量子非局域性(quantum nonlocality，又稱為Bell nonlocality)、量子操控性(quantum steering，又稱為 Einstein-Podolsky-Rosen steering)、量子糾纏(quantum entanglement)、以及量子傳輸適用性(usefulness of quantum teleportation)是四個各自互異的量子效應。有鑑於最近量子非局域性的超啟動(superactivation)被證明，是否量子操控性以及量子傳輸適用性亦具備超啟動特性便成為了一個有趣的問題。在這篇碩士論文之中，我嘗試將上述四種不同的量子效應之間的關係釐清，以及更進一步證明與探討量子操控性的超啟動特質。另外，我亦會於此篇論文之中介紹我所證明出的量子非局域性以及量子操控性之超啟動特質的充分條件(sufficient condition)，並將會介紹此充分條件的可能應用與推廣，其中包含使用此充分條件所得到對於非負貝爾泛函(nonnegative Bell functional)的最大貝爾違反量(the largest Bell violation)以及飛覆操控泛函(nonnegative steering functional)的最大操控違反量(the largest steering violation)的上界估計(upper bounds)。

Einstein-Podolsky-Rosen (EPR) steering, together with Bell nonlocality and entanglement, are three important nonlocal feature of Quantum Mechanics. They are crucial in their own ways: entanglement is the core of many novel phenomena in quantum computation and quantum information; Bell nonlocality is not only one of the strongest experimental verification of Quantum Mechanical phenomena, but also admitting many applications in quantum cryptography, communication complexity, and many other topics. EPR steering, having its strength between nonlocality and entanglement, has its own unique role. For instance, in a quantum key distribution scenario between two parties Alice and Bob, a violation of steering inequality allows certification of quantum correlation shared by them even if Bob doesn’t trust Alice’s apparatus. It also has potential application in quantum communication. Moreover, being a nonlocal property weaker than nonlocality, EPR steering has potential to reveal quantumness easier than nonlocality. These motivation encouraged people to verify and study EPR steering experimetnally and theoretically. However, unlike nonlocality and entanglement, the current understanding of EPR steering is limited. One such example is the superactivation of EPR steering. To capture the idea of superactivation, suppose there is a state which doesn’t have a given physical property (e.g. nonlocality, entanglement, EPR steerability, etc.). One can ask the question: How about we consider several copies of that state? Can the resulting state obtain the physical property? If a given physical property can be superactivated, it means one can construct the given physical property from a collection of states without that property. In 2012, Palazuelos established nontrivial superactivation of Bell nonlocality. This important breakthrough not only enables people to see more about the theoretical structure such as the relation between nonlocality and teleportation, it also sheds new lights on the experimental demonstration of nonlocality from a collection of local states. Hence, whether EPR steering can be superactivated becomes an interesting question both theoretically and experimentally.
In this thesis, I will give a brief introduction of Bell nonlocality, EPR steering, entanglement, quantum teleportation, their superactivation property, and how they are related to each other. This thesis is constructed as follows. Chapter 1 is for a conceptual introduction for Bell nonlocality, quantum teleportation, EPR steering, entanglement, and their nontrivial relations. In Chapter 2, two mathematical tools called fully entangled fraction (FEF) and quantum twirling will be introduced. Both of them are crucial for Chapter 3, and rather than simply stating known results, readers can find my progress on the study of relation between FEF and two generalized versions of quantum twirling defined by me. In Chapter 3, I will firstly state the mathematical formalism for Bell nonlocality, EPR steering, and the nontrivial superactivation of Bell nonlocality. After the discussion of those known results, I will demonstrate nontrivial superactivation of EPR steering, with an exact form of the steering functional achieving the superactivation, and I will also derive new sufficient condition for k-copy nonlocality and k-copy steerability (from Alice to Bob). Using a simple physical argument, those two sufficient conditions will give us upper bounds for the largest Bell/steering violation of maximally entangled state with a given nonnegative Bell/steering functional. Furthermore, by considering projective measurement, better upper bounds can be found.

In this thesis, I will give a brief introduction of Bell nonlocality, EPR steering, entanglement, quantum teleportation, their superactivation property, and how they are related to each other. This thesis is constructed as follows. Chapter 1 is for a conceptual introduction for Bell nonlocality, quantum teleportation, EPR steering, entanglement, and their nontrivial relations. In Chapter 2, two mathematical tools called fully entangled fraction (FEF) and quantum twirling will be introduced. Both of them are crucial for Chapter 3, and rather than simply stating known results, readers can find my progress on the study of relation between FEF and two generalized versions of quantum twirling defined by me. In Chapter 3, I will firstly state the mathematical formalism for Bell nonlocality, EPR steering, and the nontrivial superactivation of Bell nonlocality. After the discussion of those known results, I will demonstrate nontrivial superactivation of EPR steering, with an exact form of the steering functional achieving the superactivation, and I will also derive new sufficient condition for k-copy nonlocality and k-copy steerability (from Alice to Bob). Using a simple physical argument, those two sufficient conditions will give us upper bounds for the largest Bell/steering violation of maximally entangled state with a given nonnegative Bell/steering functional. Furthermore, by considering projective measurement, better upper bounds can be found.

CHAPTER 1: INTRODUCITON......1
1.1 Bell Nonlocality, Quantum Teleportation, and Entanglement......2
1.2 Bell Nonlocality, Einstein-Podolsky-Rosen Steering, and Entanglement......12
CHAPTER 2: PRELIMINARY......17
2.1 Fully Entangled Fraction (FEF)......18
2.2 Quantum Twirling......23
CHAPTER 3: SUPERACTIVATION......31
3.1 Notations......32
3.2 Khot-Vishnoi Game and Superactivation of Bell Nonlocality......35
3.3 Superactivation of Einstein-Podolsky-Rosen Steering......49
3.4 Superactivation of Quantum Teleportation?......61
SUMMARY......63
APPENDIX......64
Appendix A: Einstein-Podolsky-Rosen Paradox......65
Appendix B: W is a Werner State......67
Appendix C: Teleportation with Arbitrary Local Dimension......70
Appendix D: No-Cloning Theorem......74
Appendix E: Detailed Proof of Proposition 2.22......75
Appendix F: Detailed Proof of Lemma 3.7......76
Appendix G: Ho ̈lder’s Inequaltiy for Trace Norm and Hilbert-Schmidt Norm ......81
REFERENCES......82

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