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Uncertainty Relations for Quantum Particles

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Uncertainty Relations for Quantum Particles Uncertainty Relations for Quantum Particles Spyridon Kechrimparis PhD University of York Mathematics December, 2015 2 Abstract The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncer- tainty relations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. Specifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances in terms of the commutators. Furthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schr¨odinger inequality is found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our find- ings are shown to extend to more than one spatial degree of freedom and we derive various types of inequalities. Finally, we propose a generalisation of the Arthurs-Kelly model of joint measure- ment of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of three canonical observables and suitable definitions of error and disturbance we ob- tain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one. 3 Contents Abstract . .3 Contents . .6 Acknowledgements . .7 Declaration . .8 1 Introduction 9 2 Mathematical Preliminaries 16 2.1 Fractional Fourier transform . 17 2.2 Wigner function . 18 2.3 Measures of uncertainty . 19 2.4 The symplectic group and symplectic transformations . 21 2.5 Gateauxˆ differential and variational calculus . 24 3 Uncertainty relations for a canonical triple 27 3.1 Introduction . 27 3.2 Results . 29 3.3 Symmetry of the triple . 31 3.4 Experiments . 32 3.5 Minimal triple uncertainty . 33 3.6 Discussion . 36 4 Uncertainty relations for canonical structures and beyond 39 4.1 Introduction . 39 4.2 Regular canonical polygons . 40 4.3 Symmetry of N canonical operators and entropic inequalities . 44 4 Contents 4.4 N observables of arbitrary “length” . 47 4.5 An inequality for the integral of variances of rotated observables . 52 4.6 Three operators in more than one degree of freedom . 55 4.7 N observables with M degrees of freedom in a product state . 57 4.8 Summary . 61 5 Uncertainty relations for one degree of freedom 62 5.1 Introduction and main result . 62 5.2 Minimising uncertainty . 65 5.2.1 Extrema of uncertainty functionals . 67 5.2.2 Universality . 69 5.2.3 Consistency conditions . 72 5.2.4 Geometry of extremal states . 74 5.2.5 Mixed states . 78 5.2.6 Known uncertainty relations . 79 5.3 New uncertainty relations . 84 5.3.1 Generalizing known relations . 84 5.3.2 Uncertainty functionals with permutation symmetries . 88 5.4 Summary and discussion . 91 6 Uncertainty relations for multiple degrees of freedom and states 96 6.1 Introduction . 96 6.2 Extremal Uncertainty . 97 6.2.1 Solving the eigenvalue equation . 97 6.2.2 Consistency conditions . 99 6.2.3 Convexity . 101 6.2.4 Mixed states . 102 6.3 Applications and examples . 102 6.3.1 One spatial degree of freedom . 102 6.3.2 N spatial degrees of freedom, product states . 103 6.3.3 An inequality including correlations in two spatial degrees of freedom . 106 5 Contents 6.4 Uncertainty in multiple states and degrees of freedom . 108 6.4.1 Uncertainty functionals in more than one states . 108 6.4.2 Functionals for a quantum system in multiple dimensions in more than one states . 111 6.5 Discussion . 113 7 Arthurs-Kelly process for more than two observables 114 7.1 Introduction . 114 7.1.1 Preliminaries . 116 7.2 Joint measurement inequalities for the statistics of the probes . 117 7.2.1 Uncorrelated probes . 117 7.2.2 The effect of correlations between the probes . 125 7.2.3 The analysis of Arthurs and Goodman . 128 7.3 Inequalities for the error and disturbance . 130 7.3.1 Error and disturbance inequalities according to Appleby . 130 7.4 Extending the model of Busch . 135 7.4.1 Joint measurement inequalities . 136 7.5 Discussion . 139 8 Conclusions 140 Appendices 145 A Appendices of Chapter 2 146 A.1 A Baker-Campbell-Hausdorff identity . 146 A.2 Convexity of the uncertainty region . 147 A.3 Extrema of functionals of various general forms . 148 A.3.1 Functions of the form f = f (xy, w) .................. 148 A.3.2 Functions of the form f = f (axm + byk, w) ............. 149 Table of Notation 152 Bibliography 153 6 Acknowledgments There is a long list of people that have contributed one way or the other to the realisa- tion of this thesis and to whom I would like to express my gratitude. First of all, I wish to thank my parents, without their unconstrained support when- ever I was in need, this thesis would not be possible; I will be forever in debt for your contribution. In extension, my brother and all my family members that have been supportive and understanding throughout this time. I also wish to express my gratitude to my supervisor, Stefan Weigert, for his guid- ance throughout these years, his patience, support, the numerous discussions and insightful suggestions. Moreover, I would like to thank him for granting me the free- dom to explore my own ideas, while at the same time making sure I was not getting off track. It was a great experience working with him and his contribution in shaping my current thinking as a researcher is invaluable. In addition, I acknowledge financial support from the State Scholarship Founda- tion (I.K.Y.) in Greece, for awarding me a scholarship covering the first three and a half years of my studies, and a grant from the WW Smith Fund in York, that covered part of my expenses of the last six months. Last but not least, I wish to thank all my friends and fellow travellers in space- time, in York and around the globe, through the interaction with whom the last few years I have been a complete and happy person. Cheers to James, Fabian, Dan, Dave H., Leon, Dave B., Dave C., Umberto, Tom, Ben, Alejandro, Jesus, Emilio, Oliver and Eli, Pablo and Alessi, Ella, Kimi, Vasilis, Christopher, and of course Mr. John Smith. Special thanks to Dave Hunt for useful comments on parts of this thesis. Thank you all. 7 Declaration I declare that the work presented in this thesis, except where otherwise stated, is based on my own research and has not been submitted previously for any degree at this or other university. Other sources are acknowledged by explicit references. Chapters 3 and 5 of this thesis have been published in: S. Kechrimparis and S. Weigert, Phys. Rev. A 90, 062118 (2014). S. Kechrimparis and S. Weigert, arXiv:1509.02146 (2015) (under review). Part of Chapter 6 is also under review while Chapters 4 and 7 contain original research that has not been published. 8 Chapter 1 Introduction Quantum theory is one of the most fundamental and successful models of physical reality known to man. It has far reaching applications in technology and everyday ex- perience; the laser and magnetic resonance imaging (nMRI) are only some examples of its influence on modern life. At a time when it was widely accepted that all major phenomena observed in nature were already described by the contemporary phys- ical theories, Planck’s solution to the black body radiation problem, one of the few remaining obstacles to the perceived complete description of nature, and Einstein’s explanation of the photoelectric effect planted the seeds for the development of a new theory of microscopic phenomena. Its first coherent formulations were developed in the first half of the twentieth century and in the subsequent decades it was put on more solid mathematical grounds. Bohr, Dirac, Heisenberg, Schrodinger,¨ Einstein, Wigner and von Neumann are some of the physicists and mathematicians who substantially contributed to the understanding of the new theory. However, regardless of its exceptional success as a mathematical framework to make testable predictions concerning our experience of interaction with the world, it was obvious, from an early stage on, that there were a number of obstacles challeng- ing some of the standard underlying assumptions of classical physical theories; few would question the implicitly assumed interpretation of what is happening when an apple falls on a physicist’s head, when such a process is described by Newton’s law of gravitation. However, quantum theory brought to the limelight a type of philosophi- cal questions that could no longer be easily suppressed. Even today, a little less than 9 Chapter 1. Introduction a century from the theory’s first formulations, significant foundational problems ex- ist within quantum mechanics. This claim can be demonstrated, for example, by the disagreement of experts in the field, on the meaning of different aspects of the theory [61, 51] and the wide variety of different interpretations [24].
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