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Bell Inequalities and Many-Body Quantum Correlations Computational Perspectives on Bell Inequalities and Many-body Quantum Correlations Matthew Joseph Hoban A thesis submitted to University College London for the degree of Doctor of Philosophy Department of Physics and Astronomy University College London April 26, 2012 I, Matthew Joseph Hoban confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. 2 Abstract The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated measurements on entangled quantum systems violate a Bell inequality [Bell1964]. We put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate particular maps, or functions to particular theories. The violation of a Bell inequality then implies the ability to perform some functions, or computations that classical, or more generally, local hidden variable (LHV) theories cannot. We derive an infinite class of Bell inequalities that establish a link to so- called \non-local games" [Cleve2004]. We then make the connection between Raussendorf and Briegel's formulation of Measurement-based Quantum Com- puting (MBQC) [Raussendorf2001], and these non-local games. Not only can we show that a quantum violation implies a computational advantage in this model, we show that adaptive measurements are required to perform all quantum com- putations. Finally, we explore post-selection of data in Bell tests from both a practical and conceptual point-of-view, with particular consideration to so-called \loopholes". Loopholes allow LHV theories to simulate quantum correlations through post- selection. We give a computational description of how loopholes can emerge in different post-selection scenarios. This motivates us to find a form of post- selection that does not lead to loopholes. Central again to this discussion is the description of LHV theories in terms of computations. Interestingly, quantum correlators can be made more \non-classical" with this loophole-free post-selection. This method of post-selection also can simulate information processing tasks, such as MBQC, that have time-like separated com- ponents. This opens up new avenues for the study of time-like tasks studied within the space-like separated scenario of the Bell test. 3 Publications The majority of the work in this thesis is based on the following publications: M. J. Hoban, E. T. Campbell, K. Loukopoulos, and D. E. Browne, Non-adaptive Measurement-based Quantum Computation and Multi-party Bell Inequalities, New J. Phys. 13 023014 (2011). M. J. Hoban and D. E. Browne, Stronger Quantum Correlations with Loophole- Free Postselection, Phys. Rev. Lett. 107, 120402 (2011). M. J. Hoban, J. J. Wallman, and D. E. Browne, Generalized Bell-inequality ex- periments and computation, Phys. Rev. A 84, 062107 (2011). 4 Acknowledgements I am indebted to many different people for many things throughout the process of completing my PhD. First of all, I desperately need to thank my supervisor, Dan Browne. To paraphrase Winston Churchill, \Never was so much owed by one PhD student to one tireless supervisor." I thank him for his patience, enthusiasm, insight and for sharing his great ideas. Not only has he been a great supervisor, he has been a good friend. My examiners Jon Barrett and Sougato Bose need to be thanked for their thorough reading of this thesis. Their experience and insight has only improved this document. It remains to be said that any remaining inaccuracies result from me. A lot of the work contained in this tome would never have seen the light of day if I had not had such good collaborators. Earl Campbell was instrumental in the first two years of my PhD, sharing his great insights and discussing his new ideas. I am very proud of the paper we wrote together with Klearchos Loukopoulos, whose contribution to many discussions I am also grateful for. Joel Wallman has also been a great collaborator for discussing very many ideas in our time-zone defying email discourses. I also thank Bob Coecke and Samson Abramsky for allowing me to join their group at Oxford. They have given me an independence that has allowed the continuation of some of the ideas resulting from the work contained herein. I would also like to thank the Quantum Information group at UCL for pro- ducing a delightful environment in which to do research. In particular, I am indebted to Janet Anders for showing me around on my first day and always be- ing around for good conversation, regardless of the topic. Also, my office mates at any one particular time including Sai-Yun Ye (for long discussions on short topics), Hussain Anwar, Brad