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Postulates of Quantum Mechanics
Having set the stage for the postulates of quantum mechanics, it is time to actually introduce them. The postulates are a set of statements “pairing” physical objects and processes with abstract notions of Hilbert spaces, giving an explicit recipe for doing physical calculations in Hilbert spaces. Different textbooks give different numbers of postulates, either 4 [NC00] [ Sha94] or 5 [ DM05] or 6 [CTDL97] . Some textbooks do not list the postulates explicitly at all, mixing themathematical background with physical concepts from the beginning [ Sze04] [ SN10] [Wei12] . Even though the number of postulates differ from textbook to textbook, all the postulates, either explicitly listed or implicitly blended into the background, cover three aspects of a physical theory: what a physical system is, how to probe a physical system to extract information, and what is the dynamics governing the evolution of a physical system. The postulates listed here will mainly follow the standard textbook of quantum information [NC00] , while only working with pure states.
Postulate 1. The state space. A physical system S is represented by a (complex) Hilbert space H. A state of the system at a given instant is given by a vector in this Hilbert space. Such a state vector is called a ket, denoted by j i . A composite physical system is represented by thetensor product of its components. Ketsof thecompositesystem correspond to linear combinations of tensor products of its components. There are a few subtleties in this postulate. First of all, states of the physical system and vectors in the corresponding Hilbert space are not mapped one-to-one. A vector, together with all its scalar multiples, corresponds to the same state. To reduce such arbitrariness, all state vectors are supposed to be normalized: k j i k = 1. The second subtlety arises from this normalization requirement. Only requiring the norm of a ket to be 1 does not completely remove the arbitrariness, because any complex number with norm 1 multiplied with a normalized ket will still give a normalized ket. Because of this, a complex number with norm 1 is usually called a phase, and two kets different only by a global phase are undistinguishable.
Scratching the Surface of Quantum Entanglement
The linearity of quantum states hasmany important consequences. The most important of which is entanglement. The term, coined by Schrodinger, was originally used to describe the kind of quantum states used in the EPR thought experiment (cf. the next section). In the modern language, entanglement characterizes the ability to factorize the Hilbert space of a multiparty state into tensor products of smaller Hilbert spaces [GT09, HHHH09] . For bipartite pure states, being entangled means that a state j i 2 H can not be decomposed as j i = ji 1 ji 2 , (2.7) where j i 1 2 H1, j i 2 2 H2 and H = H1 H2. When a state can be written as (2.7), it is called a product or separable state.
Almost Probability-free Nonlocality & Hardy Paradox
Nonlocality from correlations and probability-free nonlocality represent two extremes: expectation value containing correlationsmust be built from statistics of many experimental runs, while probability-free only use perfect correlations. The third type, which can be seen as a mixture of the other two approaches, was proposed by Hardy in early 90s as a logical paradox first, then put into an inequality [Har93, Har94] . This inequality is the first inequality using probabilities instead of expectation values. It is first derived by the first half of CHSH [CH74] , and experimentally tested by the first and third (the most famous) experiments of Aspect et al [AGR81, ADR82] .
To see what the Hardy paradox is and why it is a logical paradox at all, consider two people, Alice and Bob, each can choose from two types of sealed boxes given to them individually from a referee, Charlie. One type of box contains food, upon opening, it will reveal either a baguette or a bowl of noodles. The other type contains drinks, upon opening, either a cup of coffee or a cup of tea can be found. Charlie made the following promises to Alice and Bob with regards to the type of boxes they choose to open and the contents they will find inside:
1. If they both choose the food, then it is possible that they both get baguettes.
2. If Alice chooses the food box and gets a baguette, then if Bob chooses the drink box he will never get a cup of tea.
3. If Bob chooses the food box and gets a baguette, then if Alice chooses the drink box she will never get a cup of tea.
4. If they both choose drinks, they will never both get coffee.
Table of contents :
Acknowledgements
Abstract
Re´ sume´
Publications
0 Sommaire de la the¡ se
0.1 Re¬sume¬ des chapitres
0.2 La non-localite¬ des e¬tats syme¬triques
0.3 La non-localite¬ et les classes d’intrication
0.4 Une ine¬galite¬monogame pour les e¬tats de Dicke
1 Introduction
2 Background
2.1 Essential Algebra
2.2 Postulates of Quantum Mechanics
2.3 Scratching the Surface of Quantum Entanglement
2.4 The Facets of Nonlocality
2.4.1 A Little Bit of History
2.4.2 Bell’s Inequality
2.4.3 Nonlocality from Correlations
2.4.4 Probability-free Nonlocality & Mermin Inequality
2.4.5 Almost Probability-free Nonlocality & Hardy Paradox
2.4.6 Unification of Different Approaches to Nonlocality
2.5 Semidefinite Programming
3 The Majorana Representation of Symmetric States
3.1 Geometry of Complex Numbers
3.1.1 The Complex Plane and the Riemann Sphere
3.1.2 The Mobius Transformation
3.2 From Complex Geometry to the Majorana Representation
3.3 Physical Interpretations
3.4 The Majorana Representation and Entanglement
4 Nonlocality of Symmetric States
4.1 Bipartite Hardy Paradox Revisited
4.2 Multipartite Hardy Paradox and the Inequality Pn
4.3 Violation of Pn By Almost All Symmetric States
4.4 Violation of Pn By All Symmetric States
5 Degeneracy and its Consequences
5.1 Degeneracy and Persistency of Nonlocality
5.2 Device Independent Classification of States
6 Analysis of Nonlocal Properties for Symmetric States
6.1 Large n Results for jWni and jGHZni
6.2 Monogamy of Entanglement and Monogamy of Correlations
6.3 Monogamy and Genuine Nonlocality of Dicke States
7 Applications
7.1 Application to Communication Complexity
7.2 Application to Bayesian Games
8 Summary
8.1 New Results in This Thesis
8.2 Recent Progress on Related Topics
8.3 Outlooks
Bibliography
