and A. Botero, Phys. Rev. A 72, 052111 (2005).

[6] Y. Aharonov, S. Popescu, D. Rohrlich, and L. Vaidman, Phys. Rev. A 48, 4084(1993).

[7] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, arXiv:0712.0320.

[8] Y. Aharonov and D. Rohrlich, Quantum Paradoxes (Wiley-VCH, Weibheim, 2005).

[9] Y. Aharonov and J. Tollaksen, to appear in Visions of Discovery: New Light onPhysics Cosmology and Consciousness, edited by R. Y. Chiao, M. L. Cohen, A.J. Leggett, W. D. Phillips, and C. L. Harper, Jr. (Cambridge University Press,Cambridge, in press), arXiv:0706.1232.

[10] Y. Aharonov and L. Vaidman, Phys. Rev. A 41, 11 (1990).

[11] Y. Aharonov and L. Vaidman, J. Phys. A 24, 2315 (1991).

[12] Y. Aharonov and L. Vaidman, in Time in Quantum Mechanics, Vol. 1, edited by J.G. Muga, R. Sala Mayato, and I. L. Egusquiza (Springer, Berlin Heidelberg, 2008)pp. 399-447.

[13] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space(Dover, New York, 1993).

[14] J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990).

[15] A. Arai and Y. Matsuzawa, Lett. Math. Phys. 83, 201 (2008).

103

104 BIBLIOGRAPHY

[16] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger,Nature 401, 680 (1999).

[17] A. D. Baute, I. L. Egusquiza, J. G. Muga, and R. Sala Mayato, Phys. Rev. A 61,052111 (2000).

[18] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).

[19] N. Bohr, Nature 121, 580 (1928).

[20] N. Bohr, in Albert Einstein: Philosopher-Scientist, edited by P. A. Schilpp (Cam-bridge Univ. Press, Cambridge, 1949). pp.199-242.

[21] G. Bonneau, J. Faraut and G. Valent, Am. J. Phys. 69, 322 (2001).

[22] V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge UniversityPress, Cambridge, 1992).

[23] N. Brunner, A. Acın, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91,180402 (2003).

[24] M. Brune, J. Bernu, C. Guerlin, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, I.Dotsenko, J. M. Raimond, and S. Haroche, arXiv:0809.1511.

[25] P. Busch, P. Mittelstaedt, and P. J. Lahti, Quantum Theory of Measurement(Springer-Verlag, Berlin, 1991).

[26] K. M. Case, Phys. Rev. 80, 797 (1950).

[27] C. Caves, Rev. Mod. Phys. 52, 341 (1980).

[28] C. Caves, Phys. Rev. Lett. 54, 2465 (1985).

[29] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J.Wrachtrup, P. R. Hemmer, and M. D. Lukin, Science 314, 281 (2006).

[30] T. E. Clark, R. Menioff and D. H. Sharp, Phys. Rev. D 22, 3012 (1980).

[31] G. M. D’Ariano, to appear in Philosophy of Quantum Information and Entangle-ment edited by A. Bokulich and G. Jaeger (Cambridge University Press, Cambridge)arXiv:0807.4383.

[32] B. Davies, Integral Transforms and their Applications (Springer-Verlag, New York,1978).

[33] E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).

BIBLIOGRAPHY 105

[34] D. Deutsch, Proc. R. Soc. Lond. A 400, 97 (1985).

[35] D. Deutsch, talk in ”UK-Workshop 2009 on Quantum Information” at British Em-bassy in Japan on 1.22.2009.

[36] P. A. M. Dirac, Proc. Roy. Soc. A 113, 621 (1926).

[37] M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S.Zibrov, P. R. Hemmer, and M. D. Lukin, Science 316, 1312 (2007).

[38] A. Einstein, Ann. Phys. (in Berlin) 360, 241 (1918).

[39] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).

[40] E. Farhi and S. Gutmann, Int. J. Mod. Phys. A 5, 3029-3051 (1990).

[41] R. P. Feynman, Int. J. of Theor. Phys. 21, 467 (1982).

[42] T. Fulop, T. Cheon and I. Tsutsui, Phys. Rev. A 66, 052102 (2002).

[43] T. Fulop, Ph.D. thesis, University of Tokyo, 2005.

[44] T. Fulop, arXiv:0708.0866.

[45] A. Furuta, talk in ”Recent Issues and Prospects on Quantum Theory” at KEK, on3.22.2008.

[46] C. F. Gauss, Theoria Combinationis Observationum Erroribus Minimis Obnoxiae(Apud Henricum Dieterich, Gottingae, 1823).

[47] P.-L. Giscard, arXiv:0901.0333.

[48] S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deleglise, U. B. Hoff, M. Brune, J. M.Raimond, and S. Haroche, Nature 446, 297 (2007).

[49] A. N. Gordeyev and S. C. Chhajlany, J. Phys. A 30, 6893 (1997).

[50] M. J. W. Hall, Phys. Rev. A 64, 052103 (2001).

[51] M. J. W. Hall, Phys. Rev. A 69, 052113 (2004).

