Formulation of uncertainty relation between error and disturbance in quantum measurement by using QUantum

Yu WatanabeFormulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory Doctoral Thesis accepted by The University of Tokyo

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Yu Watanabe

Formulation of Uncertainty Relation Between Error

and Disturbance in Quantum Measurement by Using

Quantum Estimation Theory

Doctoral Thesis accepted by

The University of Tokyo, Tokyo, Japan

123

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Author (Current Address)

Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic)

ISBN 978-4-431-54492-0 ISBN 978-4-431-54493-7 (eBook)

DOI 10.1007/978-4-431-54493-7

Springer Tokyo Heidelberg New York Dordrecht London

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 Springer Japan 2014

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Parts of this thesis have been published in the following journal articles:(i) Y Watanabe, T Sagawa, and M Ueda, Optimal Measurement on NoisyQuantum Systems, Phys Rev Lett 104, 020401 (2010).

(ii) Y Watanabe, T Sagawa, and M Ueda, Uncertainty Relation Revisited fromQuantum Estimation Theory, Phys Rev A 84, 042121 (2011)

(iii) Y Watanabe, M Ueda, Quantum Estimation Theory of Error and Disturbance

in Quantum Measurement, arXiv:1106.2526 (2011)

Supervisor’s Foreword

In this thesis, Dr Yu Watanabe applies quantum estimation theory to investigateuncertainty relations between error and disturbance in quantum measurement Inhis seminal work, Heisenberg discussed a thought experiment concerning theposition measurement of a particle by using a gamma-ray microscope, and dis-covered a trade-off relation between the error of the measured position and thedisturbance on the quantum-mechanically conjugate momentum caused by themeasurement process This trade-off relation epitomizes the complementarity inquantum measurements: we cannot perform a measurement of an observablewithout causing disturbance in its canonically conjugate observable However,Heisenberg’s argument was rather qualitative, and the quantitative understanding

of the trade-off relationship was elusive because in his era, quantum measurementtheory had not been established Meanwhile, Kennard and Robertson discussed adifferent type of inequality concerning inherent fluctuations of observables Thisversion of Heisenberg’s uncertainty relation is commonly described in quantummechanics textbooks and often erroneously interpreted as a mathematical formu-lation of the complementarity From the modern point of view, Heisenberg’suncertainty relation is the trade-off relation between the information gain about anobservable and the concomitant information loss about its conjugate observable Inthis thesis, Dr Watanabe argues that the best solution to this problem is to applythe estimation theory to the outcomes of the measurement for quantifying the errorand disturbance in quantum measurement He has successfully formulated theerror and disturbance in terms of the Fisher information content, which gives theupper bound of the accuracy of the estimation Moreover, Dr Watanabe hasderived the attainable bound of the error and disturbance in quantum measurement.The obtained bound is determined by the quantum fluctuations and correlationfunctions of the observables, which characterize the non-classical fluctuation ofthe observables Notably, this bound is stronger than the conventional one set bythe commutation relation of the observables I believe that this thesis provides a

vii

groundbreaking work that establishes the fundamental bound on the accuracy ofone measured observable and the disturbance on the conjugate observable in theoriginal spirit of Heisenberg, and I expect that the method developed here will beapplied to a broad class of problems related to quantum measurement.

I would like to thank my supervisor, Prof Masahito Ueda for providing helpfulcomments and suggestions I would like to thank Takahiro Sagawa for a work onerror in quantum measurement and uncertainty relations; Yuji Kurotani for guiding

me to uncertainty relations and quantum measurement theory when I was anundergraduate student; Prof Masahito Hayashi for fruitful discussions I express

my appreciation to Prof Mio Murao, Prof Akira Shimizu, Prof Kimio Tsubono,Prof Masato Koashi, and Prof Makoto Gonokami for refereeing my thesis and forvaluable discussions Finally, I am grateful to the numerous researchers who haveprovided me with opportunities for many helpful discussions

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Contents

1 Introduction 1

References 5

2 Reviews of Uncertainty Relations 7

2.1 Heisenberg’s Gamma-Ray Microscope 7

2.2 Von Neumann’s Doppler Speed Meter 9

2.3 Kennard-Robertson’s Inequality and Schrödinger’s Inequality 11

2.4 Arthurs-Goodman’s Inequality 12

2.5 Ozawa’s Inequality 14

References 17

3 Classical Estimation Theory 19

3.1 Parameter Estimation of Probability Distributions 19

3.2 Cramér-Rao Inequality and Fisher Information 23

3.3 Monotonicity of the Fisher Information and Cˇ encov’s Theorem 28

3.4 Maximum Likelihood Estimator 30

References 36

4 Quantum Estimation Theory 37

4.1 Parameter Estimation of Quantum States 37

4.2 Monotonicity of the Fisher Information in Quantum Measurement 38

4.3 Quantum Cramér-Rao Inequality and Quantum Fisher Information 39

4.4 Adaptive Measurement 42

References 44

5 Expansion of Linear Operators by Generators of Lie Algebrasu(d) 45

5.1 Generators of Lie AlgebrasuðdÞ 45

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5.2 Quantum State and Bloch Space 47

