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Informationally Complete Measurements and Optimal Representations of Quantum Theory

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Informationally Complete Measurements and Optimal Representations of Quantum Theory University of Massachusetts Boston ScholarWorks at UMass Boston Graduate Doctoral Dissertations Doctoral Dissertations and Masters Theses 8-2020 Informationally Complete Measurements and Optimal Representations of Quantum Theory John B. DeBrota Follow this and additional works at: https://scholarworks.umb.edu/doctoral_dissertations Part of the Physics Commons INFORMATIONALLY COMPLETE MEASUREMENTS AND OPTIMAL REPRESENTATIONS OF QUANTUM THEORY A Dissertation Presented by JOHN B. DEBROTA Submitted to the Office of Graduate Studies, University of Massachusetts Boston, in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2020 Applied Physics Program ©2020 by John B. DeBrota All rights reserved INFORMATIONALLY COMPLETE MEASUREMENTS AND OPTIMAL REPRESENTATIONS OF QUANTUM THEORY A Dissertation Presented by JOHN B. DEBROTA Approved as to style and content by: Christopher A. Fuchs, Professor Chairperson of Committee Alioscia Hamma, Associate Professor Member Maxim Olchanyi, Professor Member Ingemar Bengtsson, Professor Stockholm University Member Jonathan Celli, Program Director Applied Physics Program Rahul Kulkarni, Chairperson Physics Department ABSTRACT INFORMATIONALLY COMPLETE MEASUREMENTS AND OPTIMAL REPRESENTATIONS OF QUANTUM THEORY August 2020 John B. DeBrota, B.S., Indiana University Bloomington M.Sc., University of Waterloo Ph.D, University of Massachusetts Boston Directed by Professor Christopher A. Fuchs Minimal informationally complete quantum measurements (MICs) furnish probabilistic representations of quantum theory. These representations cleanly present the Born rule as an additional constraint in probabilistic decision theory, a perspective advanced by QBism. Because of this, their structure illuminates important ways in which quantum theory differs from classical physics. MICs have, however, so far received relatively little attention. In this dissertation, we investigate some of their general properties and relations to other topics in quantum information. A special type of MIC called a symmetric informationally com- plete measurement makes repeated appearances as the optimal or extremal solution in dis- tinct settings, signifying they play a significant foundational role. Once the general struc- ture of MICs is more fully explicated, we speculate that the representation will have unique advantages analogous to the phase space and path integral formulations. On the conceptual iv side, the reasons for QBism continue to grow. Most recently, extensions to the Wigner’s friend paradox have threatened the consistency of many interpretations. QBism’s resolution is uniquely simple and powerful, further strengthening the evidence for this interpretation. v ACKNOWLEDGMENTS I owe my utmost thanks for the success of this work to my advisor and friend Christopher Fuchs. I have never met anyone with Chris’s brand of intuition for the physical and philo- sophical; it has and continues to fascinate and inspire me. Since day one, five years ago, Chris has tirelessly sought opportunities for me and worked on my behalf. I would not be where I am nor who I am had anyone else taken his place. Very special thanks go to Blake Stacey, who has also been with me for the duration. Blake is a remarkable person, one of my closest friends, and most regular collaborator. His depth and breadth of knowledge never cease to amaze me. Very special thanks also go to my family, especially my parents, David and Erin De- Brota. I have had the rare gift of wonderful, supportive parents. At every turn, they made sure that I could follow my passions. I thank the QBists and QBissels, Marcus Appleby, Gabriela Baretto Lemos, Jacques Pienaar, and Rüdiger Schack for all the discussions, drinks, jokes, hikes, and company. They have each significantly shaped my view of the world. In the course of graduate school, the following groups and individuals greatly im- pacted my life. In some way or another, they all partially enabled this dissertation. I thank: The Glütenhäus, Jeffrey La, Kathryn Hausler, and Talena Gandy. My climbing friends, Adrian Collado, Mario Esquivel, Mike Fannon, Sam Lee, Carmelo Locurto, Benjamin Murphy, AJ Pualani, Dan Stuligross, Eric Wong, and each of their families. UMass friends, Benjamin Cruikshank, Vanja Dunjko, Jake Golde, Zai Hwang, Ian Mulligan, and Joanna Ruhl. Perimeter friends, Nick White and Shuyi Zhang. Harald Atmanspacher, Ingemar Bengtsson, Florian Boge, Irina-Mihaela Dumitru, David Glick, Richard Healey, Jan-Åke Larsson, Gustavo Rodrigues Rocha, Thomas Ryckman, and Christopher Timpson. vi TABLE OF CONTENTS ACKNOWLEDGMENTS.................................................................................... vi LIST OF FIGURES ............................................................................................ x CHAPTER Page 1. INTRODUCTION .................................................................................... 1 1.1 Author Contributions ...................................................................... 8 1.2 List of References........................................................................... 12 2. MIC FACTS............................................................................................. 14 2.1 Introduction ................................................................................... 14 2.2 Basic Properties of MICs................................................................. 17 2.3 Explicit Constructions of MICs........................................................ 28 2.3.1 SICs..................................................................................... 28 2.3.2 MICs from Random Bases ..................................................... 30 2.3.3 Group Covariant MICs .......................................................... 33 2.3.4 Equiangular MICs................................................................. 35 2.3.5 Tensorhedron MICs............................................................... 37 2.4 SICs are Minimally Nonclassical Reference Measurements ................ 39 2.5 Computational Overview of MIC Gramians ...................................... 46 2.6 Conclusions ................................................................................... 48 2.7 List of References........................................................................... 52 3. LTP ANALOGS ....................................................................................... 58 3.1 Appendix A: Proof of Lemma 3 ....................................................... 72 3.2 Appendix B: Proof of Theorem 10 ................................................... 79 3.3 List of References........................................................................... 83 4. LÜDERS MIC CHANNELS...................................................................... 87 4.1 Introduction ................................................................................... 87 4.2 Depolarizing Lüders MIC Channels ................................................. 93 4.3 Entropic Optimality ........................................................................ 97 4.4 Conclusions ................................................................................... 102 4.5 Appendix A ................................................................................... 105 4.6 Appendix B.................................................................................... 107 4.7 List of References........................................................................... 109 5. THE PRINCIPAL WIGNER FUNCTION................................................... 111 5.1 Introduction ................................................................................... 111 vii CHAPTER Page 5.2 Probability and Quasiprobability ...................................................... 115 5.3 Discrete Minimal Wigner Functions ................................................. 117 5.4 The Frame Operator and the Canonical Tight Frame .......................... 120 5.5 The Principal Wigner Basis ............................................................. 122 5.6 Behind Every Great Wigner Basis is a Great MIC.............................. 130 5.7 Discussion ..................................................................................... 138 5.8 List of References........................................................................... 141 6. SUM NEGATIVITY ................................................................................. 145 6.1 Introduction ................................................................................... 145 6.2 Negativity and Sum Negativity......................................................... 154 + − 6.3 Sum Negativities of fQj g and fQj g................................................ 159 6.4 Weyl-Heisenberg Q-reps in d = 3..................................................... 162 6.5 Sum Negativity Bounds in d = 3...................................................... 165 6.6 Further Observations about SIC Q-reps............................................. 173 6.7 Discussion ..................................................................................... 174 6.8 List of References........................................................................... 177 7. WIGNER’S FRIENDS .............................................................................. 181 7.1 Introduction ................................................................................... 181 7.2 Agents and QBism.........................................................................
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