UNIVERSITY OF SHEFFIELD DOCTORAL THESIS The Statistics and Security of Quantum Key Distribution Author: Supervisors: Scott E. Vinay Dr. Pieter Kok, Dr. Stefano Pirandola A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Low-Dimensional Semiconductor Devices group Department of Physics and Astronomy December 2018 iii Declaration of Authorship I, Scott Vinay, declare that this thesis titled, The Statistics and Security of Quantum Key Distribution, and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research de- gree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my- self. Signed: Date: v https://xkcd.com/538/, this work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. There once was a lass named Alice, who’s friend, Bob, lived over in Dallas. They wanted to chat, but they didn’t know that, sneaky Eve was listening with malice... vii Abstract The Statistics and Security of Quantum Key Distribution by Scott E. Vinay In this work our aim has been to elucidate our theoretical developments that bolster the efficiency of quantum key distribution systems leading to more secure commu- nication channels, as well as develop rigorous methods for their analysis. After a review of the necessary mathematical and physical preliminaries and a discussion of the present state of quantum communication technologies, we begin by investi- gating the Trojan Horse Attack, a form of side-channel attack that could threaten the security of existing key distribution protocols. We examine the secret key rates that may be achieved when an eavesdropper may use any Gaussian state in the presence of thermal noise, and prove that the coherent state is optimal in this case. We then allow the eavesdropper to use any separable state, and show that this gives a key rate bound close to that of the coherent state. We develop a protocol for a quantum repeater that makes use of the double- heralding procedure for entanglement-generation. In our analysis, we include sta- tistical effects on the key rate arising from probabilistic entanglement generation, which results in some quantum memories decohering while other sections complete their entanglement generation attempts. We show that this results in secure commu- nication being possible over thousands of kilometres, allowing for intercontinental key distribution. Finally, we investigate in more depth the statistical issues that arise in general quantum repeater networks. We develop a framework based on Markov chains and probability generating functions, to show how one may easily calculate an analytic expression for the completion time of a probabilistic process. We then extend this method to show how one may track the distribution of the number of errors that accrue in operating such a process. We apply these methods to a typical quantum repeater network to get new tight bounds on the achievable key rates. ix Publications and conferences Throughout the course of this doctorate, some of the work has been published and presented in journals and conferences. Published works Scott E. Vinay and Pieter Kok. Practical repeaters for ultralong-distance quantum communication. Physical Review A, 95(5), 2017. (Mostly discussed in Chapter5.) Scott E. Vinay and Pieter Kok. Extended analysis of the Trojan-horse attack in quan- tum key distribution. Physical Review A, 97(4), 2018. (Mostly discussed in Chapter4.) Scott E. Vinay and Pieter Kok. Statistical analysis of quantum entangled network generation." Physical Review A, 99(4), 2019. (Mostly discussed in Chapter6.) Conferences Quantum Roundabout 6th – 8th July 2016, Nottingham, UK. Poster presented: Highly loss-tolerant quantum repeaters. SPIE West 28th January – 2nd February 2017, San Francisco, USA. Talk by Pieter Kok: Practical repeaters for ultralong-distance quantum communication. Winter school on Complex Networks: From Classical to Quantum 3rd – 7th April 2017, Obergurgl, Austria. Poster presented: Highly loss-tolerant quantum re- peaters. QCrypt 18th – 22nd September 2017, Cambridge, UK. Poster presented: The Trojan Horse Attack against QKD. xi Acknowledgements The extent to which I am indebted to so many of the people in my life for sup- porting, encouraging, entertaining and uplifting me these past few years, in both an academic and a personal context, is surely too great to be expressed on this page, or indeed in this entire volume. I would, however, like to indicate a particular gratitude to a few individuals. Firstly, to my family, and especially my parents. I certainly would not have reached this stage without the incredible support and unconditional love that you have shown to me. Your unwavering selflessness is a true inspiration to me, and is a model of personhood to which I aspire. I wish to deeply thank everyone who has helped to make my time in Sheffield an immensely enjoyable one, and for standing by my side in all occasions, whether joyous or trying. A particularly notable proper subset of this may be written as fDavid Hurst, Earl Campbell, Giuseppe Buonaiuto, Jasminder Sidhu, Luke Heyfron, Mark Howard, Mark Pierce, Mike Roche, Sam Coveny, Zixin Huang, and all com- rades of the IMTg. My deepest love goes out also to all my other friends. Whether of Derby, Sheffield, or elsewhere in the world, those I have known a few weeks, to those I have known for many years, whether entering my life or leaving it: my life would contain but a fraction of its happiness were it not for the rich fabric of joy that you have all woven throughout it. And last but certainly not least, I wish to extend an appreciation of the most inestimable order to my supervisor, Pieter Kok. It is hard to imagine a person with more unshakable optimism and faith in his students, which did not falter even when I doubted myself entirely. For a busy person to consistently find so much time for his students is a rare thing indeed. I have unquestionably grown as a researcher during my time here, and our discussions on the philosophy of quantum mechanics have shaped my thinking of the world. It has been an honour to be a member of this research group, and a student at this university. I dedicate this work to all those who are mentioned here, and to all those who are not. xiii Contents Declaration of Authorship iii Abstract vii Acknowledgements xi 1 Introduction1 1.1 The need for a new kind of communication................1 1.2 The structure of this thesis..........................3 2 Quantum mechanics preliminaries5 2.1 Quantum states................................5 2.1.1 Pure states...............................5 Measurements............................7 2.1.2 Mixed states..............................8 2.1.3 Discrete-variable quantum states.................. 10 Fock states............................... 10 Qubits................................. 12 2.1.4 Continuous-variable quantum states................ 13 Coherent states............................ 14 Squeezed states............................ 16 2.1.5 State quality measures........................ 17 Fidelity................................. 17 Trace distance............................. 18 2.1.6 Entropy................................ 18 2.2 Entanglement................................. 21 2.2.1 Bell pairs and entanglement quantification............ 21 2.2.2 Entanglement swapping and teleportation............ 23 2.2.3 Graph-state entanglement...................... 24 2.2.4 Entanglement distillation...................... 27 2.2.5 The ontology of quantum states.................. 30 3 Quantum communication preliminaries 33 3.1 Quantum cryptography protocols...................... 34 3.1.1 Bennet and Brassard 1984...................... 34 Protocol description......................... 35 xiv Security check............................. 36 Non-standard attacks........................ 37 3.1.2 Ekert 1991............................... 38 Equivalence with BB84........................ 39 3.1.3 Security and secret key rates.................... 40 3.1.4 Double-heralding........................... 44 3.1.5 Side-channels............................. 48 3.2 Quantum repeaters.............................. 50 3.2.1 Functionality............................. 53 3.2.2 Innsbruck protocol.......................... 56 3.2.3 Other repeaters............................ 57 Coherent light repeater....................... 57 All-optical repeater.......................... 58 4 Defending against the Trojan-Horse side-channel attack 61 4.1 Gaussian-state attack............................. 63 4.1.1 State description........................... 63 4.1.2 Secret key rate............................. 67 4.1.3 Effect of thermal noise on Bob................... 70 4.1.4 Results of Gaussian-state analysis................. 72 4.2 General separable attacks.......................... 74 4.3 Shutter defence................................ 78 4.4 Summary...................................