Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally-Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs.

This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements. Chapter 2 and the attached papers I and II also explore the geometric properties and symmetries of aligned symmetric informationally-complete POVMs.

Chapter 3 and the attached papers III and IV look at an application of symmetric informationally-complete POVMs, the so-called Elegant Bell inequality. We use this inequality for device-independent quantum certification, the task of characterizing quantum scenarios without modelling the devices involved in these scenarios. Bell inequalities are functions that are bound in classical theories more tightly than in quantum theories, and can thus be used to probe whether a system is quantum. We characterize all scenarios in which the Elegant Bell inequality reaches its maximum quantum value. In addition, we show that this inequality can be used for randomness certification.