The objective of this thesis is to use the Majorana description of a spin-1 system to give a geometrical construction of a maximal set of Mutually Unbiased Bases (MUBs) and Symmetric Informationally Complete Positive Operator Valued Measures (SIC-POVMs) for this system. In the Majorana Approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the Reimann sphere, or a pair of unit vectors (known as Majorana vectors or M-vectors). Spin-1 states can be of three types: those whose vectors are parallel, those whose vectors are antiparallel and those whose vectors make an arbitrary angle. The types of bases possible for a spin-1 system are thus geometrically much more varied than for a spin-half system or qubit, which is the standard unit of information storage in most quantum protocols. Our derivation of the MUBs and SIC-POVMs proceeds from a recently derived expression for the squared overlap of two spin-1 states in terms of their M-vectors and the minimal additional set of assumptions that are needed. These assumptions include time-reversal invariance in the case of the MUBs and the requirement of three-fold symmetry in the case of the SIC-POVMs. The applications of these results to problems in quantum information are mentioned.

Le sujet de cette thèse est sur les propriétés non-locales des états symétriques invariant sous les permutations des systèmes et les usages potentiels de ces états dans le domaine de traitement d'information quantique. La non-localité de presque tous les états sysmétriques, hors les états de Dicke, est établie par une version étendue du paradoxe de Hardy. Grâce à la représentation de Majorana pour les états symétriques, des paramètres de mesure avec lesquels toutes les conditions du paradoxe sont satisfaites peuvent être trouvés. Une version étendue de l'inégalité de CH peut être dérivée à partir des conditions probabilistes de ce paradoxe. Cette inégalité est violée par tous les états symétriques. Les propriétés de la non-localité et les propriétés de l'intrication sont aussi discutées et comparées, notamment par rapport à la persistance et la monogamie. Des résultats indiquent que la dégénérescences de certains états symétriques est liée à la persistance, qui donne une façon d'inventer des tests qui sont indépendants des dispositifs visés pour séparer les différentes classes de non-localité. Il est aussi montré que l'inégalité utilisée pour démontrer la non-localité de tous les états symétriques n'est pas monogame dans le sens strict. Néanmoins, une nouvelle inégalité pour les états de Dicke est proposée, qui est monogame quand le nombre de participants tends vers l'infinité. Malheureusement, toutes ces inégalités sont incapables de détecter la non-localité authentique. Des applications de la non-localité à la complexité de communication et aux jeux bayésiens sont discutées. / This thesis is about the nonlocal properties of permuation symmetric states and the potential usefulness of such properties in quantum information processing. The nonlocality of almost all symmetric states, except Dicke states is shown by constructing an $n$-party Hardy paradox. With the help of the Majorana representation, suitable measurement settings can be chosen for these symmetric states which satisfy the paradox. An extended CH inequality can be derived from the probabilistic conditions of the paradox. The inequality is shown to be violated by all symmetric states. The nonlocality properties and entanglement properties of symmetric states are also discussed and compared, natbly with respect to persistency and monogamy. It is shown that te degeneracy of some symmetric states is linked to the persistency, which provides a way to use device independent tests to separte nonlocality classes. It is also shown that the inequalities used to show the nonlocality of all symmetric states are not strictly monogamous.A new inequality for Dicke states is shown to be monogamous when the number of parties goes to infinity. But all these inequalites can not detect genuine nonlocality. Applications of nonlocality to communication complexity and Bayesian game theory are also discussed.

The algebra of SU(2) is ubiquitous in physics, applicable both to the atomic spin states and the polarisation states of light. The method developed by Majorana and Schwinger to represent pure, symmetric spin-states of arbitrary value as a product of spin-1/2 states is a powerful tool that allows for a great conceptual and practical simplification. Foremost, it allows the representation of a qudit on the same geometry as a qubit, i.e., the Bloch sphere. An experimental implementation of the Majorana representation in the realm of quantum optics is presented. The technique allows the projection of arbitrary quantum states from a coherent state input. It is also shown that the method can be used to synthesise arbitrary interference patterns with unit visibility, and without resorting to quantum resources. In this context, it is argued that neither the shape nor the visibility of the interference pattern is a good measure of quantumness. It is only the measurement scheme that allows for the perceived quantum behaviour. The Majorana representation also proves useful in delineating uncertainty limits of states with a particular spin value. Issues with traditional uncertainty relations involving the SU(2) operators, such as trivial bounds for certain states and non-invariance, are thereby resolved with the presented pictorial solution. / <p>QC 20170428</p>