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

2014
ISBN: 978-4-431-54493-7
Verlag: Springer Japan
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 122 Seiten

Reihe: Springer Theses

In this thesis, quantum estimation theory is applied to investigate uncertainty relations between error and disturbance in quantum measurement. The author argues that the best solution for clarifying the attainable bound of the error and disturbance is to invoke the estimation process from the measurement outcomes such as signals from a photodetector in a quantum optical system. The error and disturbance in terms of the Fisher information content have been successfully formulated and provide the upper bound of the accuracy of the estimation. Moreover, the attainable bound of the error and disturbance in quantum measurement has been derived.
The obtained bound is determined for the first time by the quantum fluctuations and correlation functions of the observables, which characterize the non-classical fluctuation of the observables. The result provides the upper bound of our knowledge obtained by quantum measurements.
The method developed in this thesis will be applied to a broad class of problems related to quantum measurement to build a next-generation clock standard and to successfully detect gravitational waves.

Dr. Yu Watanabe
Kyoto University
Kitashirakawa Oiwake-Cho,
Sakyo-Ku, Kyoto
606-8502 Japan
[e-mail]

Autoren/Hrsg.

Watanabe, Yu

Inhaltsverzeichnis

1;Supervisor’s Foreword;7
2;Acknowledgments;9
3;Contents;10
4;1 Introduction;13
4.1;References;17
5;2 Reviews of Uncertainty Relations;19
5.1;2.1 Heisenberg's Gamma-Ray Microscope;19
5.2;2.2 Von Neumann's Doppler Speed Meter;21
5.3;2.3 Kennard-Robertson's Inequality and Schrodinger's Inequality;23
5.4;2.4 Arthurs-Goodman's Inequality;24
5.5;2.5 Ozawa's Inequality;26
5.6;References;29
6;3 Classical Estimation Theory;30
6.1;3.1 Parameter Estimation of Probability Distributions;30
6.2;3.2 Cramer-Rao Inequality and Fisher Information;34
6.3;3.3 Monotonicity of the Fisher Information and Cencov's Theorem;39
6.4;3.4 Maximum Likelihood Estimator;41
6.5;References;47
7;4 Quantum Estimation Theory;48
7.1;4.1 Parameter Estimation of Quantum States;48
7.2;4.2 Monotonicity of the Fisher Information in Quantum Measurement;49
7.3;4.3 Quantum Cramer-Rao Inequality and Quantum Fisher Information;50
7.4;4.4 Adaptive Measurement;53
7.5;References;55
8;5 Expansion of Linear Operators by Generators of Lie Algebra su(d);56
8.1;5.1 Generators of Lie Algebra su(d);56
8.2;5.2 Quantum State and Bloch Space;58
8.3;5.3 Observable;62
8.4;5.4 Quantum Measurement;64
8.4.1;5.4.1 Positive Operator-Valued Measure (POVM) Measurement;64
8.4.2;5.4.2 Projection-Valued Measure (PVM) Measurement and Spectral Decomposition;65
8.5;5.5 Quantum Operation;67
8.5.1;5.5.1 Unitary Evolution;69
8.5.2;5.5.2 Interaction with an Environment;70
8.5.3;5.5.3 Measurement Processes;72
8.6;References;81
9;6 Lie Algebraic Approach to the Fisher Information Contents;82
9.1;6.1 Classical Fisher Information;82
9.1.1;6.1.1 Positive State Model;84
9.1.2;6.1.2 Block Diagonal State Model;87
9.1.3;6.1.3 Decohered State Model;90
9.2;6.2 SLD Fisher Information;91
9.2.1;6.2.1 Positive State Model;92
9.2.2;6.2.2 Block Diagonal State Model;93
9.2.3;6.2.3 Decohered State Model;95
9.3;6.3 RLD Fisher Information;95
9.3.1;6.3.1 Positive State Model;96
9.3.2;6.3.2 Block Diagonal State Model;97
9.3.3;6.3.3 Decohered State Model;98
9.4;Reference;99
10;7 Error and Disturbance in Quantum Measurements;100
10.1;7.1 Error in Quantum Measurement;100
10.1.1;7.1.1 Comparison with the Error Defined by Arthurs and Goodman;105
10.1.2;7.1.2 Comparison with the Error Defined by Ozawa;106
10.2;7.2 Disturbance in Quantum Measurement;107
10.3;References;111
11;8 Uncertainty Relations Between Measurement Errors of Two Observables;112
11.1;8.1 Setup;112
11.2;8.2 Heisenberg-Type Uncertainty Relation;114
11.3;8.3 Attainable Bound of the Product of the Measurement Errors;115
11.4;References;124
12;9 Uncertainty Relations Between Error and Disturbance in Quantum Measurements;125
12.1;9.1 Heisenberg's Uncertainty Relation in Terms of Fisher Information Contents;125
12.2;9.2 Attainable Bound of the Product of Error and Disturbance;127
13;10 Summary and Discussion;130
13.1;References;131