5.3 Observable 51

5.4 Quantum Measurement 53

5.4.1 Positive Operator-Valued Measure (POVM) Measurement 53

5.4.2 Projection-Valued Measure (PVM) Measurement and Spectral Decomposition 54

5.5 Quantum Operation 56

5.5.1 Unitary Evolution 58

5.5.2 Interaction with an Environment 59

5.5.3 Measurement Processes 61

References 70

6 Lie Algebraic Approach to the Fisher Information Contents 71

6.1 Classical Fisher Information 71

6.1.1 Positive State Model 73

6.1.2 Block Diagonal State Model 76

6.1.3 Decohered State Model 79

6.2 SLD Fisher Information 80

6.2.1 Positive State Model 81

6.2.2 Block Diagonal State Model 82

6.2.3 Decohered State Model 84

6.3 RLD Fisher Information 84

6.3.1 Positive State Model 85

6.3.2 Block Diagonal State Model 86

6.3.3 Decohered State Model 87

Reference 88

7 Error and Disturbance in Quantum Measurements 89

7.1 Error in Quantum Measurement 89

7.1.1 Comparison with the Error Defined by Arthurs and Goodman 94

7.1.2 Comparison with the Error Defined by Ozawa 95

7.2 Disturbance in Quantum Measurement 96

References 100

8 Uncertainty Relations Between Measurement Errors of Two Observables 101

8.1 Setup 101

8.2 Heisenberg-Type Uncertainty Relation 103

8.3 Attainable Bound of the Product of the Measurement Errors 104

References 113

9 Uncertainty Relations Between Error and Disturbance

in Quantum Measurements 1159.1 Heisenberg’s Uncertainty Relation in Terms of Fisher

Information Contents 1159.2 Attainable Bound of the Product of Error and Disturbance 117

10 Summary and Discussion 121References 122

and the disturbanceη(p x ) in the momentum p xcaused by the measurement process:

This inequality epitomizes the complementarity in quantum measurements: we not perform a measurement of an observable without causing disturbance in itscanonically conjugate observable The errorε(x) in the position measurement char-

can-acterizes the accuracy of the estimation of x from the measurement outcomes The measurement process randomly changes the momentum p x, therefore the originalmomentum cannot be estimated accurately from the post-measurement particle Thedisturbanceη(p x ) characterizes the accuracy of the estimated value of the original

p xfrom the post-measurement particle

Neumann [3, 4] discussed a thought experiment on the measurement of themomentum of a particle by using the Doppler effect, and derived the trade-off rela-tion between the error in the momentum and the disturbance in the position caused

by the measurement process:

Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance 1

in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses,

DOI: 10.1007/978-4-431-54493-7_1, © Springer Japan 2014

the expectation value of an observable ˆA over the quantum state ˆρ, the square bracket

denotes the commutator[ ˆA, ˆB] := ˆA ˆB − ˆB ˆA, and σ( ˆA)2 :=  ˆA2⊂ −  ˆA⊂2 TheKennard-Robertson inequality actually implies the indeterminacy of quantum states:non-commuting observables cannot have definite values simultaneously However,sinceσ( ˆA) does not depend on the measurement process, the Kennard-Robertson

inequality reflects the inherent nature of a quantum state alone, and does not concernany trade-off relation between the error and disturbance in the measurement process

In 1988, Arthurs and Goodman [7] considered a simultaneous measurement oftwo non-commuting observables ˆA and ˆ B in a fully quantum mechanical treatment.

Because ˆA and ˆ B do not commute with each other, it is necessary to extend the

Hilbert space to make both of them simultaneously measurable This can be done byletting the system interact with another system, called the apparatus By consideringthe interaction between the system and apparatus, they considered an indirect mea-surement In order to make the outcomes of the indirect measurement meaningfulfor ˆA and ˆ B, they assumed the unbiasedness of the measurement outcomes: that is,

the expectation values of the outcomes respectively equal to ˆA⊂ and  ˆB⊂ for an

arbi-trary quantum state The unbiasedness of the measurement implies that ˆA⊂ can be

estimated directly from the distribution of the measurement outcomes They showedthat the variances of the measurement outcomes satisfy

σ∼( ˆA)σ∼( ˆB) ≥[ ˆA, ˆB]⊂. (1.5)

Comparing this result with the Kennard-Robertson inequality, we find that the lowerbound is doubled Fluctuations of the measurement outcomes originate from thesystem’s inherent fluctuations and the error in the measurement process, namely,each source of fluctuations has the lower bound of 12[ ˆA, ˆB]⊂, and the bound in

(1.5) is doubled as the total Because the measurement discussed by Arthurs andGoodman is restricted to the unbiased measurement, a natural question arises as towhat happens for the biased measurement case

Ozawa [8 10] generalized the Arthurs-Goodman inequality by removing the asedness condition, and presented the following inequality:

unbi-ε( ˆA)unbi-ε( ˆB) + unbi-ε( ˆA)σ( ˆB) + σ ( ˆA)unbi-ε( ˆB) ≥ 1

2



[ ˆA, ˆB]⊂. (1.6)

Because the errorε( ˆA) is always finite, if the error ε( ˆA) vanishes, the product of

the measurement errorsε( ˆA)ε( ˆB) also vanishes Thus, the Heisenberg-type trade-off

defini-ε( ˆA) can vanish even if we cannot estimate  ˆA⊂ Such a result originates from

ignor-ing the estimation process which must inevitably be accompanied in the unbiasedmeasurement Ozawa also defined the disturbanceη( ˆB) caused by the backaction of

the measurement, and derived the following inequality:

