Temporal Quantum Correlations and Hidden Variable Models (eBook)

Costantino Budroni (Autor)

eBook Download: PDF

2015 | 1st ed. 2016
XIII, 114 Seiten
Springer International Publishing (Verlag)
978-3-319-24169-2 (ISBN)

Infos

In this thesis, the main approach to the characterization of the set of classical probabilities, the correlation polytope approach, is reviewed for different scenarios, namely, hidden variable models discussed by Bell (local), Kochen and Specker (non-contextual), and Leggett and Garg (macrorealist). Computational difficulties associated with the method are described and a method to overcome them in several nontrivial cases is presented. For the quantum case, a general method to analyze quantum correlations in the sequential measurement scenario is provided, which allows computation of the maximal correlations.

Such a method has a direct application for computation of maximal quantum violations of Leggett-Garg inequalities and it is relevant in the analysis of non-contextuality tests. Finally, possible applications of the results for quantum information tasks are discussed.

Inhalt

Abstract 6
Supervisor's Foreword 8
Contents 11
1 Introduction 14
1.1 Hidden Variable Theories 17
1.2 Local Hidden Variables and Bell's Theorem 19
1.2.1 Local Hidden Variables 19
1.2.2 CHSH Inequality and Bell's Theorem 20
1.2.3 Experimental Tests 22
1.3 Noncontextual Hidden Variables and Kochen-Specker Theorem 22
1.3.1 Noncontextual Hidden Variable Theories 23
1.3.2 Kochen and Specker's Original Problem 25
1.3.3 Kochen-Specker Theorem 27
1.3.4 State-Independent Contextuality 28
1.4 Macrorealist Theories and Leggett-Garg Inequalities 30
1.4.1 Macrorealist Theories 31
1.4.2 Continuous Variables 32
1.4.3 Quantum Violations 33
1.5 Correlations Polytopes 34
1.5.1 Definition 34
1.5.2 Examples 36
1.6 Tsirelson Bound 38
1.6.1 Original Argument 38
1.6.2 Semidefinite Programming Approach 39
1.7 Linear and Semidefinite Programming 42
References 44
2 Noncontextuality Inequalities from Variable Elimination 47
2.1 Extension of Measures and Consistency Conditions 48
2.2 Bell Inequalities 49
2.2.1 CHSH Polytope from Bell-Wigner Polytope 49
2.2.2 Bipartite (2,n) Scenario 51
2.2.3 Two Parties, Three Settings 51
2.2.4 Bipartite (3,n) Scenario 52
2.2.5 Multipartite (m,ƒ,m,n) Scenario 52
2.2.6 Computational Results 52
2.3 Noncontextuality Inequalities: The n-Cycle Scenario 53
2.4 Discussion 56
References 56
3 Optimal Tests for State-Independent Contextuality 58
3.1 Optimization Method 59
3.2 Applications 61
3.2.1 Yu and Oh 61
3.2.2 Extended Peres-Mermin Set 63
3.2.3 Cabello, Estebaranz, and García-Alcaine's 18-Vector Proof 64
3.3 Experimental tests 65
3.4 Discussion 66
References 66
4 Quantum Bounds for Temporal Correlations 68
4.1 Sequential Projective Measurements 69
4.2 The Simplified Method 70
4.3 The General Method 71
4.4 Applications 72
4.5 Details of the Calculations 75
4.5.1 Discussion of the Simplest Leggett-Garg Scenario 75
4.5.2 Detailed Discussion of Sequential Bounds for the N-cycle Inequalities 76
4.5.3 Completeness of the General Method 78
4.5.4 Quantum Bounds for Compatible Measurements in the N-cycle Scenario 79
4.6 Discussion 81
References 82
5 Dimension Witnesses 84
5.1 Noncontextuality and Dimension Witnesses 86
5.1.1 The KCBS Inequality 86
5.1.2 The KCBS Inequality with Incompatible Observables 88
5.1.3 The Peres-Mermin Inequality 89
5.1.4 The PM Inequality with Incompatible Observables 90
5.1.5 Imperfect Measurements 91
5.1.6 Experimental Results 92
5.1.7 Generalizations 93
5.2 Sequential Measurements and Leggett-Garg Inequality 93
5.2.1 The Measurement Scheme 95
5.2.2 A Simple Example 95
5.2.3 Asymptotic Limit 97
5.2.4 Maximum Violations 97
5.2.5 Temporal Versus Spatial Correlations 98
5.3 Details of the Calculations 99
5.3.1 Alternative Proof of Observation 1 99
5.3.2 Detailed Discussion of the N-Cycle Inequalities 99
5.3.3 Detailed proof of Observation 2 101
5.3.4 Proof of Observation 3 103
5.3.5 Imperfect Measurements 108
5.3.6 Asymptotic Value of the Leggett-Garg Correlator for the Precessing Spin Model 112
5.4 Discussion 114
References 115
6 Conclusions 118
References 120
Curriculum Vitae 121

Produkt-Details

• Erscheint lt. Verlag: 22.10.2015
• Reihe/Serie: Springer Theses
• Reihe/Serie: Springer Theses
• Zusatzinfo: XIII, 114 p.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Classical versus quantum probabilities • Contextuality in Quantum Physics • Kochen-Specker Theorem • Leggett-Garg inequalities • Quantum Correlations • Quantum Nonlocality • Sequential measurements of quantum system
• ISBN-10: 3-319-24169-9 / 3319241699
• ISBN-13: 978-3-319-24169-2 / 9783319241692