Entanglement Theory and its Applications in Gaussian Quantum Information Spyros Tserkis B.Sc., M.Sc. A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2019 School of Mathematics and Physics ii Abstract Entanglement is a physical property, emerging from the superposition of composite quantum systems, and manifested through non-classical correlations of quantum observables. In the field of Quantum Information, entanglement is considered the “resource” for various quantum protocols, so the ability to quantify it would allow us to associate it with the success of those protocols. Entanglement of formation is a proper way to quantify entanglement, but an analytical expression for this measure exists only for special cases. In this thesis, we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for this measure that get tight for several special cases. We further study how quantum teleportation and entanglement distillation can be employed in order to error-correct information encoded on a Gaussian state that suffers from Gaussian noise. In particular, we derive every physical state able to simulate a given phase-insensitive Gaussian channel through teleportation with finite-energy resources, and show how error-correction of a state is related to the simulation ofa less decohering channel that the state has to pass through. We also discuss how the premise that the whole environment is under control of the adversary in quantum key distribution can be eliminated if we consider a teleportation-based eavesdropping attack. More specifically, we propose an all-optical teleportation attack that under collective measurements can reach optimality in the limit of infinite amount of entanglement, while for finite entanglement resources it outperforms the corresponding optimal individual attack. Finally, we show how using a class of finite-energy resource states we can increasingly approximate the infinite-energy bounds for decreasing purity, so that they provide tight upper bounds to the secret-key capacity of single-mode phase-insensitive Gaussian channels. Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. Thecontentof my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis. Spyros Tserkis Publications included in this thesis 1. Spyros Tserkis and Timothy C. Ralph, Quantifying entanglement in two-mode Gaussian states, Physical Review A 96, 062338 (2017). 2. Spyros Tserkis, Josephine Dias, and Timothy C. Ralph, Simulation of Gaussian channels via teleportation and error correction of Gaussian states, Physical Review A 98, 052335 (2018). 3. Spyros Tserkis, Sho Onoe, and Timothy C. Ralph, Quantifying entanglement of formation for two-mode Gaussian states: Analytical expressions for upper and lower bounds and numerical estimation of its exact value, Physical Review A 99, 052337 (2019). 4. Riccardo Laurenza, Spyros Tserkis, Leonardo Banchi, Samuel L. Braunstein, Timothy C. Ralph, and Stefano Pirandola, Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources, Physical Review A 100, 042301 (2019). 5. Spyros Tserkis, Neda Hosseinidehaj, Nathan Walk, and Timothy C. Ralph, Teleportation-based collective attacks in Gaussian quantum key distribution, Physical Review Research 2, 013208 (2020). Other publications and preprints during candidature 1. Hao Jeng, Spyros Tserkis, Jing Yan Haw, Helen M. Chrzanowski, Jiri Janousek, Timothy C. Ralph, Ping Koy Lam, and Syed M. Assad, Entanglement properties of a measurement-based entanglement distillation experiment, Physical Review A 99, 042304 (2019). 2. Cheng Jiang, Spyros Tserkis, Kevin Collins, Sho Onoe, Yong Li, Lin Tian, Switchable bipartite and genuine tripartite entanglement via an optoelectromechanical interface, arXiv:1910.13173. Contributions by others to the thesis Prof. Timothy C. Ralph has supervised all the projects/papers discussed in this thesis (presented in Chapters 6-9), and, along with Dr. Austin P. Lund, provided critical feedback for improvements and revisions of the whole thesis. I also acknowledge the contributions by Josephine Dias in Ref. [1], Sho Onoe in Ref. [2], Neda Hosseinidehaj and Nathan Walk in Ref. [3], who helped in the preparation of those manuscripts