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Springer Theses
Recognizing Outstanding Ph.D. Research

Yu Watanabe

Formulation of
Uncertainty Relation
Between Error
and Disturbance in
Quantum Measurement
by Using Quantum
Estimation Theory

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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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Supervisor
Prof. Masahito Ueda
The University of Tokyo
Tokyo
Japan

Author (Current Address)
Dr. Yu Watanabe
Kyoto University
Kyoto
Japan

ISSN 2190-5053
ISBN 978-4-431-54492-0
DOI 10.1007/978-4-431-54493-7

ISSN 2190-5061 (electronic)
ISBN 978-4-431-54493-7 (eBook)


Springer Tokyo Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013947354
Ó 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 Noisy
Quantum Systems, Phys. Rev. Lett. 104, 020401 (2010).

(ii) Y. Watanabe, T. Sagawa, and M. Ueda, Uncertainty Relation Revisited from
Quantum 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 investigate
uncertainty relations between error and disturbance in quantum measurement. In
his seminal work, Heisenberg discussed a thought experiment concerning the
position measurement of a particle by using a gamma-ray microscope, and discovered a trade-off relation between the error of the measured position and the
disturbance on the quantum-mechanically conjugate momentum caused by the
measurement process. This trade-off relation epitomizes the complementarity in
quantum measurements: we cannot perform a measurement of an observable
without 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 measurement
theory had not been established. Meanwhile, Kennard and Robertson discussed a
different type of inequality concerning inherent fluctuations of observables. This
version of Heisenberg’s uncertainty relation is commonly described in quantum
mechanics textbooks and often erroneously interpreted as a mathematical formulation of the complementarity. From the modern point of view, Heisenberg’s
uncertainty relation is the trade-off relation between the information gain about an
observable and the concomitant information loss about its conjugate observable. In
this thesis, Dr. Watanabe argues that the best solution to this problem is to apply
the estimation theory to the outcomes of the measurement for quantifying the error
and disturbance in quantum measurement. He has successfully formulated the
error and disturbance in terms of the Fisher information content, which gives the
upper bound of the accuracy of the estimation. Moreover, Dr. Watanabe has
derived the attainable bound of the error and disturbance in quantum measurement.

The obtained bound is determined by the quantum fluctuations and correlation
functions of the observables, which characterize the non-classical fluctuation of
the observables. Notably, this bound is stronger than the conventional one set by
the commutation relation of the observables. I believe that this thesis provides a

vii


viii

Supervisor’s Foreword

groundbreaking work that establishes the fundamental bound on the accuracy of
one measured observable and the disturbance on the conjugate observable in the
original spirit of Heisenberg, and I expect that the method developed here will be
applied to a broad class of problems related to quantum measurement.
Tokyo, 31 March 2012

Prof. Masahito Ueda


Acknowledgments

I would like to thank my supervisor, Prof. Masahito Ueda for providing helpful
comments and suggestions. I would like to thank Takahiro Sagawa for a work on
error in quantum measurement and uncertainty relations; Yuji Kurotani for guiding
me to uncertainty relations and quantum measurement theory when I was an
undergraduate 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 for

valuable discussions. Finally, I am grateful to the numerous researchers who have
provided me with opportunities for many helpful discussions.

ix


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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
5

2

Reviews of Uncertainty Relations . . . . . .
2.1 Heisenberg’s Gamma-Ray Microscope
2.2 Von Neumann’s Doppler Speed Meter
2.3 Kennard-Robertson’s Inequality
and Schrödinger’s Inequality . . . . . . .
2.4 Arthurs-Goodman’s Inequality . . . . . .
2.5 Ozawa’s Inequality. . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .

7

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Expansion of Linear Operators by Generators
of Lie Algebra su(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Generators of Lie Algebra suðdÞ . . . . . . . . . . . . . . . . . . . . . .