Augstein, Tahir Sharaan, Hulya Yadsan-Appleby, and Peter Burns. Outside of my office, along the corridor are the Public Engagement Unit. I would like to thank them for being just brilliant friends, especially Hilary Jackson 5 and Gemma Moore. They put up with my semi-coherent ramblings for far longer than they needed to. I will, and do already miss them. Outside of my corridor, in big old London, I have to thank my friends for distracting me from everything. In particular James Millen (who became in- tegrated into my office friends), Andy Sykes (who became integrated into my corridor friends), Philippa Stanger, Andy Webster, Keira Poland and countless others. They make it easy to miss London. Outside of London, in the United Kingdom at large, I want to thank my family. My mum and dad, Christine and Chris Hoban, for everything they have done to allow me to get this far. I doubt anyone loves their parents as much I do and defy anyone to say otherwise. Along with my parents, I want to thank my brothers Kieran, Niall and Dominic for their support and love, and for testing the water before my arrival. Kieran especially needs to be thanked for the countless lunches, excellent conversation and emotional support. Finally, and most importantly, I want to thank Francesca Richards for her kind- ness, intelligence, love and humour. Cesca (and Mwg) kept me sane throughout the whole of my PhD and I cannot begin to thank her enough. 6 Contents 1 Introductions 11 1.1 Quantum Mechanics and Entanglement . 12 1.1.1 Postulates of Quantum Mechanics . 12 1.1.2 Entanglement . 16 1.2 EPR Paradox and Bell Inequalities . 19 1.2.1 Realism and \Incompleteness" of Quantum Mechanics . 19 1.2.2 CHSH Inequality . 21 1.2.3 Geometric Construction of Bell Inequalities . 23 1.2.4 Experimental implementations for testing local realism . 24 1.2.5 The GHZ Paradox . 24 1.3 Quantum Information Processing . 27 1.3.1 Quantum Cryptography . 27 1.3.2 Quantum Computing . 29 1.3.3 Measurement-based Quantum Computing . 30 1.3.4 Entanglement as a Resource . 32 1.4 Bell Inequalities and Quantum Information . 32 1.4.1 Device-independent Quantum Information . 33 1.4.2 GHZ Paradox and Measurement-based Quantum Computing 34 1.5 Chapter Summary . 35 2 Correlators and Bell Tests 37 2.1 A General Framework for Bell tests . 38 2.1.1 Convex Polytopes and Stochastic Maps . 40 2.1.2 The Non-signalling Polytope . 42 2.1.3 Local Hidden Variable Theories and Bell Inequalities . 42 2.1.4 Facet Bell Inequalities and Computational Complexity . 44 2.2 Correlators . 47 2.2.1 Correlators as Computations . 49 2.2.2 Correlators from Physical Theories . 54 7 2.3 Non-signalling Correlations . 57 2.3.1 Generalised bipartite PR boxes . 58 2.3.2 Multipartite Generalisations of the PR box . 59 2.4 Svetlichny Correlations . 62 2.4.1 Three-party Generalised Svetlichny Correlators . 63 2.4.2 Multipartite Svetlichny Correlators . 65 2.5 Chapter Summary . 67 3 Constructing Bell Inequalities and Quantum Violations 68 3.0.1 Notation . 69 3.1 Facet Bell Inequalities . 70 3.1.1 Symmetries of the LHV Polytope . 71 3.1.2 Notation for Bell inequalities . 73 3.1.3 Bipartite facet Bell inequalities . 74 3.1.4 Tripartite facet Bell inequalities . 77 3.1.5 Multipartite facet inequalities for (n; 2; 2) . 78 3.2 Quantum Violations of Bell Inequalities . 84 3.2.1 Numerical Methods for finding Violations of Bell Inequalities 85 3.2.2 Bipartite Quantum Violations and Entanglement . 88 3.2.3 Quantum Upper Bounds of (n; 2; 2) Bell Inequalities . 91 3.3 Non-trivial Bell Inequalities . 93 3.3.1 Non-trivial Inequalities as Generalisations of the CHSH In- equality . 94 3.3.2 Non-local Games . 97 3.3.3 (n; 2; 2) scenario and the n-partite NAND function . 100 3.4 Non-adaptive Measurement-based Quantum Computing . 105 3.4.1 nMBQC, NLG and non-trivial Bell inequalities . 108 3.4.2 Generalized GHZ Paradoxes and PR boxes . 113 3.5 Chapter Summary . 116 4 Data Post-selection in Bell Tests 118 4.0.1 Notation . 120 4.1 Post-selection and the Detection Loophole . 120 4.1.1 The Detection loophole . 122 4.1.2 Rederivation of the GM detection efficiency . 126 4.1.3 Generalisation of the GM bound to Many Parties . 129 4.1.4 Summary of Loopholes . 132 8 4.2 Loophole-free Post-selection and Quantum Correlators . 133 4.2.1 Post-selection, Linearity and Loopholes . 134 4.2.2 Linear Input Post-selection in (n; 2; 2) tests . 136 4.2.3 Linear Output-Input Post-selection in (n; 2; 2) tests . 137 4.2.4 Bipartite Quantum correlators under post-selection . 138 4.2.5 Multipartite quantum correlators . 140 4.3 General settings and Input Post-selection .
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