ε( ˆA)η( ˆB) + ε( ˆA)σ( ˆB) + σ ( ˆA)η( ˆB) ≥ 1

estima-Estimation theory [11–13] provides us a description of how accurately we canestimate values and how much information we can obtain from realizations of theprobabilistic phenomena In quantum theory, measurements on the quantum systemare necessary to obtain some pieces of information about the quantum system, andthe measurement outcomes are obtained according to the probability distribution.Thus, it is necessary to involve the estimation theory for clarifying the uncertaintyrelations about the error and disturbance in quantum measurements In estimationtheory, one of the most important quantities is the Fisher information [11], whichgives the upper bound on the accuracy of the estimated value

In this thesis, we develop a general theory of error and disturbance in quantummeasurements We show that the unbiasedness is necessary not for the measurements,but for the estimation from the measurement outcomes From that analysis, we canrelax the restriction of the unbiased measurement, and define the error and disturbance

in an arbitrary measurement process By invoking the estimation theory, we showthat the measurement error can be quantified as

where J (M) is the Fisher information obtained by the measurement M, J Q is thequantum Fisher information [14] about the original quantum state, and a is a set of

parameters that determines the observable ˆA As shown in Chaps.3and4, a·J(M)−1a

gives the accuracy of the estimation, and a·J Q−1a characterizes the inherent fluctuation

of the observable Since the observable is inherently fluctuated, the accuracy of theestimation is bounded by the inherent fluctuation Therefore, ε( ˆA; M) is always

non-negative and vanishes if and only if we perform the most accurate measurement

4 1 Introduction

We also show that the disturbance caused by the measurement process K can be

quantified as

where J Sis the symmetric logarithmic derivative (SLD) Fisher information about the

original quantum state, and J∼Sis the SLD Fisher information contained in the

post-measurement state The disturbance characterizes the loss of the Fisher informationcaused by the measurement process Our definition of the measurement error reduces

to Arthurs-Goodman’s definition for the case of the unbiased measurements

By using our definition of the error and disturbance, we will prove that thefollowing uncertainty relations:

ε( ˆA)ε( ˆB) ≥ σ Q ( ˆA)2σ Q ( ˆB)2− CQS( ˆA, ˆB)2, (1.13)

ε( ˆA)η( ˆB) ≥ σ Q ( ˆA)2σ Q ( ˆB)2− CQS( ˆA, ˆB)2, (1.14)where σ Q ( ˆA) and CQS( ˆA, ˆB) are quantum fluctuation and correlation function.

As shown in Sect.8.3, the quantum fluctuation σ Q ( ˆA) and correlation function

CQS( ˆA, ˆB) characterize non-classical fluctuation and correlation in quantum

inequal-In Chap.4, we review quantum estimation theory and introduce the quantum Fisherinformation In Chap.5, we develop techniques to expand relevant operators in terms

of the generators of Lie algebrasu(d) This expansion method greatly facilitates the

calculation of the Fisher information contents, error and disturbance in quantum surement In Chap.6, we calculate various Fisher information by using the techniques

mea-of the expansion by the generators mea-of Lie algebrasu(d) These Fisher information

contents are used for defining error and disturbance and showing uncertainty tions In Chap.7, we show why estimation theory is crucial to analyze error anddisturbance in quantum measurement, and define the error and disturbance in terms

rela-of Fisher information contents In Chap.8, we derive uncertainty relations of themeasurement errors of two observables In Chap.9, we derive uncertainty relations

Chpter 4 Quantum Estimation Theory

Chpter 5 Expansion of Linear Operators by Generators of Lie Algebra (d)

Chpter 6 Lie Algebraic Approach to the Fisher Information Contents

Chpter 7 Error and Disturbance in Quantum Measurement

Chpter 8 Uncertainty Relations between

Measurement Errors of

Two Observables

Chpter 9 Uncertainty Relations between Error and Disturbance in Quantum Measurement

Fig 1.1 The flowchart of this thesis Chaps.2 4 are reviews of relevant past works Our results are shown in Chaps 5 9

between the error and disturbance In Chap.10, we summarize this thesis and discusssome outstanding issues

The results in Chaps.5and6are based on Ref [15] collaborating with Sagawa andUeda The results in Chap.6, Sect.7.1and Chap.8are based on Ref [16] collaboratingwith Sagawa and Ueda The results in Sect.7.2and Chap.9are based on Ref [17]collaborating with Ueda

References

1 W Heisenberg, Zeitschrift fr Physik A Hadrons and Nuclei 43, 172 (1927)

2 J.A Wheeler, W.H Zurek, Quantum Theory and Measurement (Princeton University Press,

New Jersey, 1983), pp 62–84

3 J von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University

Press, New Jersey, 1955), p 209

4 V Braginsky, F Khalili, K Thorne, Quantum Measurement (Cambridge University Press,

Cambridge, 1992)

