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Experiments with Entangled Photons Bell Inequalities, Non-local Games and Bound Entanglement Muhammad Sadiq Thesis for the degree of Doctor of Philosophy in Physics Department of Physics Stockholm University Sweden. c Muhammad Sadiq, Stockholm 2016 c American Physical Society. (papers) c Macmillan Publishers Limited. (papers) ISBN 978-91-7649-358-8 Printed in Sweden by Holmbergs, Malmö 2016. Distributor: Department of Physics, Stockholm University. Abstract Quantum mechanics is undoubtedly a weird field of science, which violates many deep conceptual tenets of classical physics, requiring reconsideration of the concepts on which classical physics is based. For instance, it permits per- sistent correlations between classically separated systems, that are termed as entanglement. To circumvent these problems and explain entanglement, hid- den variables theories–based on undiscovered parameters–have been devised. However, John S. Bell and others invented inequalities that can distinguish be- tween the predictions of local hidden variable (LHV) theories and quantum mechanics. The CHSH-inequality (formulated by J. Clauser, M. Horne, A. Shimony and R. A. Holt), is one of the most famous among these inequalities. In the present work, we found that this inequality actually contains an even simpler logical structure, which can itself be described by an inequality and will be violated by quantum mechanics. We found 3 simpler inequalities and were able to violate them experimentally. Furthermore, the CHSH inequality can be used to devise games that can outperform classical strategies. We explore CHSH-games for biased and un- biased cases and present their experimental realizations. We also found a re- markable application of CHSH-games in real life, namely in the card game of duplicate Bridge. In this thesis, we have described this application along with its experimental realization. Moreover, non-local games with quantum inputs can be used to certify entanglement in a measurement device indepen- dent manner. We implemented this method and detected entanglement in a set of two-photon Werner states. Our results are in good agreement with theory. A peculiar form of entanglement that is not distillable through local oper- ations and classical communication (LOCC) is known as bound entanglement (BE). In the present work, we produced and studied BE in four-partite Smolin states and present an experimental violation of a Bell inequality by such states. Moreover we produced a three-qubit BE state, which is also the first experi- mental realization of a tripartite BE state. We also present its activation, where we experimentally demonstrate super additivity of quantum information re- sources. To my wife. Acknowledgements First of all I offer my profound thanks to Mohamed Bourennane, for being such a generous supervisor. He deserves a special mention of my deep grati- tude. This work was accomplished as a result of his encouragement and efforts, providing me continuous support. After this, my most humble thanks to Ingemar Bengtsson, for helping me at every stage of my life in Sweden. I am also very thankful to my co-supervisor Hoshang Heydari for occasional help that he has provided me during his stay, here in Fysikum. I am grateful to my collaborators and colleagues, I have had pleasure to work with during my PhD , thanks Adán Cabello, Marek Zukowski, Pawel Horodecki, Marcin Pawlowski Piotr Badzia¸g, Maciej Ku- rant, , Mohamed Nawareg, Elias Amselem and Armin Tavakoli. All former and new members of KIKO including Hannes, Guillermo, Ha- tim, Massimiliano, Tewodros,Victoria, Marco, David, Adrian, Arash, Saeed, Waqas, Breno,Victor, Atia,Christian Kothe, Magnus, Amir, Ramiz and Ben- jamin, thanks for an inspiring and pleasant atmosphere. In particular, Johan, Elias, Nawareg, Hammad, Alley, Kate, Ashraf, Ole, and Ian deserve special thanks for useful and friendly discussions that we had during all these years and, for nice and pleasing memories. Finally, special thanks also go to my family and friends, for their coopera- tion and extra support that provided me confidence to present this work. Contents Abstract iii Acknowledgementsv List of Papers xi Author’s Contribution xii List of Figures xv List of Tables xvii 1 Introduction1 1.1 Outline . .3 2 Preliminary Concepts5 2.1 Pure and Mixed States . .5 2.2 The Density Operator . .6 2.3 Qubit: The Simplest Quantum System . .7 2.4 Bloch Sphere: The State Space of a Qubit . .8 2.5 Multi-Qubit Systems . 10 2.6 Entangled States . 11 2.7 EPR Paradox . 12 2.8 Bell Inequality . 14 2.8.1 Bell States . 18 2.8.2 Hardy’s Proof of Non-locality . 18 2.9 GHZ State: Bell Theorem without Inequalities . 20 2.10 Contextuality and Quantum Mechanics . 22 3 Experimental Background 25 3.1 Qubits: A Polarization Implementation . 25 3.2 Manipulation of Polarization Qubit . 26 3.2.1 Polarizers . 26 viii Contents 3.2.2 Wave-Plates . 27 3.2.3 Beam Splitters . 30 3.2.4 Polarizing Beam Splitters . 31 3.2.5 Optical Fibers . 32 3.2.6 Single Photon Detectors . 