45
45

Classical Estimation Theory . . . . . . . . . . . . . . . . . .
3.1 Parameter Estimation of Probability Distributions
3.2 Cramér-Rao Inequality and Fisher Information . .
3.3 Monotonicity of the Fisher Information
ˇ encov’s Theorem . . . . . . . . . . . . . . . . . . .
and C
3.4 Maximum Likelihood Estimator. . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantum Estimation Theory. . . . . . . . . . . . . . . .
4.1 Parameter Estimation of Quantum States. . . . .
4.2 Monotonicity of the Fisher Information
in Quantum Measurement . . . . . . . . . . . . . . .
4.3 Quantum Cramér-Rao Inequality and Quantum
Fisher Information . . . . . . . . . . . . . . . . . . . .
4.4 Adaptive Measurement . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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xii

Contents

5.2
5.3
5.4

Quantum State and Bloch Space . . . . . . . . . . . . . . . . .
Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum Measurement . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Positive Operator-Valued Measure (POVM)
Measurement . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Projection-Valued Measure (PVM) Measurement
and Spectral Decomposition . . . . . . . . . . . . . . .
5.5 Quantum Operation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Unitary Evolution . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Interaction with an Environment . . . . . . . . . . . .
5.5.3 Measurement Processes . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Lie Algebraic Approach to the Fisher Information Contents
6.1 Classical Fisher Information . . . . . . . . . . . . . . . . . . . . .
6.1.1 Positive State Model . . . . . . . . . . . . . . . . . . . . .
6.1.2 Block Diagonal State Model . . . . . . . . . . . . . . . .
6.1.3 Decohered State Model . . . . . . . . . . . . . . . . . . .
6.2 SLD Fisher Information . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Positive State Model . . . . . . . . . . . . . . . . . . . . .
6.2.2 Block Diagonal State Model . . . . . . . . . . . . . . . .
6.2.3 Decohered State Model . . . . . . . . . . . . . . . . . . .
6.3 RLD Fisher Information . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Positive State Model . . . . . . . . . . . . . . . . . . . . .
6.3.2 Block Diagonal State Model . . . . . . . . . . . . . . . .
6.3.3 Decohered State Model . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7

Error and Disturbance in Quantum Measurements . . . . .
7.1 Error in Quantum Measurement . . . . . . . . . . . . . . . . .
7.1.1 Comparison with the Error Defined by Arthurs
and Goodman . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Comparison with the Error Defined by Ozawa .
7.2 Disturbance in Quantum Measurement . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Uncertainty Relations Between Measurement Errors
of Two Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Heisenberg-Type Uncertainty Relation . . . . . . . . . . . . . . . .
8.3 Attainable Bound of the Product of the Measurement Errors .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

9


xiii

Uncertainty Relations Between Error and Disturbance
in Quantum Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Heisenberg’s Uncertainty Relation in Terms of Fisher
Information Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Attainable Bound of the Product of Error and Disturbance . . . .

115
117

10 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121
122

115


Chapter 1

Introduction

In 1927, W. Heisenberg [1, 2] discussed a thought experiment about the position
measurement of a particle by using a γ -ray microscope. He argued that more accurately the position is measured, the more strongly the momentum of the particle is
disturbed by the backaction of the measurement by using semi-classical analysis,
and derived a trade-off relation between the error ε(x) in the measured position x
and the disturbance η( px ) in the momentum px caused by the measurement process:
ε(x)η( px )


.

(1.1)

This inequality epitomizes the complementarity in quantum measurements: we cannot perform a measurement of an observable without causing disturbance in its
canonically conjugate observable. The error ε(x) in the position measurement characterizes the accuracy of the estimation of x from the measurement outcomes. The
measurement process randomly changes the momentum px , therefore the original
momentum cannot be estimated accurately from the post-measurement particle. The
disturbance η( px ) characterizes the accuracy of the estimated value of the original
px from the post-measurement particle.
Neumann [3, 4] discussed a thought experiment on the measurement of the
momentum of a particle by using the Doppler effect, and derived the trade-off relation between the error in the momentum and the disturbance in the position caused
by the measurement process:
.
(1.2)
η(x)ε( px )
By considering a sequence of measurements, i.e., first, performing the position
measurement by the γ -ray microscope, and then, performing the momentum measurement by the Doppler speed meter on the post-measurement state, the following
inequality can be expected:
.
(1.3)
ε(x)ε(
ˆ pˆ x )