[52] R. Hanson et al., Science 320, 352 (2008).

[53] S. Haroche and J,-M. Raimond, Exploring the Quantum: Atoms, Cavities, andPhotons (Oxford Univ. Press, Oxford, 2006).

[54] M. Hayashi and F. Sakaguchi, J. Phys. A 33, 7793 (2000).

106 BIBLIOGRAPHY

[55] W. Heisenberg, Z. Phys. 43, 172 (1927).

[56] W. Heisenberg, The Physical Principles of the Quantum Theory (University ofChicago Press, Chicago, 1930; Dover, New York, 1949, 1967).

[57] C. W. Helstrom, Int. J. Theor. Phys. 11, 357 (1974).

[58] P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A.J. Miller, and J. E. Nordholt, New J. Phys. 8 193 (2006).

[59] F. Hiroshima, S. Kuribayashi, and Y. Matsuzawa, arXiv:0810.2350.

[60] H. F. Hofmann, arXiv:0805.0029.

[61] A. S. Holevo, Rep. Math. Phys. 13, 379 (1978).

[62] A. S. Holevo, Rep. Math. Phys. 16, 385 (1979).

[63] A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North-Holland,Amsterdam, 1982).

[64] A. S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2001).

[65] O. Hosten and P. Kwiat, Science 319, 787 (2008).

[66] M. Hotta and M. Ozawa, Phys. Rev. A 70, 022327 (2004).

[67] R. A. Hulse and J. H. Taylor, Astrophysical Journal 195, L51 (1975).

[68] C. J. Isham, Lectures on Quantum Theory – Mathematical and Structural Founda-tions – (Imperial College Press, London, 1995).

[69] S. Ishii, Heisenberg’s microscopy (Nikkei Business Publication, Tokyo, 2006) (inJapanese).

[70] S. Ishikawa, Int. J. Theor. Phys. 30, 401 (1991).

[71] S. Ishikawa, Rep. Math. Phys. 29, 257 (1992).

[72] K. Ito, An Introduction to Probability Theory (Cambridge University Press, Cam-bridge, 1978).

[73] F. Jelezko et al., Phys. Rev. Lett. 92, 076401 (2004).

[74] L. M. Johansen, Phys. Lett. A 322, 298 (2004).

[75] P. Jordan, Z. Phys. 37, 383 (1926).

BIBLIOGRAPHY 107

[76] R. Jozsa, Phys. Rev. A 76, 044103 (2007).

[77] S. Kagami, Master Thesis at Tokyo Institute of Technology (2009).

[78] E. H. Kennard, Z. Phys. 44, 326 (1927).

[79] N. Konno, Quantum Inf. Process. 1, 345 (2002).

[80] N. Konno, J. Math. Soc. Jpn. 57, 1179 (2005).

[81] N. Konno, Mathematics on Quantum Walk (Sangyo Tosho, Tokyo, 2008) (inJapanese).

[82] K. Koshino and A. Shimizu, Phys. Rep. 412 191 (2005).

[83] A. M. Krall, J. Diff. Eq. 45, 128 (1982).

[84] K. Kraus, Ann. Phys. 64, 311 (1971).

[85] L. D. Landau and E. M. Lifschitz, Quantum Mechanics Non-Relativistic Theory(Pergamon Press, 1977) Third edition.

[86] U. Larsen, J. Phys. A 23, 1041 (1990).

[87] J. Maddox, Nature 331, 559 (1988).

[88] L. Mandelstam and I. Tamm, J. Phys.(USSR) 9, 249 (1945).

[89] N. Margolus and L. B. Levitin, Physica D 120, 188 (1998).

[90] K. Mφlmer, Phys. Lett. A 292, 151 (2001).

[91] G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007).

[92] J. von Neumann, Math. Ann. 102, 49 (1929).

[93] J. von Neumann, Mathematische Grundlagen der Quantumechanik (Springer,Berlin, 1932), [ Eng. trans. by R. T. Beyer, Mathematical foundations of quantummechanics (Princeton University Press, Princeton, 1955). ]

[94] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2000).

[95] I. Ojima, in Proceeding of ”Quantum Bio-Informatics – From Quantum Informationto Bio-Information –” at Tokyo University of Science on 3.14-17.2007. pp. 217–228.arXiv:0705.2945.

108 BIBLIOGRAPHY

[96] M. Onoda, S. Murakami, and N. Nagaosa, 93, 083901 (2004).

[97] M. Ozawa, J. Math. Phys. 25, 79-87 (1984).

[98] M. Ozawa, Phys. Rev. Lett. 60, 385 (1988).

[99] M. Ozawa, in Squeezed and Nonclassical Light, edited by P. Tombesi and E. R. Pike,(Plenum, New York, 1989), pp. 263- 286.

[100] M. Ozawa, in Quantum Aspects of Optical Communications (Springer, Heidelberg,1991) pp. 1-17.

[101] M. Ozawa, Phys. Rev. A 67, 042105 (2003).

[102] M. Ozawa, J. Quant. Inf. 1, 569 (2003).

