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Quantum State Discrimination with Bosonic Channels and Gaussian States by Si Hui Tan

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Quantum State Discrimination with Bosonic Channels and Gaussian States by Si Hui Tan Quantum State Discrimination with Bosonic ARCHNE~S Channels and Gaussian States ASSAQHUSETTS INSTITUTE by Si Hui Tan LBRAR S Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctoral of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2010 @ Massachusetts Institute of Technology 2010. All rights reserved. A uthor ....... ......................................... Department of Physics September 3, 2010 Certified by.... .. ........... .............. .1 Seth Lloyd 2so I/Mechanical Engineering Thesis Supervisor Certified by.... .. .... ............ ........ Edward Farhi Cecil a d Ida Green Professor of Physics Thesis Supervisor A ccepted by...... .......................... Krishna Rajagopal Associate Head for Education, Physics 2 Quantum State Discrimination with Bosonic Channels and Gaussian States by Si Hui Tan Submitted to the Department of Physics on September 10, 2010, in partial fulfillment of the requirements for the degree of Doctoral of Philosophy in Physics Abstract Discriminating between quantum states is an indispensable part of quantum infor- mation theory. This thesis investigates state discrimination of continuous quantum variables, focusing on bosonic communication channels and Gaussian states. The specific state discrimination problems studied are (a) quantum illumination and (b) optimal measurements for decoding bosonic channels. Quantum illumination is a technique for detection and imaging which uses entanglement between a probe and an ancilla to enhance sensitivity. I shall show how entanglement can help with the discrimination between two noisy and lossy bosonic channels, one in which a target re- flects back a small part of the probe light, and the other in which all probe light is lost. This enhancement is obtained even though the channels are entanglement-breaking. The main result of this study is that, under optimum detection in the asymptotic limit of many detection trials, 6 dB of improvement in the error exponent can be achieved by using an entangled state as compared to a classical state. In the study of optimal measurements for decoding bosonic channels, I shall present an alternative measurement to the pretty-good measurement for attaining the classical capacity of the lossy bosonic channel given product coherent-state inputs. This new measurement has the feature that, at each step of the measurement, only projective measurements are needed. The measurement is a sequential one: the number of steps required is exponential in the code length, and the error rate of this measurement goes to zero in the limit of large code length. Although not physically practical in itself, this new measurement has a simple physical interpretation in terms of collective energy mea- surements, and may give rise to an implementation of an optimal measurement for lossy bosonic channels. The two problems studied in my thesis are examples of how state discrimination can be useful in solving problems by using quantum mechanical properties such as entanglement and entangling measurements. Thesis Supervisor: Seth Lloyd Title: Professor of Mechanical Engineering 3 Thesis Supervisor: Edward Farhi Title: Cecil and Ida Green Professor of Physics 4 Acknowledgments It is often said that a man can be judged by the company he keeps. I have the fortune of a great company of people to whom I am grateful to regardless of how I am to be judged. First and foremost, I would like to thank my advisor, Seth Lloyd, who has shared his ideas and enthusiasm over so many years in the pursuit of building a quantum- computer universe. I would also like to thank him for providing funding opportunities for my Ph.D studies. Special thanks go to the W.M. Keck Foundation Center for Extreme Information Theory which has provided the funding for my Ph.D. studies. Next, I would like to thank members of the Lloyd group and collaborators who have taught me the ropes and provided so many stimulating conversations. They are Jeffrey Shapiro, Saikat Guha, Lorenzo Maccone, Vittorio Giovannetti, Baris Erkmen, Stefano Pirandola, Mankei Tsang, Yutaka Shikano, Christian Weedbrook, Masoud Mohseni and Raul Garcii-Patr6n. Special thanks go to Daniel Oi, Silvano Garnerone and Prabha Mandayam who arranged for my visit to the University of Strathcldye, University of Southern California and Caltech respectively. I had so much fun visiting your groups at these universities. Then I would like to thank my long-time friends, Xianhui Fam and Xuanzhen Koh, who have kept our friendship alive after all these years I have been away from Singapore. I would also like to thank my friends from MIT: Andy Lutomirski, you have been a great source of inspiration, support and the quantum-money T-shirt. Darius Torchinsky, Jialu Yeh, Wanwan Yang, Jeffrey Edlund, Jian Yuan Thum, Then Kim, Jiahao Chen, Yingkai Ouyang, Scott Perlman, Justyna Pytel, and Valerie vande Panne. Graduate school has its rough spots and you have made it so much better. Last but not least, I would like to thank my parents for providing all the love and support that a daughter can ask for and also for being so patient with me while I pursue a dream that is not easily understood to be practical outside of academia. I dedicate this thesis to them. 5 6 Contents 1 Introduction 13 1.1 The notion of a quantum state. .. ... .. .. .. .. .. .. 