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Design of Quantum Repeaters Design of Quantum Repeaters by Yi Li A thesis submitted for the degree of Doctor of Philosophy of The Australian National University July 2017 ⃝c Copyright by Yi Li 2017 All Rights Reserved Page ii Statement of Originality This work contains no material that has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of the thesis, when deposited in the University Library, being available for loan, photocopying and dissemination through the library digital thesis collection. The author of this thesis acknowledges that copyright of published work contained within this thesis (as listed in the publications page) resides with the copyright holder(s) of that work. Signed Date Page iii Page iv Publications Paper published [1] Y. Li, A. R. R. Carvalho, and M. R. James, \Continuous-mode operation of a noiseless linear amplifier,” Phys. Rev. A, vol. 93, p. 052312, May 2016. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.93.052312 [2] Y. Li, \Generation of distributed w-states over long distances," Optics Communications, vol. 396, pp. 19 { 22, 2017. [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0030401817301979 [3] Y. Li, \Single photon modulation," in 2016 35th Chinese Control Conference (CCC), July 2016, pp. 9124{9127. [4] Y. Li and M. R. James, \Equalization for linear quantum channels," in the 56th IEEE Con- ference on Decision and Control, 2017. Page v Page vi Acknowledgments First and foremost, I must recognize my principal supervisor Prof. Matthew James for his constant and patient guidance during the period of my PhD study. His meticulous approach to learning and tolerance towards others will influence me for the rest of my life. Talking with him not only in professional areas but also on other topics such as culture has always been enjoyable and beneficial. I would also like to wholeheartedly thank my co-supervisors Dr. Andre Carvalho and my adviser Prof. Ping Koy Lam for providing valuable suggestions to my research. Sincere thanks also to my classmates and colleagues in the Research School of Engi- neering for their unselfish help in extending my knowledge. They are Thien Nguyen, Oliver Thearle, James Dannatt, Shuangshuang Fu, Syed Assad, Jiayin Chen and Yu Pan. I am indebted for the work done by the staff in the Research School of Engineering and the College of Engineering and Computer Science. A very special gratitude goes out to my neighbours in Graduate House for their constant support and encouragement during my PhD studies. Finally, my deepest thanks to my mother and brother, who always encourage and support me in my life. Thanks for all your encouragement! Yi Li (July 2017) Page vii Page viii Abstract Quantum communication holds the promise of achieving long-distance secure mes- sage transmission by exploiting quantum entanglement between remote locations. Quantum repeaters are indispensable to the realization of quantum networks for long-distance quantum communication. Similar to its classical analogue, a good quantum repeater should be able to compensate channel attenuation with a quan- tum amplifier, and to combat channel distortion through a quantum equaliser. This quantum repeater should also operate by an efficient and robust protocol. The first part of this project researches the continuous mode operation of a noiseless linear amplifier (NLA). We develop a dynamical model to describe the operation of the nondeterministic NLA in the regime of continuous-mode inputs. Both the quantum scissor based NLA and the photon addition-subtraction based NLA are analysed. Simulation results are also presented to confirm theoretical analysis. The second part proposes two quantum protocols. An atomic ensemble based quan- tum protocol is developed to generate distributed W-states. These generated dis- tributed W-states could be considered as an entanglement resource between more than two distant nodes and would be useful in quantum communication and dis- tributed quantum computation in the future. We also propose a protocol by which quantum key distribution can be achieved deterministically between multiple nodes. This deterministic quantum key distribution scheme may be used to guarantee se- cure communication for wireless sensor networks and Internet of Things. The last project analyses distortion of quantum channels and develops physically realisable modules to combat it. The minimum phase channel and non-minimum phase all pass channel are discussed separately. Page ix Page x Contents Statement of Originality iii Publications v Acknowledgments vii Abstract ix Contents xi Conventions xv Abbreviations xvii List of Figures xix Chapter 1. Introduction and Motivation 1 1.1 Research area . 2 1.1.1 Introduction to quantum amplification . 