Y. Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance
in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses,
DOI: 10.1007/978-4-431-54493-7_1, © Springer Japan 2014

1



2

1 Introduction

In the early days of quantum mechanics, the Kennard-Robertson inequality [5, 6]
ˆ ( B)
ˆ ≥
σ ( A)σ

1
ˆ B]⊂
ˆ
[ A,
2

(1.4)

was erroneously interpreted as a mathematical formulation of the trade-off relation
ˆ := T r [ρˆ A]
ˆ is
between error and disturbance in a quantum measurement, where A⊂
ˆ
the expectation value of an observable A over the quantum state ρ,
ˆ the square bracket
ˆ B]
ˆ := Aˆ Bˆ − Bˆ A,
ˆ and σ ( A)
ˆ 2 := Aˆ 2 ⊂ − A⊂

ˆ 2 . The
denotes the commutator [ A,
Kennard-Robertson inequality actually implies the indeterminacy of quantum states:
non-commuting observables cannot have definite values simultaneously. However,
ˆ does not depend on the measurement process, the Kennard-Robertson
since σ ( A)
inequality reflects the inherent nature of a quantum state alone, and does not concern
any trade-off relation between the error and disturbance in the measurement process.
In 1988, Arthurs and Goodman [7] considered a simultaneous measurement of
two non-commuting observables Aˆ and Bˆ in a fully quantum mechanical treatment.
Because Aˆ and Bˆ 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 by
letting the system interact with another system, called the apparatus. By considering
the interaction between the system and apparatus, they considered an indirect measurement. In order to make the outcomes of the indirect measurement meaningful
ˆ they assumed the unbiasedness of the measurement outcomes: that is,
for Aˆ and B,
ˆ and B⊂
ˆ for an arbithe expectation values of the outcomes respectively equal to A⊂
ˆ can be
trary quantum state. The unbiasedness of the measurement implies that A⊂
estimated directly from the distribution of the measurement outcomes. They showed
that the variances of the measurement outcomes satisfy
ˆ B]⊂
ˆ .
ˆ ∼ ( B)
ˆ ≥ [ A,
σ ∼ ( A)σ

(1.5)


Comparing this result with the Kennard-Robertson inequality, we find that the lower
bound is doubled. Fluctuations of the measurement outcomes originate from the
system’s inherent fluctuations and the error in the measurement process, namely,
ˆ B]⊂
ˆ , and the bound in
each source of fluctuations has the lower bound of 21 [ A,
(1.5) is doubled as the total. Because the measurement discussed by Arthurs and
Goodman is restricted to the unbiased measurement, a natural question arises as to
what happens for the biased measurement case.
Ozawa [8–10] generalized the Arthurs-Goodman inequality by removing the unbiasedness condition, and presented the following inequality:
ˆ B)
ˆ + ε( A)σ
ˆ ( B)
ˆ + σ ( A)ε(
ˆ B)
ˆ ≥
ε( A)ε(

1
ˆ B]⊂
ˆ .
[ A,
2

(1.6)

ˆ is always finite, if the error ε( A)
ˆ vanishes, the product of
Because the error ε( A)
ˆ

ˆ
the measurement errors ε( A)ε( B) also vanishes. Thus, the Heisenberg-type trade-off


1 Introduction

3

relation can be violated:
ˆ B)
ˆ = 0 ≤ 1 [ A,
ˆ B]⊂
ˆ .
ε( A)ε(
2

(1.7)

