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

Dr. Yu Watanabe Kyoto University Kitashirakawa Oiwake-Cho, Sakyo-Ku, Kyoto 606-8502 Japan yuwata@yukawa.kyoto-u.ac.jp

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