Entanglement quantification and quantum benchmarking of optical communication devices by Nathan Killoran A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2012 c Nathan Killoran 2012 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In this thesis, we develop a number of operational tests and tools for benchmarking the quantum nature of optical quantum communication devices. Using the laws of quantum physics, ideal quantum devices can fundamentally outperform their classical counterparts, or even achieve objectives which are classically impossible. Actual devices will not be ideal, but they may still be capable of facilitating quantum communication. Benchmark- ing tests, based on the presence of entanglement, can be used to verify whether or not imperfect quantum devices offer any advantage over their classical analogs. The general goal in this thesis is to provide strong benchmarking tools which simultaneously require minimal experimental resources but also offer a wide range of applicability. Another major component is the extension of existing qualitative benchmarks (`Is it quantum or classi- cal?') to more quantitative forms (`How quantum is it?'). This thesis is structured as follows. Chapters 1 and 2 provide an introduction and background material on quantum theory and quantum optics. Specifically, we outline the basic notions of quantum mechanics from a quantum information perspective, define some theoretical concepts which are important throughout the thesis, and detail two different methods for encoding quantum information in states of light, namely single-photon and continuous-variable encodings. In Chapter 3, we develop our first benchmarking results, for remote state preparation (RSP) protocols. To do this, we first distinguish quantum RSP protocols from classical RSP protocols by allowing the use of a pre-shared entangled state in the quantum case. Using the average fidelity between the desired target states and the actual remotely-prepared states to measure performance, we show how to calculate the best value achievable by a classical protocol. We apply these results to a number of different ensembles of target states, including pure states, mixed states, discrete ensembles, and continuous ensembles. From this, we propose a number of simple benchmarks based on Platonic solids. Finally, the proposed benchmarks are used to certify a quantum RSP experiment. In Chapters 4-6, we deal with the problem of extending existing continuous-variable (CV) quantum benchmarks to provide useful quantitative information. In Chapter 4, af- ter reviewing the known CV benchmarking results, in particular the entanglement-based approach, we outline a quantification scheme based on entanglement measures and pro- jections of infinite-dimensional optical systems to more managable finite subspaces. From this, two complementary paths are taken for the choice of projection. In Chapter 5, we pur- sue a two-qubit projection, which gives good quantitative results when the tested devices are near their ideal operating point. On the other hand, Chapter 6 follows the approach of iii projecting onto low-energy Fock states. This provides a better capacity to certify devices with larger degrees of imperfection. Finally, in Chapter 7, we show how to perform quantum benchmarking using arbitrary test states. Since many existing benchmarks are still limited to special cases, this provides a much-needed way to certify devices under more realistic testing conditions. We also study how to strengthen such benchmarks, leading to the best chance of certifying quantum devices. We use this approach, along with the quantitative tools developed in Chapters 4 and 6, to certify a real CV quantum memory. Together, the results contained in this thesis comprise a comprehensive suite of tools which can be used to perform efficient benchmarking tests on quantum devices using realistic experimental resources. iv Acknowledgements First of all, I would like to thank my supervisor Norbert L¨utkenhaus for his support and advice over the past four years. He was always willing to discuss my research with me, yet he also gave me space to work independently. As well, I appreciate the numerous opportunities he provided to attend international conferences and visit overseas research institutes. My thanks go as well to the members of my advisory and defense committees, Andrew Childs, Joseph Emerson, David Kribs, Kevin Resch, and to Alex Lvovsky for agreeing to serve as my external examiner. The Institute for Quantum Computing provided a fantastic environment for my grad- uate studies. In particular, I would like to thank the numerous members of the OQCT group who have come and gone during my tenure: Razieh Annabestani, Juan Miguel Ar- razola, Normand Beaudry, Agnes Ferenczi, Oleg