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Uwaterloo Latex Thesis Template View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Waterloo's Institutional Repository Geometry of Quantum States from Symmetric Informationally Complete Probabilities by Gelo Noel M. Tabia A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics - Quantum Information Waterloo, Ontario, Canada, 2013 c Gelo Noel M. Tabia 2013 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 It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system corre- spond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called den- sity operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex ge- ometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. This thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equian- gular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elu- cidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a per- mutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a proce- dure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly. iii Acknowledgements There is no one way the world is because the world is still in creation, still being hammered out. { Christopher A. Fuchs, letter to Howard Barnum and Tony Sudbury, August 18, 2003, quoted in Scientific American, Sept. 2004 I wish to express my sincerest gratitude to my advisor Chris Fuchs, for having provided me the opportunity and the privilege of working with him in my pursuit of a doctoral degree. I will always feel greatly indebted to him, since he was willing to take a chance on me four years ago when I had been looking for a supervisor for almost a year to no avail. Chris is an excellent advisor, who cares not just about how much work his students get done but also in their general well-being. I have personally benefitted from his valuable insights and sage advice on what problems to look at or what next steps to consider. His knowledge, guidance, patience and support have been invaluable to my research and studies. But I think just as important is his overall influence in my way of thinking and approach to physics and life. I feel some of the important lessons I have learned from him, which are worth mentioning here: (i) Do not be afraid to ask silly questions. True, they are usually a waste of other people's time but occasionally they turn out not to be silly at all. (ii) Be constructive, not just productive. Do not fall into the trap of busywork. (iii) When you obtain a result that is simpler than you expected, find out why. (iv) Do not be satisfied with vague claims based on intuition. Always demand precision and some rigorous mathematics. (v) But while mathematics is important, remember that often the most profound ideas in physics can be expressed meaningfully with a few, well-chosen words. It is my sincere hope that these lessons, taken to heart, bring me a few steps closer to the kind of physicist he is. I would like to extend my special thanks to Marcus Appleby, Asa˚ Ericsson, Hoan Dang, and Matthew Graydon, friends and comrade-in-arms in research topics related to SICs. To Marcus and Asa,˚ I am grateful for their willingness to discuss many of my ideas, most of which did not work, and for sharing their valuable insight into various problems that iv I have struggled with in my research. To Hoan and Matthew, I am grateful for all the time we spent as students of Chris, for all the enlightening and enjoyable conversations on both physics and non-physics matters alike, and for the tradition of attending APS March Meetings, which I hope continues for many more years. I would like to thank Norbert L¨utkenhaus, Kevin Resch, Joseph Emerson, and Andrew Childs for serving in my PhD committee and for always making sure I was always on track in fulfilling the requirements of my program. I would also like to thank Ingemar Bengtsson for acting as the external examiner for my thesis defense. His work on the geometry of quantum states has been a source of inspiration to my research and it is an honor and a privilege to have him evaluate my contributions to the field. Each of the following have in one way or another, influenced this thesis, sometimes directly in its content but even if only in little ways that affected its emphasis on certain ideas: Berge Englert, Zach Medendorp, Christoph Schaeff, Bill Wootters, Rob Spekkens, Lucien Hardy, Jeff Bub, John Watrous, Ben Reichardt, Frank Wilhelm, and Huangjun Zhu. I am grateful to the Perimeter Institute for providing a stimulating environment for research, and its staff in particular, for the numerous times they have assisted me on administrative matters. I would like to mention in particular my officemates Jeff Hnybida and Katja Reid, for many delightful and stimulating conversations about their own work and other things, which provide a short reprieve from the mostly tedious part of research that happens on my desk. Special thanks to Alfonso Cesar Albason, a companion from distance in my journey through my graduate studies. Getting a PhD is hard when you have many concerns and distractions beyond research and having a friend to talk to who understands the experience is both helpful and encouraging. My continuing education would not be possible without the unwavering support and understanding of my dearest family and closest friends, for the many times they have encouraged me to carry on when things became difficult, and for relieving some of the pressures of those hard and trying moments. Many thanks to them. v To the loving memory of my father, Angel Garrucha Tabia, Jr. vi Table of Contents List of Tables xi List of Figures xii 1 Introduction1 1.1 Historical background and motivation.....................2 1.2 Research scope and objectives.........................7 1.3 Roadmap for the thesis.............................9 1.4 List of specific contributions.......................... 10 2 Overview of quantum mechanics 11 2.1 Outline of the quantum formalism....................... 12 2.2 Qubits and the Bloch ball........................... 16 2.3 The set of density operators.......................... 17 3 SIC-POVMs and Weyl-Heisenberg qutrit SICs 22 3.1 Characterizing SIC-POVMs.......................... 23 3.1.1 Equiangular lines............................ 23 3.1.2 Complex projective design....................... 25 3.1.3 Maximal equiangular tight frame................... 26 3.2 Weyl-Heisenberg SICs and the Clifford group................. 28 vii 3.3 Classifying qutrit SICs into inequivalent families............... 30 3.4 Geometric invariants of qutrit SICs...................... 32 3.5 Linear dependency structure of qutrit SICs.................. 40 3.6 The Zauner subspace for qutrit SICs..................... 45 3.7 Lie algebraic properties of qutrit SICs..................... 47 4 Practical implementations of SIC-POVMs 53 4.1 Experimental SICs using optimal polarimetry................ 54 4.2 Implementing SICs by successive measurements............... 56 4.3 Storage loop SIC- POVM experiment with weak projections........ 59 4.4 Multiport scheme from Naimark dilation of SIC-POVMs.......... 62 4.4.1 Multiport Qubit SIC-POVM.....................
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