[103] M. Ozawa, Ann. Phys. 311, 350-416 (2004).

[104] M. Ozawa, J. Opt. B Quantum Semiclass. Opt. 7, S672 (2005).

[105] A. Pais, ”Subtle is the Lord... The Science and Life of Albert Einstein” (OxfordUniversity Press, Oxford, 1982).

[106] A. D. Parks, D. W. Cullin, and D. C. Stoudt, Proc. R. Soc. Lond. A 454, 2997(1998).

[107] A. D. Parks, J. Phys. A 41, 335305 (2008).

[108] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (Mcgraw-Hill,New York, 1935).

[109] A. Peres and P. Scudo, J. Mod. Opt. 49, 1235 (2002).

[110] G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, Phys.Rev. Lett. 94, 220405 (2005).

[111] M. Redei and S. J. Summers, Stud. Hist. Philos. Sci. B 38, 390 (2007).

[112] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Anal-ysis, Self-Adjointness (Academic Press, New York, 1975).

[113] F. Rellich, Math. Ann. 122, 343-368 (1950).

[114] K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Lett. A 324, 125 (2004).

[115] K. J. Resch, Science 319, 733 (2008).

BIBLIOGRAPHY 109

[116] B. Reznik and Y. Aharonov, Phys. Rev. A 52, 2538 (1995).

[117] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (1991).

[118] H. P. Robertson, Phys. Rev. 34, 163-164 (1929).

[119] D. Rohrlich and Y. Aharonov, Phys. Rev. A 66, 042102 (2002).

[120] R. A. Sack, Phil. Mag. 3, 497 (1958).

[121] T. Sagawa, Master Thesis, Tokyo Institute of Technology, 2008.

[122] H. Salecker and E. P. Wiger, Phys. Rev. 109, 571 (1958).

[123] L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1965) Third edition.

[124] E. Schrodinger, Sitz. Ber. Preuss. Akad. Wiss., Phys.-Math. KI. (Berlin), 296 (1930).

[125] A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singa-pore, 1988).

[126] Y. Shikano and A. Hosoya, J. Math. Phys. 49, 052104 (2008).

[127] Y. Shikano and A. Hosoya, arXiv:0812.4502.

[128] E. Sjoqvist, Phys. Lett. A 359, 187 (2006).

[129] E. Sjoqvist, Acta Phys. Hung. B 26, 195 (2006).

[130] A. M. Steinberg, Phys. Rev. Lett. 74, 2405 (1995).

[131] A. Tanaka, Phys. Lett. A 297 307 (2002).

[132] A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, Am. J. Phys. 57,117 (1989).

[133] I. Tsutsui, T. Fulop and T. Cheon, J. Phys. A 36, 275-287 (2003).

[134] J. Twamley and G. J. Milburn, New J. of Phys. 8, 328 (2006).

[135] A. Uhlmann, Rep. Math. Phys. 24, 229 (1986).

[136] A. Uhlmann, Lett. Math. Phys. 21, 229 (1991).

[137] A. Uhlmann, in Proceedings of the First Max Born Symposium, edited by R. Giel-erak, J. Lukierski, and Z. Popowicz (Kluwer Academics Press, Netherlands, 1992)pp. 267-274. mp-arc:92-44.

110 BIBLIOGRAPHY

[138] L. Vaidman, Am. J. Phys. 60, 183 (1992).

[139] L. Vaidman, Found. Phys. 26, 895 (1996).

[140] M. S. Wang, Phys. Rev. Lett. 79, 3319 (1997).

[141] H. Weyl, Math. Ann. 68, 220-269 (1910).

[142] H. Weyl, Gryppentheorie und Quantenmechanik (S. Hirzel, Leipzig, 1928), [ Eng.Trans. H. P. Robertson, The Theory of Groups and Quantum Mechanics (Dover,New York, 1950). ].

[143] N. S. Williams and A. N. Jordan, Phys. Rev. Lett. 100, 026804 (2008).

[144] H. M. Wiseman, Phys. Rev. A 65, 032111 (2002).

[145] H. P. Yuen, Phys. Rev. Lett. 51, 719 (1983).

[146] K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, arXiv:0811.1625.

[147] ”The Quantum and the Foundations of Physics Conference” at the University ofTexas organized by J. A. Wheeler on 12.17-20.1982.

Epilogue

Michael Nielsen, who was a famous researcher on quantum information but left the re-search field on quantum information, said

An ideal collaboration market will enable just such an exchange of questionsand ideas. It will bake in metrics of contribution so participants can demon-strate the impact their work is having. Contributions will be archived, times-tamped, and signed, so it’s clear who said what, and when. Combined withhigh quality filtering and search tools, the result will be an open culture oftrust which gives scientists a real incentive to outsource problems, and con-tribute in areas where they have a great comparative advantage. This willchange science.

Michael Nielsen ”Future of Science”

Which future direction of science will really be driven? At least in quantum information,I believe that we will go back to the original questions a la the Renaissance.

111

master thesis 2009 weak values in quantum measurement ...shikano/master_thesis.pdf · fessor...

Documents