Abstract
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.