Bell Nonlocality

Cover
Bell Nonlocality
Copyright
Dedication
Preface
Acknowledgment
Contents
Part I. Classic Bell Nonlocality
1. First Encounter with Bell Nonlocality
1.1 Three Roles for Bell Nonlocality
1.2 Introducing Bell Nonlocality
1.2.1 Setting Bell tests: Laboratories and games
1.2.2 The definition of Bell nonlocality
1.2.3 On resources and semantics
1.3 My First Bell Test: Clauser-Horne-Shimony-Holt (CHSH)
1.4 Four more Classic Bell Tests
1.4.1 Mermin’s outreach criterion
1.4.2 Greenberger-Horne-Zeilinger (GHZ) test
1.4.3 Hardy’s test
1.4.4 The Magic Square
1.5 A Closer Scrutiny: Addressing Loopholes
1.5.1 The “memory loophole,” or doing proper statistics
1.5.2 The “detection loophole(s),” or the dangers of post-selection
1.5.3 The “free-will loophole,” or measurement independence
1.5.4 The “locality loophole,” or hidden communication channels
1.5.5 The unknown loophole: Skepticism
1.6 Experimental Metaphysics?
1.6.1 Group 1: Nonlocal hidden variables
1.6.2 Group 2: Superdeterminism and its friends
1.6.3 Group 3: Only statistics are speakable, a.k.a. the “orthodox”
1.6.4 Group 4: Hoping for collapse
1.6.5 Additional remarks
EXERCISES
2. Formalizing Bell Nonlocality
2.1 Bell Scenarios, Processes, and Behaviors
2.1.1 Bell scenarios
2.1.2 Processes and strategies
2.1.3 The observed behavior
2.2 No-Signaling Processes and Behaviors
2.2.1 Definition and motivation
2.2.2 Description of no-signaling statistics
2.3 Local Behaviors
2.3.1 Definition and motivation
2.3.2 Local deterministic processes
2.3.3 Characterization of local behaviors: Fine’s theorem
2.3.4 Examples of local behaviors
2.4 The Local Polytope and Bell Inequalities
2.4.1 The local set as a polytope
2.4.2 Assessing the locality of a behavior
2.4.3 The notion of Bell inequality
2.4.4 Multiple versions, liftings, and no-signaling forms of an inequality
2.5 The CHSH Scenario
2.5.1 The local polytope and the CHSH inequality
2.5.2 The CH form of the CHSH inequality
2.6 Proper Statistics: Beyond i.i.d. and Finite Samples
2.6.1 Non-i.i.d. behaviors and the local polytope
2.6.2 Certifying nonlocality on real data
EXERCISES
3. Bell Nonlocality in Quantum Theory
3.1 Quantum Behaviors
3.1.1 Quantum and nonlocality: A preamble
3.1.2 Definitions
3.1.3 Relations between states, measurements, and behaviors
3.2 CHSH in Quantum Theory
3.2.1 The CHSH Bell operator
3.2.2 The Tsirelson bound
3.2.3 CHSH for two-qubit states and projective measurements
3.2.4 All pure bipartite entangled states violate CHSH
3.3 Mixed Entangled States as Nonlocal Resources
3.3.1 Single-copy nonlocality:Werner states
3.3.2 Many-copy nonlocality
EXERCISES
4. Review of Bipartite Bell Scenarios
4.1 Unanticipated Complexity
4.2 Several Inputs, Two Outputs
4.2.1 The (3, 2; 3, 2) scenario and the inequality I3322
4.2.2 The Mermin outreach criterion
4.2.3 Chained inequalities
4.3 Two Inputs, Several Outputs
4.3.1 CGLMP for m=3
4.3.2 CGLMP for any m
4.4 Hardy’s Test and the Magic Square
4.4.1 Hardy’s test
4.4.2 The Magic Square
EXERCISES
5. Multipartite Bell Nonlocality
5.1 Definition and Systematic Results
5.1.1 Multipartite local behaviors
5.1.2 All pure entangled states are nonlocal resources
5.1.3 The simplest multipartite Bell scenario
5.1.4 All correlation inequalities for two inputs, two outputs, n players
5.2 Examples of Multipartite Bell Inequalities
5.2.1 Construction of the MABK family
5.2.2 Maximal violation in quantum theory