Gittsovich, Hauke H¨aseler,Mattias Heid, Xiongfeng Ma, Will Matthews, Sergei Mikheev, Tobias Moroder, Geir-Ove Myhr, Varun Narasimhachar, Marco Piani, Dave Pitkanen, and Mohsen Razavi. As well, I would like to thank my many friends and colleagues in the ever-growing IQC community, in particular the diehards who showed up every week for ball hockey. I would also like to thank the folks at the Max Planck Institute for the Science of Light in Erlangen for providing a welcoming and congenial environment throughout my semester there. During my graduate studies, I lived in three places - the school, the grocery store and the skating rink. I thank my teammates on Smooth Liquid Thunder and Maxwell's Demons for many memorable seasons. I also want to thank Guelph Ultimate and the Waterloo Organization of Disc Sports for facilitating my activities on many summer and fall evenings. Finally, I'd like to thank all the other people who had an impact on me during my graduate studies. Thanks to my erstwhile roommates Devon, Chris, Mike, Osama (and Gina) for many memorable nights. Thanks to my parents for their continuing support and for always welcoming me for visits home. Most importantly, my gratitude to Emma for keeping me well during the thesis-writing process and for her enduring companionship throughout my studies. v Dedication To my fellow scientists. vi Table of Contents List of Figures x List of Tables xii 1 Introduction 1 2 Background 5 2.1 Preparation . 6 2.2 Evolution . 8 2.3 Measurement . 9 2.4 Quantum information theory . 11 2.5 Quantum optical implementations . 14 2.5.1 Single-photon encoding . 15 2.5.2 Continuous-variable encoding . 18 3 Remote state preparation benchmarks 23 3.1 Benchmarking framework . 25 3.1.1 Quantum RSP . 26 3.1.2 Classical RSP . 28 3.2 Threshold calculations . 31 3.2.1 Finite ensembles . 31 vii 3.2.2 Continuous ensemble . 35 3.2.3 Mixed states . 36 3.3 Experimental test . 38 4 Benchmarking continuous variable quantum communication devices 42 4.1 Quantum benchmarking framework . 44 4.1.1 Fidelity-based benchmarking . 45 4.1.2 Entanglement-based benchmarking . 47 4.2 Quantitative benchmarking . 56 4.2.1 Benchmarking with entanglement measures . 56 5 Quantitative benchmarking I: two-qubit projection 61 5.1 Quantification procedure . 62 5.1.1 Choosing the projection . 63 5.1.2 Estimating the constraints . 64 5.1.3 Minimizing the entanglement . 68 5.2 Results . 71 6 Quantitative benchmarking II: Fock-state projection 77 6.1 Quantification procedure . 78 6.1.1 Choosing the projection . 78 6.1.2 Estimating the constraints . 79 6.1.3 Minimizing the entanglement . 84 6.2 Results . 86 7 Benchmarking with realistic states 93 7.1 Benchmarking framework for arbitrary states . 95 7.1.1 Entanglement-based benchmarking with arbitrary states . 95 7.1.2 Optimizing the purifications . 97 viii 7.1.3 Fidelity-based benchmarking with arbitrary states . 102 7.1.4 Benchmarking with phase-symmetric ensembles . 107 7.2 Certifying a quantum memory . 110 7.2.1 Finding the Gram matrix . 111 7.2.2 Computing the entanglement . 113 8 Final remarks 119 APPENDICES 123 A Supplement for remote state preparation benchmarks 124 A.1 Proof of bound for continuous case . 124 A.2 Proof of optimal average fidelity for mixed states . 126 B Supplement for two-qubit projection scheme 128 B.1 Derivation of overlap bounds . 128 B.2 Derivation of supplementary diagonal bounds . 130 B.3 Derivation of off-diagonal bounds . 131 B.4 Constraints for two-qubit optimization . 132 C Supplement for Fock-state projection scheme 133 C.1 Derivation of first order constraint . 133 C.2 Derivation of second order constraint . 134 D Supplement for benchmarking with arbitrary states 136 D.1 Convexity proofs . 136 D.2 Proof of standard form . 137 D.3 Proof of trace norm in standard form . 138 References 141 ix List of Figures 2.1 Representation of a single-qubit state as a vector in the Bloch sphere . 17 2.2 Schematic of balanced homodyne measurement . 21 3.1 Remote state preparation schematic . 26 3.2 Example partitioning in a remote state preparation strategy . 30 3.3 Optimal classical fidelity thresholds for Platonic solid ensembles . 34 3.4 Comparison of experimentally-achieved mean fidelities with benchmark val- ues for pure states . 40 3.5 Comparison of experimentally-achieved mean fidelities with benchmark val- ues for mixed states . 41 4.1 Schematic of a measure and prepare strategy/Example CV test states . 46 4.2 Illustrations of entanglement-based benchmarking and effective entangle- ment concepts . 48 4.3 Quantum domain for test ensemble containing two coherent states |±αi . 53 4.4 Quantum domain for test ensemble containing M = 3; 4; 5; 6 symmetrically k distributed coherent states α!M ....................... 54 4.5 Quantum domain for a squeezed/antisqueezed pair |±ri .