Digital Quantum Information Processing with Continuous-Variable Systems (Springer Theses)

✍ Scribed by Takaya Matsuura

• Publisher: Springer
• Tongue: English
• Leaves: 172
• Category: Library

✦ Synopsis

The book provides theoretical methods of connecting discrete-variable quantum information processing to continuous-variable one. It covers the two major fields of quantum information processing, quantum communication and quantum computation, leading to achievement of a long-sought full security of continuous-variable quantum key distribution (QKD) and proposal of a resource-efficient method for optical quantum computing.

Firstly, the book provides a security of continuous-variable QKD against arbitrary attacks under a realistic condition such as finite communication rounds and the use of digitized information processing. 

The book also provides the unified view for conventionally used approximate Gottesman-Kitaev-Preskill (GKP) codes, which encodes qudits on a continuous-variable system, enabling direct comparison between researches based on different approximations.  The book finally proposes a resource-efficient method to realize the universal optical quantum computation using the GKP code via the direct preparation of the GKP magic state instead of GKP Pauli states. Feasibility of the proposed protocol is discussed based on the existing experimental proposals for the GKP state preparation.

✦ Table of Contents

Supervisor’s Foreword
Preface
List of Publications
Acknowledgements
Contents
1 Introduction
1.1 Background of the Research
1.2 What is Studied in This Thesis
1.3 The Organization of This Thesis
References
2 Preliminaries
2.1 The Basic Principles of Quantum Information Theory
2.1.1 Linear Operator and Quantum State
2.1.2 Quantum Operation and Quantum Channel
2.1.3 Quantum Measurement and Quantum Instrument
2.1.4 Qubit as an Information Unit
2.2 Measures for the Closeness of Quantum States
2.2.1 The Trace Norm and The Trace Distance
2.2.2 The Quantum Fidelity
References
3 Continuous-Variable Quantum System
3.1 Basics of the Continuous-Variable Quantum System
3.1.1 Position and Momentum Operators
3.1.2 Characteristic Function and Wigner Function
3.1.3 Gaussian States, Gaussian Channels, and Gaussian Measurements
3.1.4 Gaussian Operations
3.2 Quantum Optical System as a Continuous-Variable System
3.2.1 Quantization of Electromagnetic Field
3.2.2 Operations in Quantum Optics
References
4 Quantum Key Distribution with Continuous-Variable Systems
4.1 Introduction for This Chapter
4.2 Notations and Preliminaries
4.2.1 Finite Field mathbbF2
4.2.2 Classical Linear Information Processing as a Quantum Operation
4.2.3 Definitions and Properties of the Entropic Quantities
4.2.4 Concentration Inequalities
4.3 The Basics of the QKD
4.3.1 The Goal of the QKD
4.3.2 The General Procedures of the QKD
4.3.3 The Security Condition of the Key in the QKD
4.3.4 An Approach to Prove the Security Condition
4.3.5 The Privacy Amplification Using Dual Universal2 Hashing
4.3.6 Key Rate of the QKD Protocol
4.4 Finite-Size Security of Continuous-Variable QKD with Digital Signal Processing
4.4.1 Estimation of the Fidelity to a Coherent State
4.4.2 Proposed Protocol
4.4.3 Security Proof
4.4.4 Derivation of the Operator Inequality
4.4.5 Numerical Simulations
4.4.6 Discussion
4.5 Finite-Size Analysis for the Binary-Modulation Protocol Based on the Reverse Reconciliation
4.5.1 Alternative Protocol
4.5.2 Security Proof Based on the Reverse Reconciliation
4.5.3 Phase Error Operator
4.5.4 Proof of the Operator Inequality
4.5.5 Numerical Simulations
4.5.6 Discussion
4.6 Conclusion for This Chapter
References
5 Quantum Computation with Continuous-Variable Systems
5.1 Introduction for This Chapter
5.2 Notations and Preliminaries
5.2.1 Qudit, The Pauli Group, and The Clifford Group
5.2.2 The Gottesman-Kitaev-Preskill Code
5.3 On the Equivalence of Approximate Gottesman-Kitaev-Preskill Codes
5.3.1 Position and Momentum Representations
5.3.2 Explicit Relations Among the Three Approximations
5.3.3 The Standard Form
5.3.4 Explicit Expressions of the Wigner Function, Inner Products, and Average Photon Number
5.3.5 Discussion
5.4 Cost-Reduced All-Gaussian Universality with the GKP Code
5.4.1 Deterministic All-Gaussian Universality Using the GKP Magic States
5.4.2 A Resource-Theoretical Analyses for Fundamental Limitations on GKP State Conversion
5.4.3 Feasibility of Preparing a GKP Magic State
5.4.4 Discussion
5.5 Conclusion for This Chapter
References
6 Conclusion
6.1 Conclusion
Appendix Appendix
A.1 The Grid Representation
A.2 Proofs of the Propositions and Lemmas in Sect. 5.3摥映數爠eflinksec:equivalencespsapproxspsgkp5.35
A.2.1 Proof of Proposition 5.3.2摥映數爠eflinkprop:positionspsrep5.3.25 and Lemma 5.3.3摥映數爠eflinklem:convspstospsthetaspsfunc5.3.35
A.2.2 Proof of Proposition 5.3.10摥映數爠eflinkprop:wignerspsgkp5.3.105
A.2.3 The Proof of Proposition 5.3.12摥映數爠eflinkprop:avespsphoton5.3.125
A.3 Alternative Expressions for the Wigner Function, Inner Products, and Average Photon Number of the Approximate GKP Code
A.3.1 An Aternative Expression of the Wigner Function
A.3.2 Alternative Expressions of Normalization Constant and Inner Product
A.3.3 Alternative Expression of Average Photon Number
References