Violation of an Augmented Set of Leggett-Garg Inequalities and the Implementation of a Continuous in Time Velocity Measurement

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Violation of an Augmented Set of Leggett-Garg Inequalities and the Implementation of a Continuous in Time Velocity Measurement Violation of an augmented set of Leggett-Garg inequalities and the implementation of a continuous in time velocity measurement by Shayan-Shawn Majidy A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Masters of Science in Physics (Quantum Information) Waterloo, Ontario, Canada, 2019 c Shayan-Shawn Majidy 2019 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Macroscopic realism (MR) is the view that a system may possess definite properties at any time independent of past or future measurements, and may be tested experimentally using the Leggett-Garg inequalities (LGIs). In this work we advance the study of LGIs in two ways using experiments carried out on a nuclear magnetic resonance spectrometer. Firstly, we addresses the fact that the LGIs are only necessary conditions for MR but not sufficient ones. We implement a recently-proposed test of necessary and sufficient conditions for MR which consists of a combination of the original four three-time LGIs augmented with a set of twelve two-time LGIs. We explore different regimes in which the two- and three-time LGIs may each be satisfied or violated. Secondly, we implement a recent proposal for a measurement protocol which determines the temporal correlation functions in an approximately non-invasive manner. It employs a measurement of the velocity of a dichotomic variable Q, continuous in time, from which a possible sign change of Q may be determined in a single measurement of an ancilla coupled to the velocity. This protocol involves a significantly different set of assumptions to the traditional ideal negative measurement protocol and a comparison with the latter is carried out. iii Acknowledgements First and foremost, I would like to thank my supervisor Prof. Raymond Laflamme. Ray's insistent curiosity has been contagious and the way he conducts himself on a daily basis has been inspiring. I feel very fortunate to have him as a role model, as a scientist and as a person. Secondly, I would like to express my deep gratitude to Prof. Jonathan Halliwell. Had I known that the initial email I sent him would lead to the relationship I have with him today, I would of sent it much sooner. His wisdom and encouragement have been crucial to navigating through the different stages of my research. Lastly, I would like to thank my examination committee for their time reading my thesis, the Department of Physics and Astronomy and the Institute for Quantum Computing for funding my research and my peers for creating a warm and supportive environment. Lastly, I would like to thank my family and in particular my uncle Payam. If there is anything I accomplish in this life, it's because of him. iv Table of Contents List of Tables viii List of Figures ix 0 Introduction1 1 Violating realism3 1.1 The Bell framework...............................4 1.1.1 Local realism..............................4 1.1.2 Bell's theorem..............................6 1.1.3 The Bell and CHSH inequalities....................7 1.2 Fine's theorem................................. 11 1.2.1 Marginal probabilities......................... 11 1.2.2 Statement of the theorem, definitions and notation......... 13 1.2.3 The arguments of Fine's theorem................... 17 1.3 The Leggett-Garg framework......................... 24 1.3.1 Macroscopic realism.......................... 24 1.3.2 The Leggett-Garg inequalities..................... 27 2 Nuclear magnetic resonance quantum information processing 32 2.1 Fundamentals of NMR............................. 32 v 2.2 Implementation as a quantum computer................... 34 2.2.1 The qubit................................ 34 2.2.2 Initialization............................... 38 2.2.3 Universal set of gates.......................... 39 2.2.4 Measurement.............................. 43 2.2.5 Decoherence............................... 46 3 Violation of an augmented set of Leggett-Garg inequalities 49 3.1 The augmented set of Leggett-Garg inequalities............... 49 3.1.1 Two regimes of interest......................... 50 3.2 Experimental design.............................. 55 3.2.1 Ideal negative measurements...................... 55 3.2.2 Pulse sequence components...................... 59 3.2.3 First set of experiments........................ 61 3.2.4 Second set of experiments....................... 62 3.3 Experimental results.............................. 62 3.3.1 Error bars................................ 65 3.4 Conclusion.................................... 