by giving feedback and editing parts of the manuscripts. Contributions by Stefano Pirandola, Leonardo Banchi, Riccardo Laurenza, and Samuel L. Braunstein regarding Ref. [4], are discussed in the beginning of Chapter 9. Statement of parts of the thesis submitted to qualify for the award of another degree No works submitted towards another degree have been included in this thesis. Research involving human or animal subjects No animal or human subjects were involved in this research. Acknowledgements I would like to acknowledge, and express my gratitude to my principal advisor, Tim Ralph, for the patient guidance, encouragement, and advice he has provided me throughout my PhD candidature. I would also like to thank my associate advisor, Austin Lund, for his support and the fruitful discussions during my studies. My sincere acknowledgement goes to my colleagues and co-authors, Josephine Dias, Nedasadat Hosseibnidehaj and Sho Onoe. I am also grateful for meeting and collaborating with Stefano Pirandola, Riccardo Laurenza, Samuel Braunstein, Syed Assad, Hao Jeng, Ping Koy Lam, Nathan Walk, and Leonardo Banchi. I would like to thank our group administrator, Kaerin Gardner, and the postgrad administrator, Murray Kane, for being so helpful to me during my studies. Special thanks to Farid Shahandeh and Anatoly Kulikov for our interesting discussions. I am also thankful for meeting all the members of the Quantum Optics and Quantum Information group. Financial support This research was supported by: • the University of Queensland International Scholarship (tuition fee award and living allowance stipend), awarded by The University of Queensland. • the Research Higher Degree Scholarship (living allowance top-up), awarded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technol- ogy (CE110001027 and CE170100012) and the Australian Department of Defence, Innovation Hub. Keywords Keywords: quantum information, quantum communication, quantum entanglement, quantum correla- tions, quantum teleportation, quantum error correction, quantum key distribution, quantum channel capacities Australian and New Zealand Standard Research Classifications (ANZSRC) • ANZSRC code: 020603 Quantum Information, Computation and Communication 80% • ANZSRC code: 020604 Quantum Optics, 20% Fields of Research (FoR) Classification • FoR code: 0206 Quantum Physics, 90% • FoR code: 0105 Mathematical Physics, 10% Table of contents List of figures xxxi I Introduction1 1 Introductory Remarks3 1.1 Historical Note . .3 1.2 Goal of the Thesis . .4 1.3 Structure of the Thesis . .5 II Literature Review9 2 Mathematical Preliminaries 11 2.1 Vector Spaces . 12 2.1.1 Hilbert Space . 13 2.1.2 Fock Space . 15 2.2 Linear Operators . 15 2.2.1 Normal Operators . 18 2.2.2 Convexity and Concavity . 20 2.2.3 Composite operators . 21 2.2.4 Superoperators . 22 3 Quantum Mechanics and Quantum Information 23 3.1 Quantum State . 24 3.1.1 Pure and Mixed States . 24 3.1.2 Interpreting the Quantum State . 26 3.1.3 Quantum Entropy of a State . 26 3.2 Quantum Operation . 28 3.2.1 Evolution of a Closed Quantum System . 29 3.2.2 Evolution of an Open Quantum System . 29 3.2.3 Measurement of a Quantum State . 32 3.3 Distinguishing Quantum States . 34 3.3.1 Trace Distance . 34 xxviii Table of contents 3.3.2 Fidelity . 35 3.3.3 Relative Entropy . 36 3.4 Examples of Quantum Systems . 36 3.4.1 Finite-dimensional System . 36 3.4.2 Infinite-dimensional System . 38 3.5 Quantum Mechanics in Phase Space . 40 4 Gaussian Systems 43 4.1 Gaussian States and Operations . 43 4.1.1 Vacuum States . 45 4.1.2 Coherent States and the Displacement Operator . 46 4.1.3 Squeezed States and the Squeezing Operator . 49 4.1.4 Thermal States . 51 4.1.5 Phase Rotation Operator . 52 4.1.6 Beam Splitter Operator . 52 4.1.7 Two-mode Squeezing . 53 4.2 Symplectic Analysis of Gaussian States . 54 4.2.1 Decomposition of a Gaussian State . 54 4.2.2 Purification of a Gaussian State . 55 4.2.3 Two-mode Gaussian States . 56 4.3 Gaussian Measurements . 58 4.3.1 Homodyne Detection . 59 4.3.2 Heterodyne Detection . 60 4.3.3 Partial Measurements . 60 4.4 Distinguishing Gaussian States . 61 4.4.1 Fidelity . 61 4.4.2 Relative Entropy .