5.3 Various Scenarios of Multipartite Nonlocality
5.3.1 Richness of nonlocal scenarios
5.3.2 Svetlichny scenarios
5.3.3 Scenarios with directional signaling
EXERCISES
Part II Nonlocality as a Tool for Certification
6. The Set of Quantum Behaviors
6.1 Device-Independent Certification: A First Introduction
6.2 Definition and Geometry of the Quantum Set
6.2.1 The definition
6.2.2 Another set, and the “Tsirelson problem”
6.2.3 Geometry
6.2.4 An example of boundary
6.3 Semidefinite Relaxations of the Quantum Set
6.3.1 The construction
6.3.2 Example 1: Membership in the quantum set
6.3.3 Example 2: The TLM bound
6.3.4 Example 3: Maximal quantum violation of a Bell inequality
EXERCISES
7. Device-Independent Self-Testing
7.1 Self-Testing of the Maximal Violation of CHSH
7.1.1 The proof
7.1.2 Consolidation
7.1.3 Two worked out examples
7.2 The Mayers-Yao Self-Testing Behavior
7.2.1 Inferences from the behavior
7.2.2 Swapping the relevant qubit into a controlled one
7.3 Formal Definition and its Consequences
7.3.1 Formal definition of self-testing
7.3.2 Which behaviors, and which states
7.4 Approximate Self-Testing: Robustness Bounds
7.4.1 Generic analytical bounds with poor robustness
7.4.2 Generic SDP techniques
7.4.3 A highly robust analytical result for specific cases
EXERCISES
8. Certifying Randomness
8.1 Introduction to Randomness
8.1.1 Product and process randomness
8.1.2 Random for whom? The predictor
8.1.3 The notion of secret randomness
8.1.4 The disruptive role of nonlocality
8.2 Quantification of Randomness
8.2.1 Guessing probability: Definition
8.2.2 Guessing probability and randomness extraction
8.2.3 Randomness in i.i.d. strategies and beyond
8.2.4 Warm up: Randomness in characterized quantum systems
8.3 Device-Independent Certification of Randomness
8.3.1 Generic formulation
8.3.2 Case study: Randomness in Alice’s output from CHSH
8.3.3 Optimizing the amount of randomness
8.4 Device-Independent Quantum Key Distribution
8.4.1 Introduction to QKD
8.4.2 Differences between QKD and randomness
EXERCISES
Part III. Foundational Insights from Nonlocality
9. Nonlocality in the No-Signaling Framework
9.1 The No-Signaling Polytope
9.2 The PR-Box
9.2.1 The PR-behavior and the hypothetical PR-box
9.2.2 Feats and failures of the PR-box as unit of nonlocality
9.2.3 An “implausible consequence” of having a PR-box
9.3 Device-Independent Certification Reloaded
9.3.1 Randomness generation
9.3.2 Key distribution
9.4 Refinements on Quantum Indeterminacy
9.4.1 The notion of local fraction
9.4.2 The randomness of the marginal distributions
EXERCISES
10. The Quest for Device-Independent Quantum Principles
10.1 Context: The Definition of Quantum Theory
10.2 Information Causality
10.2.1 The basic task: Random access code
10.2.2 The power of PR-boxes, again
10.2.3 Definition of Information Causality
10.2.4 Recovering the Tsirelson bound with IC
10.2.5 Information Causality as a physical principle?
10.3 Macroscopic Locality
10.3.1 Motivation and formulation
10.3.2 The set of ML behaviors is Q1
10.3.3 Summary of Macroscopic Locality
10.4 Local Orthogonality, a.k.a Consistent Exclusivity
10.5 The Current Barrier: The “Almost-Quantum” Set
EXERCISES
11. Signaling and Measurement Dependence
11.1 Motivation: Towards Ultimate Relaxations
11.2 Signaling Models: The Information in the Communication
11.2.1 Fine tuning
11.2.2 Signaling in Bell scenarios
11.2.3 Simulating state-behaviors with communication
