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The Role of Entanglement in Quantum Communication, and Analysis of the Detection Loophole

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The Role of Entanglement in Quantum Communication, and Analysis of the Detection Loophole The Role of Entanglement in Quantum Communication, and Analysis of the Detection Loophole Thomas Cope Doctor of Philosophy University of York Computer Science September 2018 Dedication To Ellen, Judith and Andrew Abstract Entanglement is a feature at the heart of quantum information. Its enablement of unusual correlations between particles drives a new wave of communication and computa- tion. This thesis explores some of the ways in which the tools for studying entanglement can be used to quantify the transmission of quantum information, and compares the use of different techniques. We begin this thesis by expanding the technique of teleportation simulation, which adds noise to the entangled resource state to mimic channel effects. By introducing classical noise in the communication step, we show it is possible to simulate more than just Pauli channels using teleportation. This new class is characterised, and studied in detail for a particular resource state, leading to a family of simulable channels named “Pauli-Damping channels” whose properties are analysed. Also introduced are a new family of quantum states, “phase Werner” states, whose entanglement properties relate to the interesting conjecture of bound entangled states with a negative partial transpose. Holevo-Werner channels, to which these states are connected, are shown to be teleportation covariant. We exploit this to present several interesting results, including the optimal estimation of the channel-defining parameter. The minimal binary-discrimination error for Holevo-Werner channels is bounded for the first time with the analytical form of the quantum Chernoff bound. We also consider the secret key capacity of these channels, showing how different entanglement measures provide a better upper bound for different regions of these channels. Finally, a method for generating new Bell inequalities is presented, exploiting non- physical probability distributions to obtain new inequalities. Tens of thousands of new inequivalent inequalities are generated, and their usefulness in closing the detection loop- hole for imperfect detectors is examined, with comparison to the current optimal construc- tion. Two candidate Bell inequalities which may equal or beat the best construction are presented. iii Contents Abstract iii List of figures vi List of tables viii Acknowledgements xi Declaration xiii 1 Foreword and Preliminaries 1 1.1 Foreword . 1 1.2 Structure of this Chapter . 4 1.3 The Mathematics of Quantum Information . 5 1.4 Measures of Entanglement . 19 1.5 Capacities of Quantum Channels . 30 1.6 Summary . 34 2 Simulation of Non-Pauli Channels 35 2.1 Structure of this Chapter . 35 2.2 The History of Channel Simulation . 35 2.3 Channel Simulation via Teleportation . 42 2.4 Introducing a Noisy Teleportation Protocol . 46 2.5 Pauli-Damping Channels . 50 2.6 Discussion and Further Directions . 69 3 Werner States and Phase Werner States 73 3.1 Structure of this Chapter . 73 v Contents 3.2 Werner States and their Role in Quantum Theory . 73 3.3 Introducing a Phase to Werner States . 83 3.4 Another Generalisation of Werner States . 94 3.5 Discussion and Further Directions . 96 4 Holevo-Werner Channels 99 4.1 Structure of this Chapter . 99 4.2 Holevo-Werner Channels . 99 4.3 Metrology of Quantum Channels . 101 4.4 Metrology of Holevo-Werner Channels . 104 4.5 Capacities of Holevo-Werner Channels . 115 4.6 Discussion and Further Directions . 127 5 Bell Polytopes and the Detection Loophole 131 5.1 Structure of this Chapter . 131 5.2 Bell Correlations and Non-locality: An Introduction . 131 5.3 Preliminaries . 133 5.4 Completely Known Scenarios . 152 5.5 An Algorithm for Generating Bell Inequalities . 155 5.6 The Detection Loophole . 174 5.7 Summary . 193 6 Conclusion and Summary 197 Abbreviations 203 References 205 vi List of Figures 2.1 LOCC simulation . 39 2.2 Teleportation simulation . 43 2.3 The Bloch sphere . 44 2.4 The Pauli tetrahedron, T ............................ 45 2.5 Noisy teleportation simulation . 47 2.6 Possible forms of Sij ............................... 48 2.7 The decomposition of Pauli-damping channels . 60 2.8 Analytical capacity bounds for Pauli-damping channels . 68 2.9 Numerical capacity bounds for Pauli-damping channels . 69 3.1 Negativity of Wη;4 and Iη;4 ........................... 78 3.2 Logarithmic negativity of Wη;4 and Iη;4 .................... 78 θ 3.3 Range of valid Iη;d states . 88 3.4 Negativity of Werner and π-Werner states . 89 θ 3.5 Distillable states Wη;8 .............................. 93 π 3.6 Werner twirling of Wη;d ............................. 94 4.1 Optimal variance of n probes of the Holevo-Werner channel . 106 4.2 α-estimation of Holevo-Werner channels, using a single probe . 107 4.3 A plot of ∆S ...................................109 4.4 A comparison of pmin for Holevo-Werner channel discrimination . 