However, if the measurement does not satisfy the unbiasedness condition, his definition of the measurement error does not correspond to the accuracy of the estimation.
For example, if the outcome of a measurement is fixed for an arbitrary state ρ,
ˆ the error
ˆ can vanish even if we cannot estimate A⊂.
ˆ Such a result originates from ignorε( A)
ing the estimation process which must inevitably be accompanied in the unbiased
ˆ caused by the backaction of
measurement. Ozawa also defined the disturbance η( B)
the measurement, and derived the following inequality:
ˆ B]⊂
ˆ .
ˆ B)

ˆ + ε( A)σ
ˆ ( B)
ˆ + σ ( A)η(
ˆ B)
ˆ ≥ 1 [ A,
ε( A)η(
2

(1.8)

However, his definition of the disturbance again does not correspond to the estimation accuracy of the original state either. We consider that to analyze the error and
disturbance in quantum measurement, it is crucial to clarify the role of the estimation
which must be made on the measurement outcomes.
Estimation theory [11–13] provides us a description of how accurately we can
estimate values and how much information we can obtain from realizations of the
probabilistic phenomena. In quantum theory, measurements on the quantum system
are necessary to obtain some pieces of information about the quantum system, and
the measurement outcomes are obtained according to the probability distribution.
Thus, it is necessary to involve the estimation theory for clarifying the uncertainty
relations about the error and disturbance in quantum measurements. In estimation
theory, one of the most important quantities is the Fisher information [11], which
gives the upper bound on the accuracy of the estimated value.
In this thesis, we develop a general theory of error and disturbance in quantum
measurements. We show that the unbiasedness is necessary not for the measurements,
but for the estimation from the measurement outcomes. From that analysis, we can
relax the restriction of the unbiased measurement, and define the error and disturbance
in an arbitrary measurement process. By invoking the estimation theory, we show
that the measurement error can be quantified as
ˆ M) = a · [J (M)−1 − J −1 ]a,
ε( A;

Q

(1.9)

where J (M) is the Fisher information obtained by the measurement M, J Q is the
quantum Fisher information [14] about the original quantum state, and a is a set of
ˆ As shown in Chaps. 3 and 4, a· J (M)−1 a
parameters that determines the observable A.
gives the accuracy of the estimation, and a·J Q−1 a characterizes the inherent fluctuation
of the observable. Since the observable is inherently fluctuated, the accuracy of the
ˆ M) is always
estimation is bounded by the inherent fluctuation. Therefore, ε( A;
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
ˆ K) = b · [J ∼−1 − J −1 ]b,
(1.10)
η( B;
S
S
where JS is the symmetric logarithmic derivative (SLD) Fisher information about the
original quantum state, and J ∼ S is the SLD Fisher information contained in the postmeasurement state. The disturbance characterizes the loss of the Fisher information
caused 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 the
following uncertainty relations:
2
1
ˆ B]⊂
ˆ
[ A,
,
4
2
ˆ B]⊂
ˆ
ˆ B)
ˆ ≥ 1 [ A,
ε( A)η(
.
4

ˆ B)
ˆ ≥
ε( A)ε(

(1.11)
(1.12)

In these inequalities, the lower bounds cannot be attained in general. We present new
inequalities whose lower bounds are attainable:
ˆ 2 σ Q ( B)
ˆ 2 − CQS ( A,
ˆ B)

ˆ 2,
ˆ B)
ˆ ≥ σ Q ( A)
ε( A)ε(

(1.13)

ˆ B)
ˆ ≥ σ Q ( A)
ˆ σ Q ( B)
ˆ − CQS ( A,
ˆ B)
ˆ ,
ε( A)η(

(1.14)

2

2

2

ˆ and CQS ( A,
ˆ B)
ˆ are quantum fluctuation and correlation function.
where σ Q ( A)
ˆ and correlation function
As shown in Sect. 8.3, the quantum fluctuation σ Q ( A)
ˆ