5 E.H Kennard, Zeitschrift fr Physik A Hadrons and Nuclei 44, 326 (1927)

6 H.P Robertson, Phys Rev 34, 163 (1929)

7 E Arthurs, M.S Goodman, Phys Rev Lett 60, 2447 (1988)

8 M Ozawa, Phys Rev A 67, 042105 (2003)

9 M Ozawa, Phys Lett A 320, 367 (2004)

10 M Ozawa, Ann Phys 311, 350 (2004)

11 R Fisher, Math Proc Cambridge Philos Soc 22, 700 (1925)

6 1 Introduction

12 H Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946)

13 E Lehmann, G Casella, Theory of Point Estimation (Springer Verlag, New York, 1983)

14 C.W Helstrom, Phys Lett A 25, 101 (1967)

15 Y Watanabe, T Sagawa, M Ueda, Phys Rev Lett 104, 020401 (2010)

16 Y Watanabe, T Sagawa, M Ueda, Phys Rev A 84, 042121 (2011)

17 Y Watanabe, M Ueda, arXiv:1106.2526 (2011)

Chapter 2

Reviews of Uncertainty Relations

In this chapter, we provide a brief overview of various uncertainty relations First,

we review historical uncertainty relations: Heisenberg’s gamma-ray microscopeand von-Neumann’s Doppler speed meter These uncertainty relations epitomizetrade-off relation between error and disturbance in quantum measurement process.Next, we review a different type of uncertainty relations: Kennard-Robertson’sinequality and Schrödinger’s inequality These characterize trade-off relations ofinherent fluctuations of observables Finally, we review Arthurs-Goodman’s inequal-ity and Ozawa’s inequality that based on modern quantum measurement theory

2.1 Heisenberg’s Gamma-Ray Microscope

As described in the Introduction, Heisenberg [1,2] discussed a thought experimentabout the position measurement of a particle by using aγ -ray microscope, and found

the following trade-off relation between the errorε(x) in the measured position x

and the disturbanceη(p x ) in the momentum p xcaused by the measurement process:

In this section, we follow Heisenberg’s orignal discussion and show the importance

of the estimation process

Let us consider that we measure the position x of a particle By irradiating the

γ -ray on the particle, a photon of the γ -ray is scattered by the particle The scattered

photon passes through a lens, impinges on a screen, and makes a blip on the screen

We measure the position x≥of the blip, and infer the position x of the particle by the

following relation:

x= L1

L2

Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance 7

in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses,

DOI: 10.1007/978-4-431-54493-7_2, © Springer Japan 2014

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where L1and L2are the distance between the lens and the particle, and that between

the lens and the screen, respectively It may seem that by determining x≥accurately,

we can also determine x accurately However, because of the wave property of the photon, even if we assume that the position x≥of the blip can be determined with an

arbitrary accuracy,we cannot estimate the position x of the particle accurately If the

particle shifts byΔx from the focal point P, the difference between the optical path

the difference between the path lengths is larger than the wavelengthλ Therefore,

the distinguishable minimal shift of the position is

2 sinθ . (2.4)

Fig 2.1 Heisenberg’sγ -ray microscope If the particle shifts its position by Δxλ/2 sin θ, we

cannot distinguish the shift

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2.1 Heisenberg’s Gamma-Ray Microscope 9

Therefore, the estimated position x involves the error

even if we determine x≥accurately.

Next, we consider the disturbance caused by the measurement process Afterthe scattering of the photon, the momentum of the particle is changed However,

we cannot determine the angle about which direction the photon is scattered Thus,

we cannot estimate the momentum changeΔp x accurately The uncertainty of themomentum change is given by

η(p x ) = 2λsinθ. (2.6)Therefore, the error and disturbance satisfy the trade-off relation (2.1)

Heisenberg’s uncertainty relation (2.1) is based on a specific model of the positionmeasurement and the semi-classical analysis of the quantum measurement: that is,the particle was assumed to possess definite position and momentum To rigorouslyprove the complementarity in quantum measurements, we need to use quantum mea-surement theory [3,4] However, at the time Heisenberg found the trade-off relation,quantum measurement theory was not established yet Quantum measurement theorywas established in the 1970s by Davies and Lewis [3]

2.2 Von Neumann’s Doppler Speed Meter

Heisenberg’sγ -ray microscope measures the position of a particle and causes the

disturbance in the momentum Von Neumann [5, 6] considered a thought ment of the momentum of a particle by using a Doppler speed meter,and found thefollowing trade-off relation between the errorε(p x ) of the measured momentum p x

experi-and the disturbanceη(x) in the position caused by the measurement process:

Note that the roles of x and p xare exchanged in comparison with Heisenberg’s tainty relation (2.1) This inequality shows that we cannot measure the momentumwithout causing disturbance in the position of the particle (Fig.2.2)

uncer-Suppose that we measure the momentum p x of a particle with mass m First, we

prepare a photon with frequencyω and duration τ that propagates to the particle If

the particle reflects the photon, then the frequency of the reflected photon changes

δω due to the Doppler effect The frequency change δω is calculated to be

10 2 Reviews of Uncertainty Relations

Fig 2.2 Von Neumann’s Doppler speed meter

wherev x is the velocity of the particle, and c is the speed of light By measuring the

frequency of the reflected photon, we can estimate the velocityv xand the momentum

the exact time of the reflection, and the uncertainty of the reflection time isτ The

uncertainty of the position is calculated to be

2.3 Kennard-Robertson’s Inequality and Schrödinger’s Inequality 11

2.3 Kennard-Robertson’s Inequality and Schrödinger’s

where σ ( ˆx) := ∃ ˆx2 − ∃ ˆx2, and∃ ˆx := Tr[ ˆρ ˆx] Kennard’s inequality implies the

indeterminacy of the quantum state, that is, the position and momentum cannot bedefinite simultaneously In the early days of quantum mechanics, this inequality waserroneously interpreted as a mathematical formulation of the Heisenberg’s uncer-tainty relation However,σ ( ˆx) implies the inherent fluctuation of the observable ˆx

and depends only on the quantum state ˆρ Kennard’s inequality does not concern any

trade-off relation between the error and disturbance in the quantum measurement.Robertson [8] generalized Kennard’s inequality for arbitrary observables, andfound the following inequality:

σ ( ˆA)σ( ˆB) ≤1

2



∃[ ˆA, ˆB], (2.14)

where the square brackets denote the commutator:[ ˆA, ˆB] := ˆA ˆB − ˆB ˆA Moreover,

Schrödinger [9] generalized Robertson’s inequality as

From Schrödinger’s inequality, Kennard’s inequality and Robertson’s inequality aredirectly derived Thus, we prove Schrödinger’s inequality here

Let C( ˆA, ˆB) be a non-symmetrized correlation function of the observables

defined as

and K ∈ C2 ×2be a Hermitian matrix defined as

K :=



σ ( ˆA)2 C( ˆA, ˆB) C( ˆB, ˆA) σ ( ˆB)2



12 2 Reviews of Uncertainty Relations

For an arbitrary complex vector z= (z1, z2)T∈ C2, where T denotes the transpose,

Heisenberg’s uncertainty relation and von Neumann’s uncertainty relation are based

on the semi-classical analysis of quantum measurements Arthurs and Kelly [10]considered a simultaneous measurement of the position and momentum in fullyquantum-mechanical analysis, and Arthurs and Goodman [11] generalized the mea-surement scheme for two arbitrary non-commuting observables

To make both observables simultaneously measurable, it is necessary to extendthe Hilbert space This can be done by letting the system interact with another system,called the apparatus Let us consider that we want to measure observables ˆA and ˆ B.

Suppose that the initial state of the system is ˆρ First, we prepare the state of the

apparatus as ˆρapp, and interact the system and apparatus with the unitary operator ˆU

After the interaction, we measure the observables ˆA≥ and ˆB≥ of the apparatus To

measure both observables simultaneously, ˆA≥and ˆB≥must commute with each other.

In order to make the outcomes of the indirect measurement meaningful for ˆA and ˆ B,

they assumed

∃ ˆA := Tr[ ˆρ ˆA] = Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†( ˆI ⊗ ˆA≥)], (2.21a)

∃ ˆB := Tr[ ˆρ ˆB] = Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†( ˆI ⊗ ˆB≥)] (2.21b)for an arbitrary state ˆρ, where ˆI is a identity operator Hereforth, we denote ˆI⊗ ˆA≥as

ˆA≥for simplicity These conditions are called unbiasedness conditions of the

measure-ment, and measurements that satisfy the unbiasedness conditions are called unbiased

2.4 Arthurs-Goodman’s Inequality 13

measurements.Note that for arbitrary observables ˆA and ˆ B, there always exists a set

of ˆU , ˆρapp, ˆA≥ and ˆB≥ that satisfies the unbiasedness condition.The unbiasedness

conditions (2.21) imply that the expectation values∃ ˆA and ∃ ˆB can directly be

esti-mated from the measurement outcomes.The variances of the measurement outcomesare given by

σ≥( ˆA≥) := Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†ˆA≥2] − Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU† ˆA≥]2

= Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†ˆA≥2] − ∃ ˆA2, (2.22a)

σ≥( ˆB≥) := Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†ˆB≥2] − Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†ˆB≥]2

= Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†ˆB≥2] − ∃ ˆB2. (2.22b)Let ˆN ˆAbe a “noise” operator defined as

ˆN ˆA := ˆU†ˆA≥ ˆU − ˆA. (2.23)From the unbiasedness condition, the noise operator satisfies

Tr[( ˆρ ⊗ ˆρapp) ˆN ˆA] = 0 (2.24)for an arbitrary state ˆρ From this equation, the following equation can be derived:

Trapp[(I ⊗ ˆρapp) ˆN ˆA ] = ˆ0, (2.25)where Trapp denotes the partial trace over the apparatus system, and ˆ0 is the nulloperator Thus, we have

Tr[ ˆU( ˆρ ⊗ ˆρapp) ˆU†ˆA≥2] = Tr[( ˆρ ⊗ ˆρapp) ˆN2

ˆA ] + Tr[ ˆρ ˆA2], (2.26)and

σ≥( ˆA≥)2= Tr[( ˆρ ⊗ ˆρapp) ˆN2

ˆA ] + σ( ˆA)2, (2.27)whereσ ( ˆA)2:= ∃ ˆA2 − ∃ ˆA2 Therefore, we can find that the variance of the mea-surement outcome consists of two types of error: inherent fluctuationσ( ˆA), and error

in the measurementεAG( ˆA)2 := Tr[( ˆρ ⊗ ˆρapp) ˆN2

ˆA] To clarify the role of the error

εAG( ˆA) in the variance σ≥( ˆA≥), let us consider the commutation relation of the noise

operators It follows from the fact that ˆA≥and ˆB≥commute with each other that

[ ˆN ˆA , ˆN ˆB ] + [ ˆN ˆA , ˆB] + [ ˆA, ˆN ˆB ] = −[ ˆA, ˆB]. (2.28)

14 2 Reviews of Uncertainty Relations

σ≥( ˆA≥)σ≥( ˆB≥) ≤ σ ( ˆA)σ( ˆB) + εAG( ˆA)εAG( ˆB) ≤∃[ ˆA, ˆB]. (2.31)

