Exploring the Elegant Bell Inequality
Abstract: In quantum information, device independent certification is a matter of both practical and fundamental interest. In this thesis, we explore the use of a particular Bell inequality, known as the elegant Bell inequality, in device independent certification. We first characterize all states and measurements that can lead to a maximal violation of the elegant Bell inequality. It turns out that, in all cases, the state involved in a maximal violation is a generalized singlet state, and the measurements of one of the parties are always maximally spread out on the Bloch sphere, forming a complete set of mutually unbiased bases. The measurements involved on the other party form two pairs of symmetric informationally complete vectors. The elegant Bell inequality, then, can be used to certify the presence of these elements.We also explore the usefulness of the elegant Bell inequality in randomness certification, in particular in a protocol for certification of maximal randomness from one entangled bit.The last part of this thesis is dedicated to a study of some of the special geometricstructures involved in the maximal violation of the elegant Bell inequality, namely the symmetric informationally complete vectors. The problem of the existence and of the construction of these sets of vectors in Hilbert spaces of any dimension is open, but there are solutions available in many dimensions. We look at these structures from both a geometric and an algebraic number theory perspective, and conjecture a relation between vectors in different dimensions. We introduce the relation of "alignment" between such sets of vectors.
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