ˆ
CQS ( A, B) characterize non-classical fluctuation and correlation in quantum
systems.
This thesis is organized as follows (The flowchart of this thesis is shown in
Fig. 1.1). In Chap. 2, we review several uncertainty relations: Heisenberg’s γ -ray
microscopy, von Neumann’s Doppler speed meter, Kennard-Robertson’s inequality, Schrödinger’s inequality, Arthurs-Goodman’s inequality, and Ozawa’s inequality. In Chap. 3, we review estimation theory and introduce the Fisher information.
In Chap. 4, we review quantum estimation theory and introduce the quantum Fisher
information. In Chap. 5, we develop techniques to expand relevant operators in terms
of the generators of Lie algebra su(d). This expansion method greatly facilitates the
calculation of the Fisher information contents, error and disturbance in quantum measurement. In Chap. 6, we calculate various Fisher information by using the techniques
of the expansion by the generators of Lie algebra su(d). These Fisher information
contents are used for defining error and disturbance and showing uncertainty relations. In Chap. 7, we show why estimation theory is crucial to analyze error and
disturbance in quantum measurement, and define the error and disturbance in terms
of Fisher information contents. In Chap. 8, we derive uncertainty relations of the
measurement errors of two observables. In Chap. 9, we derive uncertainty relations


1 Introduction
Chpter 2
Reviews of
Unvertainty Relations

5
Chpter 3
Classical Estimation Theory

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

Lie Algebra

Chpter 4
Quantum Estimation Theory

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 discuss
some outstanding issues.
The results in Chaps. 5 and 6 are based on Ref. [15] collaborating with Sagawa and
Ueda. The results in Chap. 6, Sect. 7.1 and Chap. 8 are based on Ref. [16] collaborating

with Sagawa and Ueda. The results in Sect. 7.2 and Chap. 9 are 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
12.
13.
14.
15.
16.
17.

1 Introduction
H. Cramér, Mathematical Methods of Statistics. (Princeton University Press, Princeton, 1946)

E. Lehmann, G. Casella, Theory of Point Estimation. (Springer Verlag, New York, 1983)
C.W. Helstrom, Phys. Lett. A 25, 101 (1967)
Y. Watanabe, T. Sagawa, M. Ueda, Phys. Rev. Lett. 104, 020401 (2010)
Y. Watanabe, T. Sagawa, M. Ueda, Phys. Rev. A 84, 042121 (2011)
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 microscope
and von-Neumann’s Doppler speed meter. These uncertainty relations epitomize
trade-off relation between error and disturbance in quantum measurement process.
Next, we review a different type of uncertainty relations: Kennard-Robertson’s
inequality and Schrödinger’s inequality. These characterize trade-off relations of
inherent fluctuations of observables. Finally, we review Arthurs-Goodman’s inequality 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 experiment
about the position measurement of a particle by using a γ -ray microscope, and found
the following trade-off relation between the error ε(x) in the measured position x
and the disturbance η( px ) in the momentum px caused by the measurement process:
ε(x)η( px )

.

(2.1)


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:
L1 ≥
x=
x,
(2.2)
L2

Y. Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance
in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses,
DOI: 10.1007/978-4-431-54493-7_2, © Springer Japan 2014

7


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8

2 Reviews of Uncertainty Relations

where L 1 and L 2 are 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 by Δx from the focal point P, the difference between the optical path
lengths AB and AC is
AB − AC =

L 21 + (a + Δx)2 −
2aΔx
L 21 + a 2

L 21 + (a − Δx)2

= 2Δx sin θ,

(2.3)

where a is the aperture of the lens, and θ is the angle described in Fig. 2.1. To
resolve the shift Δx by determining the position x ≥ of the blip, it is necessary that
the difference between the path lengths is larger than the wavelength λ. Therefore,
the distinguishable minimal shift of the position is
2Δx sin θ = λ

⊂

Δx =

λ
.
2 sin θ

Fig. 2.1 Heisenberg’s γ -ray microscope. If the particle shifts its position by Δx
cannot distinguish the shift


www.Ebook777.com

(2.4)

λ/2 sin θ, we


2.1 Heisenberg’s Gamma-Ray Microscope

9

Therefore, the estimated position x involves the error
ε(x) =

λ
2 sin θ

(2.5)

even if we determine x ≥ accurately.
Next, we consider the disturbance caused by the measurement process. After
the 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 change Δpx accurately. The uncertainty of the
momentum change is given by
η( px ) =

2
sin θ.

