Characterizing classical and quantum systems from marginal correlations[PhD Thesis]

Title page......Page 1
Abstract......Page 3
Zusammenfassung......Page 5
Contents......Page 7
Symbols......Page 10
1 Introduction......Page 11
2.1.1 Bipartite entanglement......Page 14
2.1.2 Multipartite entanglement......Page 19
2.1.3 SLOCC classification......Page 21
2.1.4 Graph and hypergraph states......Page 22
2.2.1 Bell's theorem......Page 25
2.2.2 Bell inequalities......Page 27
2.2.3 Correlation polytopes......Page 28
2.3.1 Directed and undirected graphs......Page 29
2.3.2 Cause-effect relations......Page 30
2.3.3 Bayesian networks......Page 32
2.3.4 Markov random fields......Page 33
2.3.5 Hidden variables......Page 34
2.3.7 Indefinite causal order......Page 35
2.4 Marginal problem......Page 38
2.5 Entropic inequalities......Page 39
2.5.1 Entropy cone......Page 40
2.5.2 Probability structures......Page 41
2.5.3 The entropic characterization of Bayesian networks......Page 42
2.5.4 The entropic characterization of counterfactuals......Page 43
2.6 Semidefinite Programming......Page 45
3.1 Emergence of entanglement......Page 48
3.2 Statement of the problem......Page 50
3.3 Construction method......Page 51
3.4.1 Three qubits......Page 53
3.4.2 Four and five qubits......Page 54
3.4.3 Generalization to more particles......Page 55
3.5.1 Separability of the higher-order marginals......Page 56
3.5.2 Proving entanglement from a subset of marginals......Page 57
3.5.3 Higher-dimensional systems......Page 59
3.5.4 No localizable entanglement in the marginals......Page 60
3.6 Conclusions......Page 61
4.1 Introduction......Page 63
4.2 Definition of qudit hypergraph states......Page 64
4.3 SLOCC and LU classes of hypergraphs......Page 65
4.3.1 Elementary hypergraphs......Page 66
4.3.2 Tools for SLOCC classification......Page 67
4.4 SLOCC classification of tripartite hypergraph states in dimensions 3 and 4.......Page 70
4.4.1 Classification of 333......Page 71
4.4.2 Classification of 444......Page 72
4.5 Conclusions......Page 77
5.0.1 Properties of graphs and hypergraphs......Page 78
5.1 Adhesivity and independence constraints associated with a marginal scenario......Page 81
5.1.1 Adhesivity of probabilities......Page 82
5.1.2 Marginal scenarios admitting a global extension......Page 83
5.1.3 Maximal set of independence conditions associated with a marginal scenario......Page 84
5.2 Optimal characterization of the marginal scenario for probabilities and entropies......Page 85
5.2.2 Probabilities......Page 86
5.2.3 Entropies......Page 87
5.2.4 Outer approximations of the entropy cone......Page 88
5.3 Indistinguishability of causal structures......Page 89
5.4.1 Inclusions in Obs. 5.4......Page 92
5.4.2 Three cases in Theorem 5.5......Page 94
5.5 Conclusions......Page 97
6 The entropic approach to causal correlations......Page 99
6.2 Bipartite entropic causal inequalities......Page 100
6.2.1 Characterization based on causal Bayesian networks......Page 101
6.2.2 Characterization based on counterfactual variables......Page 110
6.3 Multipartite entropic causal inequalities......Page 114
6.3.1 Causal Bayesian network method......Page 115
6.3.2 Counterfactual variable method......Page 117
6.4 Information bounds in causal games......Page 118
6.5 Discussion......Page 120
Conclusions......Page 122
Appendix A - An Appendix to Chapter 6......Page 124
Bibliography......Page 129
List of publications......Page 142
Acknowledgments......Page 143