113 4.5 Secret key bounds of Wη;d, using the subadditive REE . 122 4.6 An upper bound on Esq (Wη;d) . 124 4.7 A comparison of Esq (Wη;d) bounds . 125 (a) d = 4 ....................................125 (b) d = 5 ....................................125 4.8 Comparison of the bounds on Q2(Wη;4) . 126 vii List of Figures 4.9 Comparison of the bounds on K(Wη;4) . 126 4.10 Our best bound on K(Wη;4) . 127 4 5.1 I4422: lifting 1 . 184 4 5.2 I4422: lifting 2 . 184 viii List of Tables 2.1 Cardinality of S30-constrained sets . 58 3.1 Comparison of representations of Werner states . 83 5.1 Solved local polytopes . 154 5.2 The linear programming search space for (2; 2; 2; 2) . 160 5.3 The linear programming search space for (3; 3; 2; 2) . 161 5.4 An alternative (2; 2; 2; 2) search space . 161 5.5 An alternative (3; 3; 2; 2) search space . 162 5.6 Equivalence of two Bell inequality affine tallies . .168 5.7 Facet analysis of the (4; 4; 2; 2) local polytope . 170 5.8 A survey of % success of obtaining a facet-defining solution . 172 5.9 Fundamental bounds on the detection threshold . 181 5.10 Detection thresholds obtained from CVX . 182 5.11 Detection thresholds for I3522 . 185 5.12 High precision CVX thresholds . 187 5.13 Best precision CVX thresholds . 187 5.14 CVX solver comparison . 188 ix Acknowledgements I am very grateful to my supervisors, Stefano Pirandola and Roger Colbeck - without them this thesis would not be possible, and their insights have been more than invalu- able. My thanks also go to my peers - Riccardo, Panos, Vicky, Peter and Oliver, whose discussions and support have been helpful at every step of the process. This gratitude is extended to Carlo, Cosmo, Gaetana, Sammy and Mirjam, whose experience has been wholly beneficial. This too is accurate of my co-authors Leon, Leonardo and Kenneth; thank you for working with me. Finally, the support of my friends and especially my family has been vital, and never taken for granted. xi Declaration I declare that the research described in this thesis is original work, which I undertook at the University of York during 2015 - 2018. This work has not previously been presented for an award at this, or any other, university. Except where stated, all of the work contained within this thesis represents the original contribution of the author. Some of the material in this thesis has been published in journals. The author of this thesis acknowledges the input of his collaborators, and has credited them appropriately throughout. A list of papers which overlap with this thesis are presented here. • Simulation of non-Pauli Channels, Thomas Cope, Leon Hetzel, Leonardo Banchi and Stefano Pirandola. Published in Physical Review A 96 (2017). [30] • Adaptive estimation and discrimination of Holevo-Werner channels, Thomas Cope and Stefano Pirandola. Published in Quantum Measurements and Quantum Metrol- ogy 4 (2017). [31] • Converse bounds for quantum and private communication over Holevo-Werner chan- nels, Thomas Cope, Kenneth Goodenough and Stefano Pirandola. Published in Journal of Physics A 51 (2018). [29] Also relevant is the publication, • Theory of channel simulation and bounds for private communication, Stefano Pi- randola, Samuel Braunstein, Riccardo Laurenza, Carlo Ottaviani, Thomas Cope, Gaetana Spedalieri and Leonardo Banchi. Published in Quantum Science and Tech- nology 3 (2018). [82] xiii Chapter 1 Foreword and Preliminaries 1.1 Foreword The field of quantum mechanics emerged around a century ago, a reaction to newexperi- ments which defied accepted physical theories. The work of Maxwell and Hertz hadshown how light behaved as a wave, building on Young’s double slit experiment. Yet observed phenomena such as black body radiation and the photoelectric effect contradicted this stance. Furthermore, discoveries were made at the atomic level suggesting a Newtonian planetary-like structure, but with strictly discrete orbital energies. From these observa- tions emerged the idea that energy came in discrete “quanta” and that these quanta could show wave-like behaviour; an idea then extended to matter particles. From these concepts emerged mathematical frameworks, such as Schrödinger’s wave functions and Heisenberg’s uncertainty principle, to form the basis of quantum theory. The theory predicted many unusual and counter-intuitive effects, with several con- cepts controversial even amongst the pioneers of quantum mechanics. Einstein, a No- bel prize winner for his resolution of the photoelectric effect, along with Podolsky and Rosen presented the “EPR paradox” [39], showing the predictions of quantum theory ap- peared to violate the principles of relativity; measurement of the position of one particle could seemingly disturb another particle distantly separated, such that a measurement of the second particle’s momentum became uncertain. They proposed underlying “hidden” variables, with the uncertainty emerging only due to our lack of knowledge of the true physical description. This argument was investigated by John Bell [8], who proved the physical assumptions made by Einstein could not match all quantum behaviours.
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