Device-Independent Quantum Information Processing: A Simplified Analysis

Supervisor’s Foreword
Abstract
Acknowledgements
Contents
1 Introduction
	1.1 Motivation
		1.1.1 Device-Independent Information Processing
		1.1.2 Reductions to IID
	1.2 Content of the Thesis
		1.2.1 Reductions
		1.2.2 Showcases
	1.3 How to Read the Thesis
	References
2 Preliminaries: Basics and Notation
	2.1 General Notation
	2.2 Probability Distributions and Random Variables
		2.2.1 Independent and Identical Random Variables
		2.2.2 Concentration Inequalities
	2.3 Quantum Formalism
		2.3.1 Operators
		2.3.2 Hilbert Spaces
		2.3.3 Quantum States
		2.3.4 Quantum Operations
	2.4 Distance Measures
	2.5 Entropies
		2.5.1 Shannon and von Neumann Entropy
		2.5.2 Min- and Max-Entropies
	References
3 Preliminaries: Device-Independent Concepts
	3.1 Black Boxes
		3.1.1 Non-signalling Boxes
		3.1.2 Quantum Boxes
		3.1.3 Classical Boxes
		3.1.4 Correlations' Space
	3.2 Bell Inequalities
		3.2.1 Non-local Games
		3.2.2 The CHSH Game
	3.3 Untrusted Devices
	References
4 Introduction to the Showcases
	4.1 Introduction to Non-signalling Parallel Repetition
		4.1.1 Parallel Repeated Games
		4.1.2 Threshold Theorems
	4.2 Introduction to Device-Independent Quantum Cryptography
		4.2.1 DIQKD Security Definitions
		4.2.2 DIQKD Protocol
		4.2.3 Main Task of a Security Proof
		4.2.4 The Honest Implementation
		4.2.5 Model of an Arbitrary Device
	References
5 Single-Round Box
	5.1 The Model
		5.1.1 Quantum Single-Round Boxes
	5.2 Showcase: Device-Independent Quantum Cryptography
	References
6 Multi-round Box
	6.1 Parallel Interaction
		6.1.1 Non-signalling Parallel Boxes
	6.2 Sequential Interaction
		6.2.1 Without Communication Between the Rounds
		6.2.2 With Communication Between the Rounds
	References
7 Working Under the IID Assumption
	7.1 The IID Assumption
	7.2 Asymptotic Equipartition Property
		7.2.1 Classical Asymptotic Equipartition Property
		7.2.2 Quantum Asymptotic Equipartition Property
	7.3 Using the IID Assumption
		7.3.1 Showcase: Non-signalling Parallel Repetition
		7.3.2 Showcase: Device-Independent Quantum Cryptography
	7.4 Beyond IID
	References
8 Reductions to IID: Parallel Interaction
	8.1 Permutation Invariance
	8.2 de Finetti Reductions for Correlations
	8.3 Ways of Using the Reductions
		8.3.1 Post-selecting Permutation Invariant Boxes
		8.3.2 Failure Probability of a Test
		8.3.3 Diamond Norm
	8.4 Impossibility Results
		8.4.1 Restricted de Finetti Box
		8.4.2 Extension to an Adversary
		8.4.3 Other de Finetti-Type Theorems
	References
9 Reductions to IID: Sequential Interaction
	9.1 Sequential Quantum Processes
	9.2 Entropy Accumulation Theorem
		9.2.1 Conceptual Difficulties to Overcome
		9.2.2 Prerequisites of the Theorem
		9.2.3 Statement of the Theorem
	References
10 Showcase: Non-signalling Parallel Repetition
	10.1 Main Challenge and Goal
	10.2 Approximately Non-signalling Marginals
		10.2.1 Single-Round Boxes from Frequencies
		10.2.2 Signalling Test
		10.2.3 Guessing Game
	10.3 Threshold Theorem
		10.3.1 Winning Probability of Approximately Non-signalling Strategies
		10.3.2 Final Result
	10.4 Open Questions
	References
11 Showcase: Device-Independent Quantum Cryptography
	11.1 Main Challenge and Goal
	11.2 Device-Independent Entropy Accumulation
		11.2.1 EAT Channels
		11.2.2 Min-Tradeoff Function
		11.2.3 Smooth Min-Entropy Rate
	11.3 Device-Independent Quantum Key Distribution
		11.3.1 Completeness
		11.3.2 Soundness
		11.3.3 Key Rate Analysis
	11.4 Open Questions
		11.4.1 Experimental Realisations
		11.4.2 Possible Extensions
		11.4.3 Bounding the von Neumann Entropy
	References
12 Outlook
	12.1 Two-Party Device-Independent Quantum Cryptography
	12.2 Parallel Device-Independent Quantum Cryptography
	12.3 Device-Independent Tomography
	References
Appendix A Additional Proofs: de Finetti Reductions
A.1  Bounding the de Finetti Box
A.2  Diamond Norm Reduction
Appendix B Additional Proofs: Non-signalling Parallel Repetition
B.1  Signalling Measure and Test
B.2  Sensitivity Analysis
Appendix C Additional Proofs: Device-Independent Quantum Cryptography
C.1  Single-Round Statement
C.2  An Improved Dependency on the Test Probability
C.2.1  Modified Entropy Accumulation Protocol
C.2.2  Modified Min-tradeoff Function
C.2.3  Modified Entropy Rate
C.2.4  Modified Key Rate
C.3  Summary of Parameters and Variables