λ

(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 position
measurement and the semi-classical analysis of the quantum measurement: that is,
the particle was assumed to possess definite position and momentum. To rigorously
prove the complementarity in quantum measurements, we need to use quantum measurement theory [3, 4]. However, at the time Heisenberg found the trade-off relation,
quantum measurement theory was not established yet. Quantum measurement theory
was established in the 1970s by Davies and Lewis [3].

2.2 Von Neumann’s Doppler Speed Meter
Heisenberg’s γ -ray microscope measures the position of a particle and causes the
disturbance in the momentum. Von Neumann [5, 6] considered a thought experiment of the momentum of a particle by using a Doppler speed meter,and found the
following trade-off relation between the error ε( px ) of the measured momentum px
and the disturbance η(x) in the position caused by the measurement process:
η(x)ε( px )

.

(2.7)

Note that the roles of x and px are exchanged in comparison with Heisenberg’s uncertainty relation (2.1). This inequality shows that we cannot measure the momentum
without causing disturbance in the position of the particle (Fig. 2.2).
Suppose that we measure the momentum px of a particle with mass m. First, we
prepare a photon with frequency ω 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

2vx /c
δω
=
ω
1 + vx /c

2vx
,
c

(2.8)

where vx 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 velocity vx and the momentum
px as
c δω
mc δω
,
px =
.
(2.9)
vx =
2ω

2ω
However, we can prepare the photon with only a limited accuracy about the frequency.
The accuracy of the frequency is Δω ∼ τ −1 . Therefore, the error in the estimated
momentum px is
mc
ε( px ) =
.
(2.10)
2ωτ
If we use a photon with the higher frequency and shorter duration, the measured
position can be more accurate.
Next, we consider the disturbance in the position of the particle caused by the
measurement process. After reflecting the photon, the momentum of the particle
changes with 2 ω/c. Because the photon emits with duration τ , we cannot find
the exact time of the reflection, and the uncertainty of the reflection time is τ . The
uncertainty of the position is calculated to be
η(x) =

2 ω
τ.
mc

(2.11)

Therefore, we obtain (2.7).
By considering a sequence of measurements, i.e., first, performing the position
measurement by using the γ -ray microscope, and then, performing the momentum
measurement by using the Doppler speed meter with a high frequency and a short
duration, on the post-measurement state, the following inequality can be derived:
ε(x)ε( px )


.

(2.12)

This inequality shows that we cannot simultaneously perform a measurement of
the position and momentum accurately. If the position is measured accurately, the
accuracy of the measured momentum decreases, and vice versa.


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

11

2.3 Kennard-Robertson’s Inequality and Schrödinger’s
Inequality
In 1927, Kennard [7] proved that the inherent fluctuations of the position and momentum are bounded by the Plank constant:
σ (x)σ
ˆ ( pˆ x ) ≤

2

,

(2.13)

ˆ Kennard’s inequality implies the
where σ (x)
ˆ := ∃xˆ 2 − ∃xˆ 2 , and ∃xˆ := Tr[ρˆ x].
indeterminacy of the quantum state, that is, the position and momentum cannot be

definite simultaneously. In the early days of quantum mechanics, this inequality was
erroneously interpreted as a mathematical formulation of the Heisenberg’s uncertainty relation. However, σ (x)
ˆ implies the inherent fluctuation of the observable xˆ
and depends only on the quantum state ρ.
ˆ 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, and
found the following inequality:
ˆ ( B)
ˆ ≤
σ ( A)σ

1
ˆ B]
ˆ ,
∃[ A,
2

(2.14)