13 1.2 Measuring a quantum state .. ... .. .. .. .. .. .. ... .. 13 1.3 State Discrimination . .. .. .. .. .. .. .. .. .. .. - 17 1.3.1 State Discrimination with Minimum Error . .. .. .. 18 1.3.2 Unambiguous State Discrimination .. .. .. .. .. .. .. 18 1.3.3 Pretty-Good Measurement . .. .. .. .. .. .. .. .. 20 1.3.4 Other Measurement Strategies . .. .. .. .. .. .. .. 20 1.4 The Role of State Discrimination in Quantum Information Theory . 21 1.5 Continuous Variables in Quantum Mechanics .. .. .. .. .. 23 1.6 Scope of the Thesis . .. .. .. .. .. .. .. .. .. .. .. 24 1.7 Layout of the Thesis .. .. .. .. .. .. .. .. .. .. ... .. 26 2 Continuous Variable Quantum Information Theory 29 2.1 Introduction ...... ............................. .. 29 2.2 Quantization of the Electromagnetic Field .. .. .. .. .. 30 2.3 Bosonic Modes from the Harmonic Oscillator .. .. .. .. ... 32 2.4 The Symplectic Structure of Phase Space . .. .. .. .. .. .. 36 2.4.1 Sp(2n, R ) .. .. .. .. .. .. .. .. .. .. 36 2.4.2 Sp(2n, R) and Canonical Linear Transformations .. .. 37 2.4.3 Quadratic Hamiltonians and Symplectic Transformations .. 37 2.5 Quantum Gates in Continuous Variables .. .. .. .. .. ... 38 2.5.1 Displacement Operator .. .. .. .. .. .. .. .. ... 40 7 2.5.2 Single-mode Squeezing Operator ....... 41 2.5.3 Two-mode Squeezing Operator ........ 42 2.5.4 Two-mode Mixing (Beam-splitter Operator) . 43 2.6 Representations in Phase Space ........... 44 2.6.1 Characteristic Functions ........... 44 2.6.2 Wigner function ................ 46 2.6.3 P distribution ........ ........ 46 2.6.4 Trace rules in Phase Space .......... 47 2.7 Uncertainty Principle ................. 48 2.8 Quantum States in Continuous Variables ...... 49 2.8.1 Coherent states . ............... 49 2.8.2 Squeezed States ........ ........ 50 2.8.3 Thermal States . ............... 51 2.8.4 Fock States ....... ........ ... 52 2.8.5 EPR States .... ........ ...... 52 2.8.6 Creating a Gaussian EPR pair ....... 54 2.9 Summary and Further Reading ........... 56 3 State Discrimination with Gaussian States 59 3.1 Introduction ........ ......... ..... 59 3.2 Normal form decomposition ... ......... 60 3.3 Distinguishability of States ....... ....... 61 3.3.1 Helstrom's Optimal Discrimination .... 61 3.3.2 Fidelity ...... ........ ...... 63 3.3.3 Quantum Chernoff Bound ..... ..... 64 3.4 Fidelity for Gaussian States ............. 66 3.4.1 Single-mode Gaussian States ..... .... 66 3.4.2 Multimode Gaussian States ....... .. 66 3.5 Quantum Chernoff Bound for Gaussian States .. 67 8 4 Quantum Illumination 69 4.1 Introduction . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 69 4.2 A Channel Description of Quantum Illumination .... ... .. .. 73 4.2.1 Use of ancillae ....................... .. ... ... 74 4.2.2 Thermal-Noise Model . .. .. .. .. .. .. ... ... 75 4.3 State Preparation ... .. .. .. .. .. .. .. .. .. .. ... ... 77 4.3.1 Choosing the Quantum States . .. .. .. .. ... .. ... 77 4.3.2 Displacing versus Squeezing the Vacuum . .. .. ...... 78 4.4 Covariance Matrices of States at the Receiver .. .. ... ... .. 80 4.4.1 Coherent-State Input ... .. .. .. .. .. .. ... ... 80 4.4.2 Two-Mode Entangled Signal-Idler Input . .. .. .. .. 82 4.5 Error Bounds .... ....................... ....... 84 4.5.1 Coherent-State Input . ... .. ... .. .. .. ... .. ... 84 4.5.2 Two-Mode Entangled Signal-Idler Input . .. .. .. .. 86 4.6 D iscussion .. ... .. .. ... .. .. ... .. .. .. .. .. .. 87 5 Optimal Measurement for Bosonic Channels 91 5.1 Introduction . ... ... ... ... ... .... ... .. .. .. .. 91 5.2 Communication with Bosonic Channel ... .. .. .. .. .. .. 91 5.3 Statistical Properties of the Encoding Ensemble . .. .. .. .. 93 5.4 T ypicality .. ... ... ... .. ... ... .. ... .. 94 5.4.1 Asymptotic Equipartition Property Theorem (AEP). .. 95 5.4.2 Classical Typical Subset .. ... .... ... .. 95 5.4.3 Quantum Typical Subspace . ... .. ... .. .. .. 96 5.4.4 Typical Subspace Theorem . .. ... ... ... .. 98 5.5 Decoding for the Lossy Channel .. .. ... ... .. .. 100 5.5.1 The Sequential Decoder ... ... .. ... .. .. .. 100 5.5.2 Projections onto codewords with Typicality .. .. .. 106 5.5.3 Levy's Lemma applied to Overlap
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