Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Historical perspective
- 2 Present situation, remaining conceptual difficulties
- 3 The theorem of Einstein, Podolsky, and Rosen
- 4 Bell theorem
- 5 More theorems
- 6 Quantum entanglement
- 7 Applications of quantum entanglement
- 8 Quantum measurement
- 9 Experiments: quantum reduction seen in real time
- 10 Various interpretations
- 11 Annex: Basic mathematical tools of quantum mechanics
- Appendix A Mental content of the state vector
- Appendix B Bell inequalities in non-deterministic local theories
- Appendix C An attempt for constructing a “separable” quantum theory (non-deterministic but local)
- Appendix D Maximal probability for a state
- Appendix E The influence of pair selection
- Appendix F Impossibility of superluminal communication
- Appendix G Quantum measurements at different times
- Appendix H Manipulating and preparing additional variables
- Appendix I Correlations in Bohmian theory
- Appendix J Models for spontaneous reduction of the state vector
- Appendix K Consistent families of histories
- References
- Index
5 - More theorems
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Historical perspective
- 2 Present situation, remaining conceptual difficulties
- 3 The theorem of Einstein, Podolsky, and Rosen
- 4 Bell theorem
- 5 More theorems
- 6 Quantum entanglement
- 7 Applications of quantum entanglement
- 8 Quantum measurement
- 9 Experiments: quantum reduction seen in real time
- 10 Various interpretations
- 11 Annex: Basic mathematical tools of quantum mechanics
- Appendix A Mental content of the state vector
- Appendix B Bell inequalities in non-deterministic local theories
- Appendix C An attempt for constructing a “separable” quantum theory (non-deterministic but local)
- Appendix D Maximal probability for a state
- Appendix E The influence of pair selection
- Appendix F Impossibility of superluminal communication
- Appendix G Quantum measurements at different times
- Appendix H Manipulating and preparing additional variables
- Appendix I Correlations in Bohmian theory
- Appendix J Models for spontaneous reduction of the state vector
- Appendix K Consistent families of histories
- References
- Index
Summary
The Bell theorem can take the form of several inequalities, as we have seen in §4.2. Moreover, since it was discovered, the theorem has stimulated the discovery of several other mathematical contradictions between the predictions of quantum mechanics and those of local realism. We review a few of them in this chapter: GHZ contradictions (§5.1) and their generalization (§5.2), Cabello's inequality (§5.3), and Hardy's impossibilities (§5.4). Finally, in §5.5, we discuss the notion of contextuality and introduce the BKS theorem.
GHZ contradiction
For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism [188, 189]. They involve a sign contradiction (100% violation) for perfect correlations, while the BCHSH inequalities are violated by about 40% (Cirelson bound) and deal with situations where the results of measurements are not completely correlated. In this section, we discuss three-particle systems, but generalizations to N particles are possible (§5.2).
Derivation
GHZ contradictions may occur in various systems, not necessarily involving spins. Initially, they were introduced in the context of entanglement swapping (§6.3.2) for four particles [188] or entanglement of three spinless particles [189].
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- Do We Really Understand Quantum Mechanics? , pp. 100 - 119Publisher: Cambridge University PressPrint publication year: 2012