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E-grāmata: Finite Sample Analysis in Quantum Estimation

  • Formāts: PDF+DRM
  • Sērija : Springer Theses
  • Izdošanas datums: 15-Jan-2014
  • Izdevniecība: Springer Verlag, Japan
  • Valoda: eng
  • ISBN-13: 9784431547778
  • Formāts - PDF+DRM
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  • Formāts: PDF+DRM
  • Sērija : Springer Theses
  • Izdošanas datums: 15-Jan-2014
  • Izdevniecība: Springer Verlag, Japan
  • Valoda: eng
  • ISBN-13: 9784431547778

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In this thesis, the author explains the background of problems in quantum estimation, the necessary conditions required for estimation precision benchmarks that are applicable and meaningful for evaluating data in quantum information experiments, and provides examples of such benchmarks.

The author develops mathematical methods in quantum estimation theory and analyzes the benchmarks in tests of Bell-type correlation and quantum tomography with those methods. Above all, a set of explicit formulae for evaluating the estimation precision in quantum tomography with finite data sets is derived, in contrast to the standard quantum estimation theory, which can deal only with infinite samples. This is the first result directly applicable to the evaluation of estimation errors in quantum tomography experiments, allowing experimentalists to guarantee estimation precision and verify quantitatively that their preparation is reliable.



This book explains the background of problems in quantum estimation, the necessary conditions required for estimation precision benchmarks that are applicable for evaluating data in quantum information experiments, and examples of such benchmarks.
1 Introduction
1(6)
References
4(3)
2 Quantum Mechanics and Quantum Estimation: Background and Problems in Quantum Estimation
7(6)
2.1 Quantum Mechanics: Operational Approach
7(2)
2.2 Quantum Estimation
9(1)
2.3 Problems in Quantum Estimation
10(3)
References
11(2)
3 Mathematical Statistics: Basic Concepts and Theoretical Tools for Finite Sample Analysis
13(14)
3.1 Preliminaries
13(8)
3.1.1 Probability Theory
13(2)
3.1.2 Statistical Parameter Estimation
15(4)
3.1.3 Figure of Merit
19(2)
3.2 Asymptotic Theory
21(3)
3.2.1 Normal Conditions
21(1)
3.2.2 Expected Loss
22(2)
3.2.3 Error Probabilities
24(1)
3.3 Finite Theory for Sample Mean
24(3)
3.3.1 Mean Squared Error
25(1)
3.3.2 Tail Probability
25(1)
References
26(1)
4 Evaluation of Estimation Precision in Test of Bell-Type Correlations
27(10)
4.1 Quantum Non-locality
27(2)
4.2 CHSH Inequality
29(3)
4.3 Test of the CHSH Inequality
32(3)
4.3.1 Estimation Setting
33(1)
4.3.2 Expected Loss
34(1)
4.3.3 Error Probability
35(1)
4.4 Summary
35(2)
References
36(1)
5 Evaluation of Estimation Precision in Quantum Tomography
37(52)
5.1 Estimation Setting
37(11)
5.1.1 Estimation Objects
37(1)
5.1.2 Quantum State Tomography
38(10)
5.2 Expected Loss
48(14)
5.2.1 Extended Linear Estimator
48(4)
5.2.2 Extended Norm-Minimization Estimator
52(1)
5.2.3 Maximum-Likelihood Estimator
53(9)
5.3 Error Probability
62(9)
5.3.1 Extended Linear Estimator
63(3)
5.3.2 Extended Norm-Minimization Estimator
66(3)
5.3.3 Maximum-Likelihood Estimator
69(2)
5.4 History and Related Works
71(1)
5.5 Summary
72(17)
References
85(4)
6 Improvement of Estimation Precision by Adaptive Design of Experiments
89(24)
6.1 Terminology
89(4)
6.1.1 Measurement Update Criterion
89(1)
6.1.2 Notation
90(1)
6.1.3 A-Optimality Criterion
91(2)
6.2 Adaptive One-Qubit State Estimation
93(10)
6.2.1 Estimation Setting
93(2)
6.2.2 Results and Analysis
95(6)
6.2.3 Discussion
101(2)
6.3 History
103(1)
6.4 Summary
104(9)
References
111(2)
7 Summary and Outlook
113(4)
References
115(2)
Curriculum Vitae 117
Dr. Takanori Sugiyama Department of Physics, Graduate School of Science, The University of Tokyo