33 3.2.7 Polarization Analysis . 34 3.3 Polarization Entangled Photons Source . 35 3.3.1 Preparation of the Pump Laser . 35 3.3.2 Spontaneous Parametric Down-conversion (SPDC) . 36 3.3.3 Walk off compensation . 39 3.3.4 Two Photon Polarization Entanglement . 41 3.4 Two-Photon Interference . 41 3.5 Bell State Measurement . 45 3.6 GHZ-State Preparation . 46 3.7 Quantum Teleportation . 48 4 Bell Inequalities for the Simplest Exclusivity Graph 51 4.1 An Introduction to Graph Theory . 52 4.2 Exclusivity Graph of CHSH inequality . 54 4.3 Simplest Exclusivity Graph With Quantum-Classical Gap . 56 5 Non-Local Games 59 5.1 CHSH Game . 59 5.1.1 Unbiased CHSH Game . 60 5.1.2 Biased CHSH Game . 60 5.2 Quantum Duplicate Bridge: An Application of CHSH Game . 63 6 Measurement-Device-Independent Entanglement Detection 65 6.1 Quantum State Verification and Entanglement Detection . 65 6.1.1 Fidelity of a Quantum State . 66 6.1.2 Violation of Bell Inequality . 67 6.1.3 Positive Partial Transpose (PPT) Criterion . 68 6.1.4 Witness Method . 68 6.2 Measurement-Device-Independent Entanglement Witness . 70 7 Bound Entanglement: Generation and Activation 71 7.1 LOCC Operations . 71 7.2 Distillation and Bound Entanglement . 72 7.3 Smolin States . 74 7.4 Experimental bound entanglement through a Pauli channel . 75 7.5 Three-Qubit Bound Entanglement Generation . 75 7.6 Three-Qubit Bound Entanglement Activation . 76 Contents ix 8 Conclusion 79 A Correlation function for GHZ state 81 References 83 x Contents List of Papers The following papers, referred to in the text by their Roman numerals, are included in this thesis. PAPER I: Bell inequalities for the simplest exclusivity graph Muhammad Sadiq, Piotr Badzia¸g, Mohamed Bourennane, Adán Cabello Phys. Rev. A, 87, 012128 (2013). DOI: PhysRevA.87.012128 PAPER II: Quantum Bidding in Bridge Muhammad Sadiq, Armin Tavakoli, Maciej Kurant, Marcin Pawlowski, Marek Zukowski, Mohamed Bourennane, Phys. Rev. X, 2, 021047 (2014). DOI: 10.1103/PhysRevX.4.021047 PAPER III: Experimental Measurement-Device-Independent Entangle- ment Detection Nawareg Mohamed, Muhammad Sadiq, Elias Amselem, Mo- hamed Bourennane, Scientific Reports, 5, 8048 (2015). DOI: 10.1038/srep08048 PAPER IV: Experimental bound entanglement through a Pauli channel Elias Amselem, Muhammad Sadiq, Mohamed Bourennane, Sci- entific Reports, 3, 1966 (2013). DOI: 10.1038/srep01966 PAPER V: Experimental Three-Qubit Bound Entanglement Muhammad Sadiq, Mohamed Nawareg, Pawel Horodecki, Mo- hamed Bourennane, submitted (2015). PAPER VI: Superadditivity of two quantum information resources Mohamed Nawareg, Muhammad Sadiq, Pawel Horodecki, Mo- hamed Bourennane, submitted (2015). Reprints were made with permission from the publishers. xii Author’s Contribution PAPER I: I performed all the experimental work, which include designing and building the setup, measuring and analyzing the data, and helped in writing the paper. PAPER II: I designed the experiment. Built the setup and measured the data with the help of Armin Tavakoli. I analyze the data and helped in writing the paper. PAPER III: We, (M. Nawareg and I) designed the experiment, built the setup and measured the data together. I helped in writing paper. PAPER IV: Elias and I performed all the experimental work together. This includes building the experiment and measuring the data. PAPER V: I and M. Nawareg designed the experiment, built the setup and measured the data together. I helped in data analysis and writing paper. PAPER VI: M. Nawareg and I designed the experiment, built the setup and measured the data together. I helped in data analysis and writing paper. Sammanfattning Kvantmekanik är tveklöst en konstig gren av naturvetenskapen och bryter mot många grundläggande antaganden inom klassisk fysik. Den kräver ompröv- ning av de koncept som den klassiska fysiken bygger på. Exempelvis tillåts korrelationer mellan system som klassisk betraktas som åtskillda, detta kal- las snärjelse. För att kringgå dessa problem och förklara snärjelse har gömda- variabel-teorier konstruerats, dessa bygger på okända parametrar. John S. Bell och andra fysiker fann olikheter som kan urskilja mellan förutsägelser från lo- kala gömda variabel (local hidden variable, LHV) teorier and kvantmekanik. CHSH-olikheten, funnen av J. Clauser, M. Horne, A. Shimony och R.A. Holt, är en av de mest kända. Under arbetet som presenteras i denna avhandling har vi funnit att CHSH-olikheten har en ännu enklare logisk struktur, som i sig kan beskrivas med en olikhet och som bryts av kvantmekaniken. Vi har funnit tre stycken enklare olikheter och brutit dem experimentellt. Vidare kan CHSH-olikheten användas för att konstruera spel där utfallet blir bättreöm kvantmekaniska tillstånd tillåts. Vi har undersökt både viktade och oviktade CHSH-spel. Vi har även funnit en anmärkningsvärd verklighets- anknuten tillämpning av ett CHSH-spel, nämligen i kortspelet kontraktsbridge. Denna tillämpning, tillsammans med dess experimentella realisering, presen- teras i avhandlingen.
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