ˆ B]
ˆ := Aˆ Bˆ − Bˆ A.
ˆ Moreover,
where the square brackets denote the commutator: [ A,
Schrödinger [9] generalized Robertson’s inequality as
ˆ 2 σ ( B)
ˆ Bˆ
ˆ 2 − CS ( A,
ˆ B)
ˆ 2 ≤ 1 ∃ A,

σ ( A)
4

2

,

(2.15)

ˆ B)
ˆ is a symmetrized correlation function of the observables defined as
where CS ( A,
ˆ B)
ˆ :=
CS ( A,

1 ˆ ˆ
∃{ A, B} − ∃ Aˆ ∃ Bˆ ,
2

(2.16)

ˆ B}
ˆ := Aˆ Bˆ + Bˆ A.
ˆ
where the curly brackets denote the anti-commutator: { A,
From Schrödinger’s inequality, Kennard’s inequality and Robertson’s inequality are
directly derived. Thus, we prove Schrödinger’s inequality here.
ˆ B)
ˆ be a non-symmetrized correlation function of the observables

Let C( A,
defined as
ˆ B)
ˆ := ∃ Aˆ Bˆ − ∃ Aˆ ∃ Bˆ ,
C( A,
(2.17)
and K ∈ C2×2 be a Hermitian matrix defined as
K :=

ˆ B)
ˆ
ˆ 2 C( A,
σ ( A)
.
2
ˆ
ˆ
ˆ
C( B, A) σ ( B)

(2.18)


12

2 Reviews of Uncertainty Relations

For an arbitrary complex vector z = (z 1 , z 2 )T ∈ C2 , where T denotes the transpose,
we have
ˆ 2 + 2Re z 1∗ z 2 C( A,

ˆ B)
ˆ + |z 2 |2 σ ( B)
ˆ 2
z · K z = |z 1 |2 σ ( A)
ˆ † (z 1 Aˆ + z 2 B)
ˆ − ∃z 1 Aˆ + z 2 Bˆ
= ∃(z 1 Aˆ + z 2 B)

2

≤ 0.

(2.19)

Therefore, K is positive semi-definite, that is, the eigenvalues of K are all
non-negative. From the semi-positivity of K , the following inequality is derived:
ˆ 2 − C( A,
ˆ B)C(
ˆ
ˆ A)
ˆ
ˆ 2 σ ( B)
B,
det[K ] = σ ( A)
ˆ B]
ˆ
ˆ 2 − CS ( A,
ˆ B)
ˆ + 1 ∃[ A,
ˆ 2 σ ( B)

= σ ( A)
2
ˆ B]
ˆ
ˆ 2 − CS ( A,
ˆ B)
ˆ 2 − 1 ∃[ A,
ˆ 2 σ ( B)
= σ ( A)
4

ˆ B]
ˆ
ˆ B)
ˆ − 1 ∃[ A,
CS ( A,
2
2

≤ 0.

(2.20)

This completes the proof of Schrödinger’s inequality.

2.4 Arthurs-Goodman’s Inequality
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 fully
quantum-mechanical analysis, and Arthurs and Goodman [11] generalized the measurement scheme for two arbitrary non-commuting observables.

To make both observables simultaneously measurable, it is necessary to extend
the Hilbert space. This can be done by letting the system interact with another system,
ˆ
called the apparatus. Let us consider that we want to measure observables Aˆ and B.
Suppose that the initial state of the system is ρ.
ˆ First, we prepare the state of the
apparatus as ρˆapp , and interact the system and apparatus with the unitary operator Uˆ .
After the interaction, we measure the observables Aˆ ≥ and Bˆ ≥ of the apparatus. To
measure both observables simultaneously, Aˆ ≥ and Bˆ ≥ must commute with each other.
ˆ
In order to make the outcomes of the indirect measurement meaningful for Aˆ and B,
they assumed
ˆ = Tr[Uˆ (ρˆ ⊗ ρˆapp )Uˆ † ( Iˆ ⊗ Aˆ ≥ )],
∃ Aˆ := Tr[ρˆ A]