The product of the inherent fluctuations and the product of the measurement errorsare both bounded from below by|∃[ ˆA, ˆB]|/2 Therefore, the lower bound in (2.31)

is doubled

2.5 Ozawa’s Inequality

Arthurs and Goodman derived the trade-off relation between the errors of the ables in the unbiased measurement By removing the unbiasedness condition of themeasurement, Ozawa [12–14] defined the measurement error for an arbitrary mea-surement as follows:

observ-εOzawa( ˆA)2:= Tr[( ˆρ ⊗ ˆρapp) ˆN2

≤ σ≥( ˆN ˆA )σ≥( ˆN ˆB ) + σ≥( ˆN ˆA )σ( ˆB) + σ( ˆA)σ≥( ˆN ˆB )

≤ ε ( ˆA)ε ( ˆB) + ε ( ˆA)σ( ˆB) + σ ( ˆA)ε ( ˆB) (2.34)

and proved the following inequality [12–14]:

εOzawa( ˆA)ηOzawa( ˆB) + εOzawa( ˆA)σ( ˆB) + σ ( ˆA)ηOzawa( ˆB) ≤ 1

From (2.33), the product of the measurement errors is not bounded by the mutation relation of the observables Because the measurement error is always finite,

com-if the measurement errorεOzawa( ˆA) vanishes, the product of the errors also vanishes:

εOzawa( ˆA)εOzawa( ˆB) = 0 ≤ 1

vio-Let us consider the case in which the measurement process is a projective

mea-surement, and the measurement outcome is scaled by a factor c Such a measurement

can be constructed by a swapping operator ˆU and ˆ A≥= c ˆA Although we can fully

obtain information about ˆA from the measurement outcome, that is, we can estimate

∃ ˆA with the same accuracy when we perform the non-scaled PVM measurement of

ˆA, the measurement error does not vanish:

εOzawa( ˆA) = (c − 1)



∃ ˆA2. (2.41)

16 2 Reviews of Uncertainty Relations

If the measurement process is also a projective measurement, and the measurement

outcome is shifted by c In this case, we also have the complete information, but the

error is

Such a conclusion does not make sense which indicates a rather serious problem inOzawa’s definition of the error Next, let us consider the measurement that always

gives a fixed value c Such a measurement can be constructed by ˆ A≥= c ˆI According

to Ozawa’s definition, the error is calculated to be

∃ ˆA and σ( ˆA) = 0 by chance, the error vanishes From those examples, we conclude

that the error defined by Ozawa does not imply the obtained information about thesystem, that is, the error is independent of the accuracy of the estimated value of∃ ˆA.

Therefore, the definition of the error is independent of the accuracy of the estimation.Next, we consider the disturbance Let us consider the case in which the mea-surement process actually does not obtain information and just rotates the system by

a unitary operator ˆV , that is, the system does not interact with the apparatus The

apparatus system In this case, the disturbance is calculated to be

ηOzawa( ˆB) = σapp( ˆB)2+ σ ( ˆB)2+ (∃ ˆBapp− ∃ ˆB)2, (2.45)where∃ ˆBapp := Tr[ ˆρapp ˆB] and σapp( ˆB)2:= ∃ ˆB2app− ∃ ˆB2

app Because the measurement state of the system equals ˆρapp, we cannot obtain information about theoriginal state ˆρ by performing any measurement on the post-measurement state If the

post-state ˆρappis equal to ˆρ by chance and σ ( ˆB) = 0, the disturbance vanishes From those

examples, we conclude that the disturbance is independent of the information aboutthe original state ˆρ contained in the post-measurement state, that is, the disturbance

does not imply the possibility of the estimation from the post-measurement state.Therefore, Ozawa’s definition of the disturbance is independent of the accuracy ofthe estimation

In those measurements, the estimation processes are rather straightforward and seem

to be trivial To define error and disturbance for an arbitrary measurement, however,

we need to invoke quantum estimation theory [15–17]

References

1 W Heisenberg, Zeitschrift fr Physik A Hadrons Nuclei 43, 172 (1927)

2 J.A Wheeler, W.H Zurek, Quantum Theory and Measurement (Princeton Univ Press, New

Jersey, 1983)

3 E.B Davies, J.T Lewis, Commun Math Phys 17, 239 (1970)

4 M Ozawa, J Math Phys 25, 79 (1984)

5 J von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ Pr, 1955)

6 V Braginsky, F Khalili, K Thorne, Quantum Measurement (Cambridge Univ Pr, Cambridge,

1992)

7 E.H Kennard, Zeitschrift fr Physik A Hadrons Nuclei 44, 326 (1927)

8 H.P Robertson, Phys Rev 34, 163 (1929)

9 E Schrödinger, Proc Prussian Acad Sci Phys Math Sect XIX, 293 (1930)

10 E Arthurs, J.L Kelly, Bell Syst Tech J 44, 725 (1965)

11 E Arthurs, M.S Goodman, Phys Rev Lett 60, 2447 (1988)

12 M Ozawa, Phys Rev A 67, 042105 (2003)

13 M Ozawa, Phys Lett A 320, 367 (2004)

14 M Ozawa, Ann Phys 311, 350 (2004)

15 C.W Helstrom, J Stat Phys 1, 231 (1969)

16 A Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland,

Amster-dam, 1982)

17 M Hayashi, Quantum Information: An Introduction (Springer Verlag, Berlin, 2006)

Chapter 3

Classical Estimation Theory

To formulate error and disturbance in quantum measurement, the estimation processfrom the measurement outcomes has an essential role In this chapter, we reviewclassical estimation theory [1][3] and introduce Fisher information, which gives theupper bound of the accuracy of the estimation [3]