3 1.1.2 Introduction to quantum communication protocols . 5 1.1.3 Introduction of quantum channel equalisation . 7 1.2 Original Contributions . 8 1.3 Thesis Structure . 8 Chapter 2. Background 11 2.1 Quantum mechanics and quantum information . 12 2.2 Quantum amplification methodologies . 13 Page xi Contents 2.2.1 Non-deterministic NLA . 13 2.2.2 Quantum scissor based NLA . 15 2.2.3 Photon addition-subtraction based NLA . 19 2.3 Quantum networking protocols . 21 2.3.1 DLCZ protocol . 21 2.3.2 Extensions of DLCZ protocol . 21 2.4 QKD protocols . 24 2.4.1 BB84 protocol . 24 2.4.2 Photon number splitting attacks . 26 2.4.3 SARG 04 protocol . 27 2.5 Channel equalisation techniques . 31 2.5.1 MMSE equaliser . 32 2.5.2 Decision feedback equaliser . 33 2.6 Conclusion . 35 Chapter 3. Quantum Amplification 37 3.1 Introduction . 38 3.2 Continuous-mode quantum scissor based NLA . 39 3.2.1 Amplification with one NLA module . 40 3.2.2 Amplification with two NLA modules . 50 3.3 Continuous-mode photon addition-subtraction based NLA . 52 3.3.1 Conditional Amplification: perfect detection case . 55 3.3.2 Conditional Amplification: detectors with finite time resolution 56 3.4 Conclusions and Discussion . 57 Chapter 4. Generation of Distributed W-States over Long Distances 59 4.1 Introduction . 60 Page xii Contents 4.2 Generation of distributed W-states . 60 4.2.1 Atomic ensembles and photons . 60 4.2.2 Implementation . 61 4.2.3 Bell State Measurement . 62 4.3 Entanglement Swapping . 63 4.4 Teleportation via W-states . 64 4.5 Summary . 67 Chapter 5. Multiparty QKD for Wireless Sensor Networks 69 5.1 Introduction . 70 5.2 Significance . 71 5.3 Preparation of Bell states . 73 5.4 Preparation of GHZ states . 74 5.5 Bell state measurement . 75 5.6 Multiparty QKD protocol . 78 5.7 Discussion . 79 5.8 Conclusion . 80 Chapter 6. Quantum Channel Equalisation 83 6.1 Introduction . 84 6.2 Effects of beam splitters and cavities on quantum optical states . 87 6.2.1 Effects of beam splitters on quantum optical states . 87 6.2.2 Effects of cavities on quantum optical states . 88 6.3 Equalisation of minimum phase channels . 90 6.4 Equalization of non-minimum phase all pass channel . 95 6.4.1 Approximation of non-causal but stable inverse . 95 6.4.2 Phase change analysis . 97 6.4.3 Simulation results . 98 6.5 Conclusion . 99 Page xiii Contents Chapter 7. Conclusions and Future Work 101 7.1 Review of and conclusions from the work in this thesis . 102 7.2 Recommendations on future Work . 103 7.2.1 Investigation of modulation schemes with photonic pulse shapes103 7.2.2 Quantum cryptography for wireless sensor networks . 104 7.2.3 Quantum equalisation implementation with waveguide chips and GEMs . 104 7.2.4 Further investigation of quantum channel equalisation tech- niques . 105 7.3 Conclusion . 105 Bibliography 107 Page xiv Conventions Typesetting This thesis is typeset using the LATEX2e software. The fonts used in this thesis are Times New Roman and Sans Serif. Referencing Referencing and citation style in this thesis are based on the Institute of Electrical and Electronics Engineers (IEEE) Transaction style [1]. For electronic references, the last accessed date is shown at the end of a reference. Units The units used in this thesis are based on the International System of Units (SI units) [2]. Spelling The Australian English spelling is adopted in this thesis. Page xv Page xvi Abbreviations BB84 Bennett Brassard 1984 BS Beam Splitter DFE Decision Feedback Equalisation DLA Deterministic Linear Amplification DLCZ Duan{Lukin{Cirac{Zoller FIR Finite Impulse Response GHZ Greenberger{Horne{Zeilinger IOT Internet of Things ISI Intersymbol Interference MMSE Minimum Mean Square Error MSZ Mu-Seberry-Zheng NLA Noiseless Linear Amplification PNS Photon Number Splitting QND Quantum Nondemolition QKD Quantum Key Distribution Qutip Quantum Toolbox in Python RF Radio Frequency SARG Scarani-Ac´ın-Ribordy-Gisin Page xvii Abbreviations SME Stochastic Master Equation UAV Unmanned Aircraft Vehicle WSN Wireless Sensor Network Page xviii List of Figures 1.1 Three aspects of quantum communication . 3 2.1 Quantum Scissor . 15 2.2 NLA with multiple branches . 18 2.3 Photon addition-subtraction . 20 2.4 DLCZ protocol . 22 2.5 Entanglement swapping in DLCZ protocol . 22 3.1 Quantum scissor based NLA . 39 3.2 Dynamical amplification system . 41 3.3 Gain, fidelity and probability of NLA . 44 3.4 Stochastic simulation of NLA . 48 3.5 wigner function for cavity state . 49 3.6 Amplification with two NLA modules . 51 3.7 Amplification with two NLA modules . 52 3.8 Photon addition-subtraction based continuous mode NLA . 53 4.1 Structure of W-states generation . 61 4.2 Bell state measurement for W-states generation . 63 4.3 Entanglement swapping between W-states .
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