71 4 The implementation of a continuous in time velocity measurement 72 4.1 Continuous in time velocity measurements.................. 72 4.1.1 Theoretical description......................... 73 4.2 Experimental design.............................. 75 4.2.1 Finding parameters........................... 75 4.2.2 Bounds on the violation........................ 80 4.2.3 Pulse sequence components...................... 80 4.2.4 Third set of experiments........................ 81 4.3 Experimental results.............................. 81 4.4 Conclusion.................................... 83 vi 5 Conclusion and discussion 88 5.1 Review of Ch. 3 and Ch. 4 conclusions.................... 88 5.2 Discussion.................................... 89 References 91 vii List of Tables 1.1 The possible measurement outcomes for a triplet state that was generated in the (P; P; N) state...............................7 1.2 The sum of the \probabilities" that pairs of measurements made on a triplet state will return the same result........................9 1 1.3 The possible correlation functions for a spin- 2 system measured at pairs of times....................................... 31 2.1 An overview of how an NMR spectrometer satisfies each of the DiVincenzo criteria for quantum computation........................ 35 3.1 An overview of the the sets of experiments described in Section 3.1.1.... 51 3.2 The experimental data from the first and second set of experiments..... 64 3.3 The values of hQii and Cij and calculated from the data from the first and second set of experiments............................ 66 3.4 The values of the LG2s and LG3s constructed from the first set of experiments. 67 3.5 The values of the LG2s and LG3s constructed from the second set of exper- iments....................................... 68 4.1 An overview of all the experiments discussed in this thesis.......... 73 4.2 The experimental data from the third set of experiments........... 83 4.3 The values of hQii and Cij and calculated from the second and third set of experiments.................................... 84 4.4 The values of the LG2s and LG3s constructed from the third set of experi- ments....................................... 85 viii List of Figures 1.1 The general design of the EPR paradox....................5 1.2 A modification of the EPR paradox which allows for testing the existence of local realism..................................6 1.3 A schematic depiction of a particles spin being measured along one of three co-planar axis...................................8 1.4 Four co-planar axes separated by successive 45◦ angles............ 10 1.5 Different ways of denoting the marginals of a probably distribution..... 12 1.6 An example of a simple method for visualizing the marginals of a probability distribution.................................... 14 1.7 Examples of macroscopic quantum phenomena which have been experimen- tally realised................................... 26 1.8 Violation of the LG3s for measurements made at equidistant time intervals. 29 1.9 A schematic depiction of a particles spin being measured along one axis at three distinct times................................ 30 2.1 An overview of the fundamentals of an NMR experiment as outlined in section 2.1.................................... 33 2.2 The molecular structure of trichloroethylene.................. 37 2.3 A schematic depiction of the pseudo-pure state................ 39 2.4 The effect of an RF pulse being applied to a two nuclei sample....... 41 2.5 The step by step effect of the pulse sequence for implementing the CNOT gate........................................ 42 2.6 The NMR spectra for the XI, XZ, YI and YZ components of the signal... 45 ix 2.7 An example of two spectra decomposed into linear combinations of the XI, YI, XZ ans YZ spectra............................. 46 2.8 The relaxation of the magnetization due to the effect of the T1 relaxation.. 47 2.9 The relaxation of the magnetization due to the effect of the T2 relaxation.. 48 3.1 A simulation of the effect of the decoherence on the correlation functions as a function of !t.................................. 52 3.2 The LG3s plotted as functions of !t with the effect of the decoherence... 52 3.3 The results of a search over the initial states for which the LG2s are, or are not, satisfied................................... 53 3.4 One of the LG2s and the LG3s plotted with the effect of decoherence.... 54 3.5 The quantum circuit used for implementing the INM protocol........ 56 3.6 A pulse sequence for producing the pseudo-pure state............. 60 3.7 A pulse sequence for implementing the CNOT gate in NMR......... 61 3.8 A pulse sequence for implementing the time delay while undoing the effect of the J-coupling................................. 61 3.9 The pulse sequences for the first set of
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