3.1 Parameter Estimation of Probability Distributions

LetX be a random variable, and p(x) the probability distribution corresponding to

X In this thesis, for simplicity, we assume that the number of elements in X is finite

(|X | < ≥) The generalization to cases in which X is continuous or |X | = ≥ can

be made by changing summation to integral and the probability distribution to theprobability density or probability measure

Suppose that a probability distribution p (x) parameterized by

where ε is an open subset of R m which is called a parameter space Here, the

statement that p (x) is parameterized by θ means that p(x) = p(x; θ) is uniquely

determined by specifying θ, and that there exists a surjective mapping from ε to

the setP of the concerning probability distributions The set P is called a statistical

model We assume that the inverse image of an arbitrary p (x; θ) with respect to

the mappingθ ∼ p(x; θ) is arcwise connected, and the mapping θ ∼ p(x; θ) is

differentiable byθ some appropriate number of times, that is, intuitively, the mapping

is sufficiently smooth

We assume that for allθ ∈ ε, the support of p(x; θ) is the same That is, by

choosingX appropriately, we assume that

Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance 19

in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses,

DOI: 10.1007/978-4-431-54493-7_3, © Springer Japan 2014

20 3 Classical Estimation Theory

The probability distribution p (x) such that p(x) = 0 for several x ∈ X can be

considered as the limit of a sequence of p (x; θ) ∈ P Such p(x) belongs to the

closure ¯P of P and it is parameterized by θ in the closure of ε Therefore, the

assumption p (x; θ) > 0 can be made without loss of generality.

In the following we introduce examples of the statistical models

Example 1 (Coin toss).

X = {H, T}, m = 1, ε = {γ ∈ R|0 < γ < 1}, (3.3a)

where H and T denote the head and tail of the coin

Example 2 (Gaussian distribution).

Example 4 (Projection valued measure measurement on a qubit) Suppose that we

perform the projection valued measure (PVM) measurement (see Sect.5.4.2) of ˆη z

on a qubit, where ˆη i is the Pauli operator in the i -direction (i = x, y, z) The state

of the qubit is described by the Bloch vectorθ:

3.1 Parameter Estimation of Probability Distributions 21

Therefore, the statistical model of this probability distribution is

Letϕ : R m ∃∼ Rl be a function defined onε, and consider the estimation of

the valueϕ(θ) from the realization x of X The random variable ϕest:= ϕest(X ) is

called an estimator ofϕ, and the realization ϕest(x) is called an estimated value.

Since the estimator ϕest(X ) is a random variable, the estimated value ϕest(x)

is usually different from the true value ϕ(θ) Therefore, the expected value and

covariance of the estimator are important for evaluating the efficiency of the estimator.The expected valueEθ [ϕest] and covariance Varθ [ϕest] are defined as follows:

(ϕest(x) − E θ [ϕest])(ϕest(x) − E θ [ϕest])Tp (x; θ). (3.11)

Note thatEθ [ϕest] and Varθ [ϕest] depend on the true value θ The difference between

Eθ [ϕest] and ϕ(θ) is called a bias of the estimator:

bθ = Eθ [ϕest] − ϕ(θ). (3.12)

If an estimatorϕest(X ) satisfies

Eθ [ϕest] = ϕ(θ) for all θ ∈ ε, (3.13)

or equivalently bθ = 0 for all θ ∈ ε, the estimator ϕest is called an unbiasedestimator The unbiasedness condition is a desirable condition for the estimator;however, it is difficult to satisfy it and, in general, no unbiased estimator exists.Therefore, usually the condition is relaxed as follows:

22 3 Classical Estimation Theory

for a specificθ0∈ ε The estimator satisfying the condition (3.14) is called a locallyunbiased estimator onθ0 The local unbiasedness is intended that the estimator isunbiased in the neighborhood ofθ0

In a usual case, we can perform several trials for the sameX , and the random

variable of each trialX i (i = 1, , n) can be assumed independently and identically

distributed (i.i.d for short) with p (x; θ) For example, we can toss the same coin

many times, and we can prepare the same quantum state and perform the samemeasurement on each of them In such a case, the probability that a set of outcomes

n is consistent and the image of

where the summation⎩

x is taken over all x = {x1, , x n } that each element x i

where nHis a number of times that outcome H is obtained Therefore, the estimator

γest= nH/n is an unbiased estimator of γ.

3.1 Parameter Estimation of Probability Distributions 23

Example 6 (Gaussian distribution)

Eθ,n [μest

n ] = μ, E[(η2)est] = η2, (3.20c)Varθ,n [ϕest

Example 8 (PVM measurement on a qubit) Let us consider the estimation of  ˆη z∈ =

γ z from the outcome of the measurement given by (3.9)

where n+(n−) is the number of times that outcome+ (−) is obtained

3.2 Cramér-Rao Inequality and Fisher Information

The precision of the estimate, which is measured by the variance Varθ [ϕest] or

Varθ,n [ϕest

n ] for i.i.d case, is one of the most important property about the estimator

The following inequality gives the lower bound of the variance The inequality meansthat there exists an upper bound of the precision that no estimator can violate

Theorem 1 (Cramér-Rao inequality [2]) For an arbitrary estimator ϕ est , the lowing inequality is satisfied:

fol-Varθ [ϕ est] ∗ λψ

λθ J θ−1



λψ λθ

T

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and J ∈ Rm ×m is an semi-positive matrix called the Fisher information matrix [ 1 ]

whose i j -element is defined by

Moore-Penrose pseudoinverse [ 4 , 5 ] (see Appendix 3.4).

