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An Investigation of No-Go Theorems in Hidden Variable Models of Quantum Mechanics

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An Investigation of No-Go Theorems in Hidden Variable Models of Quantum Mechanics An investigation of No-Go theorems in Hidden Variable Models of Quantum Mechanics by Navid Siami BSc. Sharif University of Technology, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) March 2016 © Navid Siami, 2016 Abstract Realism defined in EPR paper as “In a complete theory there is an element corresponding to each element of reality.” Bell showed that there is a forbidden triangle (Realism, Quantum Statistics, and Locality), and we are only allowed to pick two out of three. In this thesis, we investigate other inequalities and no-go theorems that we face. We also discuss possible Hidden Variable Models that are tailored to be consistent with Quantum Mechanics and the specific no-go theorems. In the special case of the Leggett Inequality the proposed hidden variable is novel in the sense that the hidden variable is in the measurement device rather than the wave-function. ii Preface This body of work by N. Siami is independent and unpublished. iii Table of Contents Abstract ............................................................................................................................................................................................ii Preface ............................................................................................................................................................................................ iii Table of Contents .......................................................................................................................................................................... iv List of Tables.................................................................................................................................................................................. vi List of Figures ............................................................................................................................................................................... vii Dedication .....................................................................................................................................................................................viii 1. Introduction.......................................................................................................................................................................... 1 2. Body ........................................................................................................................................................................................ 5 2.1. Spin ............................................................................................................................................................................... 5 2.2. Quantum Formalism ................................................................................................................................................ 7 2.3. Constraints on reality..............................................................................................................................................10 2.4. Copenhagen Interpretation ...................................................................................................................................12 2.5. Macro-objectivation problem ................................................................................................................................14 2.6. Einstein’s critic (EPR) and Bohr’s response .....................................................................................................17 2.7. No-go Theorems.......................................................................................................................................................21 2.8. Classification of HVMs ..........................................................................................................................................23 2.9. Pusey Barret Rudolph No-go theorem and Emerson Response ....................................................................26 2.10. Epistemic models are always possible .................................................................................................................32 2.11. Leggett’s inequality .................................................................................................................................................37 iv 2.12. Bohmian Mechanics ...............................................................................................................................................52 2.13. Bell-Mermin Model .................................................................................................................................................53 2.14. Evidence for locality of Quantum Mechanics ...................................................................................................57 3. Conclusion...........................................................................................................................................................................62 Bibliography..................................................................................................................................................................................64 Appendix ........................................................................................................................................................................................71 A. Weak Measurement...............................................................................................................................................................71 A.1. Note ....................................................................................................................................................................................71 A.2. Introduction......................................................................................................................................................................71 A.3. Formalism.........................................................................................................................................................................72 A.4. Multiple Spins ..................................................................................................................................................................75 A.5. One Spin System during information gain ................................................................................................................78 B. Incompleteness of Quantum Mechanics...........................................................................................................................86 v List of Tables Table 1: The bulb has a filter that can either be set to filter Red light or Green light. As a result, each light that is observed does not simply reflect the inner structure of lamp. ........................... 34 vi List of Figures Figure 1: Classical Spin .................................................................................................................. 5 Figure 2: Forbidden Triangle ........................................................................................................ 19 Figure 3: Z, X, and I are Pauli matrices in x and y-direction and identity. IZI means that operator on first and third spin is identity and z-Pauli is acting on the second spin. This is a local observable, but ZZZ in a non-local one. ....................................................................................... 21 Figure 4:v1-v6 are eigenvalues of local observables plugged back into the Mermin star. The eigenvalue of each global observables is a multiplication of eigenvalues of corresponding local observables.................................................................................................................................... 22 Figure 5: Classification of Hidden Variable Models .................................................................... 26 Figure 6: Mermin Star................................................................................................................... 56 Figure 7: Forbidden Triangle ........................................................................................................ 63 Figure 8: If the random walk was not self-interacting one could simply estimate 훼, 훽 by multiple measurements................................................................................................................................ 80 Figure 9: When 휎 ≪ ℏ, the overlap is very small and the first value of 푥 determines how the wave-function will look like finally.............................................................................................. 81 Figure 10: When 휎 ≫ ℏ the wave-function would not change initially but the value of 푥1 does not carry much of information in order to distinguish initial states.............................................. 82 Figure 11: If there was a middle value of 푁 which would give us more information than the strong measurement instead of the tangential behaviour one expects a local maxima................. 84 vii Dedication To David Bohm viii 1. Introduction Quantum Mechanics is a jewel in the toolbox of modern physics. Many experiments have confirmed the validity of this theory. Nevertheless, there still exists many problems regarding the interpretation of this theory. There are many unresolved
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