(2.21a)

ˆ = Tr[Uˆ (ρˆ ⊗ ρˆapp )U ( Iˆ ⊗ B )]
∃ Bˆ := Tr[ρˆ B]

(2.21b)

ˆ†

ˆ≥

for an arbitrary state ρ,
ˆ where Iˆ is a identity operator. Hereforth, we denote Iˆ ⊗ Aˆ ≥ as
≥
ˆ

A for simplicity. These conditions are called unbiasedness conditions of the measurement, and measurements that satisfy the unbiasedness conditions are called unbiased


2.4 Arthurs-Goodman’s Inequality

13

ˆ there always exists a set
measurements.Note that for arbitrary observables Aˆ and B,
of Uˆ , ρˆapp , Aˆ ≥ and Bˆ ≥ that satisfies the unbiasedness condition.The unbiasedness
conditions (2.21) imply that the expectation values ∃ Aˆ and ∃ Bˆ can directly be estimated from the measurement outcomes.The variances of the measurement outcomes
are given by
2

σ ≥ ( Aˆ ≥ ) := Tr[Uˆ (ρˆ ⊗ ρˆapp )Uˆ † Aˆ ≥2 ] − Tr[Uˆ (ρˆ ⊗ ρˆapp )Uˆ † Aˆ ≥ ]
= Tr[Uˆ (ρˆ ⊗ ρˆapp )Uˆ † Aˆ ≥2 ] − ∃ Aˆ 2 ,
ˆ≥

ˆ†

(2.22a)
ˆ†

ˆ≥ 2

σ ( B ) := Tr[Uˆ (ρˆ ⊗ ρˆapp )U B ] − Tr[Uˆ (ρˆ ⊗ ρˆapp )U B ]
≥

ˆ ≥2


= Tr[Uˆ (ρˆ ⊗ ρˆapp )Uˆ † Bˆ ≥2 ] − ∃ Bˆ 2 .

(2.22b)

Let Nˆ Aˆ be a “noise” operator defined as
ˆ
Nˆ Aˆ := Uˆ † Aˆ ≥ Uˆ − A.

(2.23)

From the unbiasedness condition, the noise operator satisfies
Tr[(ρˆ ⊗ ρˆapp ) Nˆ Aˆ ] = 0

(2.24)

for an arbitrary state ρ.
ˆ From this equation, the following equation can be derived:
ˆ
Tr app [(I ⊗ ρˆapp ) Nˆ Aˆ ] = 0,

(2.25)

where Tr app denotes the partial trace over the apparatus system, and 0ˆ is the null
operator. Thus, we have

and

Tr[Uˆ (ρˆ ⊗ ρˆapp )Uˆ † Aˆ ≥2 ] = Tr[(ρˆ ⊗ ρˆapp ) Nˆ A2ˆ ] + Tr[ρˆ Aˆ 2 ],

(2.26)


ˆ 2,
σ ≥ ( Aˆ ≥ )2 = Tr[(ρˆ ⊗ ρˆapp ) Nˆ 2ˆ ] + σ ( A)

(2.27)

A

ˆ 2 := ∃ Aˆ 2 − ∃ Aˆ 2 . Therefore, we can find that the variance of the meawhere σ ( A)
ˆ and error
surement outcome consists of two types of error: inherent fluctuation σ ( A),
2
2
ˆ
ˆ
in the measurement εAG ( A) := Tr[(ρˆ ⊗ ρˆapp ) N ˆ ]. To clarify the role of the error
A
ˆ in the variance σ ≥ ( Aˆ ≥ ), let us consider the commutation relation of the noise
εAG ( A)
operators. It follows from the fact that Aˆ ≥ and Bˆ ≥ commute with each other that
ˆ + [ A,
ˆ Nˆ ˆ ] = −[ A,
ˆ B].
ˆ
[ Nˆ Aˆ , Nˆ Bˆ ] + [ Nˆ Aˆ , B]
B

(2.28)