If the estimatorϕestis locally unbiased atθ, (3.23) is reduced to the followinginequality:

n ] If the estimator ϕest

n is asymptotically unbiased, it satisfies

T

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3.2 Cramér-Rao Inequality and Fisher Information 25

The estimator that satisfies the equality of (3.29) is called an asymptotically efficientestimator

Before proving the Cramér-Rao inequality, we show several relations about the

support and image of J θ, Varθ [ϕest], and λψ λθ

Lemma 1 The variance Var θ [ϕ est ], Fisher information J θ , and λψ

λθ satisfy the

fol-lowing relations:

img



λψ λθ



⊗ supp(Var[ϕ est ]), (3.30)supp



λψ λθ

λ i ψ = 

x∈X

p (x; θ)[ϕest(x) − ψ]λ i log p (x; θ). (3.34)Because of the semi-positivity of Varθ [ϕest], for a ∈ ker(Var θ [ϕest]), we have

a· Varθ [ϕest]a = 0, ⇔ a · (ϕest(x) − ψ) = 0 for all x ∈ X ,

⇒



λψ λθ

T

and this is equivalent to (3.30)

By using a calculus similar to (3.35), we obtain for a∈ ker(J ε ),

a· J θa = 0 ⇔ a · ∇θ log p (x; θ) = 0 for all x ∈ X

26 3 Classical Estimation Theory

whereλ i := (λ1, , λ m )T, and a· ∇θ =⎩m

i=1a i λ i Therefore, we obtainker(J θ ) ⊗ ker

λψ

λθ



Proof (Proof of Theorem 1) For a ∈ Rl, b∈ Rk, by using the Cauchy-Schwarzinequality, we have

≤ (a · Var θ [ϕest]a)(b · J θb). (3.39)

Therefore, for b∈ ker(J θ ),

a· Varθ [ϕest]a ∗ (a · λψ λθb)2

is satisfied Since the left-hand side of (3.40) is independent of b, the strongest bound

is given by maximizing the right-hand side with respect to b By defining c:= J θ1b for b∈ supp(J θ ), we have

3.2 Cramér-Rao Inequality and Fisher Information 27

Lemma 2 For an arbitrary estimator ϕ est , the following inequality is satisfied:



λψ λθ

is satisfied for all a∈ ker(Var θ [ϕest]) Following a procedure similar to the proof of

the Cramér-Rao inequality, we obtain

Since b is arbitrary, (3.44) is obtained ⊥

Corollary 1 The necessary condition for the existence of locally unbiased estimators

or asymptotically unbiased estimators of ϕ(θ) is

supp



λϕ λθ



⊗ supp(J θ ). (3.47)

Proof It is given by (3.31) in Lemma 1 ⊥

Example 9 (Coin toss) The Fisher information in (3.3) is

Therefore, the estimator (3.19) is an efficient estimator ofγ.

Example 10 (Gaussian distribution) The Fisher information in (3.4) is

Therefore, the estimator μest

n in (3.20) is efficient, and (η2)est

n is asymptoticallyefficient

Example 11 (Poisson distribution) The Fisher information in (3.5) is

28 3 Classical Estimation Theory

Therefore, the estimatorμest

ez · J θ−1ez = 1 − γ2

z =  ˆη2

z ∈ −  ˆη z∈2. (3.53)Therefore, the estimator in (3.22) is efficient

3.3 Monotonicity of the Fisher Information

If an information processing is performed onX , the outcome Y is also a random

variable For example, for the case in which a dice is thrown,X = {1, 2, 3, 4, 5, 6}.

By computing whether the realization x ∈ X is odd (y = 1) or even (y = 0),

another random variableY = {0, 1} is generated The information processing loses

the information about the input data, for example, whether x = 1, 3, or 5 cannot

be distinguished from the output data y= 1 In general, the information processing

can be described as a Markov mapping fromX to Y The probability distribution

A relation between the Fisher information given by p (x) and q(y) is given by the

following theorem

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3.3 Monotonicity of the Fisher Information and ˇ Cencov’s Theorem 29

Theorem 2 (Monotonicity of the Fisher information) Suppose that p (x; θ) and

q (y; θ) are connected by a transition probability distribution κ(y; x) of a Markov

mapping The Fisher information J p and J q of each probability distribution satisfy the following inequality, called an information processing inequality:

Since a is arbitrary, (3.56) is obtained ⊥

The above theorem states that the Fisher information decreases monotonicallyunder Markov mapping ˇCencov [6,7] shows that the Fisher information is uniquelydetermined from the monotonicity condition by ignoring a constant factor

Theorem 3 ( ˇCencov’s theorem) Let P be a statistical model with a coordinate

system θ = (γ1, , γ m ) The monotone metric g θ := g[p(x; θ)] on P is given

by c J θ , where c > 0 is an arbitrary positive number Here, by the metric being

monotone, we mean that for q (y; θ) obtained by Markov mapping of p(x; θ), the

metric satisfies

g [p(x; θ)] ∗ g[q(y; θ)]. (3.59)The metric on the statistical model measures the distinguishability of two probability

distribution p (x; θ) and p(x; θ +φθ) It seems intuitively natural that the Cramér-Rao

inequality is satisfied

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