11.3 Signaling Models: The Speed of the Influence
11.3.1 The trouble with “faster than light”
11.3.2 The need for infinite speed
11.4 Relaxing Measurement Independence
11.4.1 Behaviors under measurement dependence
11.4.2 Imposing restrictions on MD
11.4.3 The MDL polytope, and one inequality
11.5 Randomness Amplification
11.5.1 DI certification and measurement dependence
11.5.2 Setting the stage
11.5.3 Derivation
EXERCISES
12. Epilogue
Appendix A. History Museum
A.1 Heisenberg’s Uncertainty Relations and the Question of Indeterminism
A.2 The EPR Argument
A.3 Von Neumann’s Observation on Pre-Established Values
A.4 Bell’s 1964 Inequality
Appendix B. Experimental Platforms: A Reading Guide
B.1 Photons
B.1.1 Sources
B.1.2 Entangled degrees of freedom
B.2 Atomic Degrees of Freedom
B.3 How to Address the Detection Loophole
B.3.1 Setting up the notions
B.3.2 Efficiency for the (2, 2; 2, 2) scenario
B.3.3 A surprising optimization
Appendix C. Notions of Quantum Theory Used in this Book
C.1 States and Measurements
C.1.1 Definition of quantum theory
C.1.2 Tomography
C.1.3 Composite systems: Partial traces and purification
C.2 Elementary Entanglement Theory
C.2.1 Entangled states
C.2.2 Witnessing entanglement
C.2.3 Entanglement as a resource
C.3 Generalized Measurements or POVMs
C.3.1 Definition
C.3.2 A semantic subtlety
C.3.3 POVMs and joint measurability
Appendix D. LV Models for Single Systems
D.1 Overview: Looking into Measurements
D.2 Bell’s Local Hidden Variable Model for One Qubit
D.3 Contextuality
D.3.1 History at a glance
D.3.2 The KCBS inequality
D.3.3 Before and after KCBS
D.3.4 Contextuality and nonlocality: A comparison
Appendix E. Basic Notions of Convex Optimization
E.1 Generalities
E.1.1 Convex programs and their Lagrange dual
E.1.2 Special case: Linear programs
E.1.3 Special case: Semidefinite programs (SDPs)
E.2 Two Explicit Examples
E.2.1 The membership problem for the local polytope
E.2.2 An example of SDP
Appendix F. Device-Independent Certification: History and Review
F.1 Bell Nonlocality and Quantum Information Science
F.1.1 Before 2005: Nonlocality as an inspiration from the past
F.1.2 The tortuous path to device-independence
F.1.3 The last step
F.2 Overview of Tasks
F.2.1 Certification of secrecy tasks: Key distribution and randomness
F.2.2 Certification of quantum resources
F.2.3 Dimensionality of the physical system
F.3 Device-Independent, Really?
F.3.1 Providers and adversaries
F.3.2 Of labels and men
F.4 Certification with Partially Characterized Devices
F.4.1 Characterizing specific devices
F.4.2 Steering: Uncharacterized Alice, characterized Bob
F.4.3 Characterized quantum inputs
F.4.4 Networks with uncorrelated sources (a.k.a. N-locality)
Appendix G. Repository of Technicalities
G.1 Analytical Solution of the Facets of the CHSH Correlation Polytope
G.2 Hardy’s Test from the Schmidt Decomposition of the State
G.3 Pitfalls in Handling Signaling Behaviors
G.4 Jordan’s Lemma
G.4.1 Proof of Jordan’s lemma
G.4.2 Usefulness of the lemma
G.5 A Case Study of Randomness with Characterised Devices
G.5.1 Introduction and basic result
G.5.2 Calculation with classical side-information
G.5.3 Calculation with quantum side-information
G.6 Simulations of the Singlet Behavior
G.6.1 Alice’s sampling
G.6.2 Application to simulation
G.7 Properties of the Variational Distance
G.8 Information Causality from Desiderata on Information Entropies
References
Index