In this thesis we investigate complete systems of MUBs and SIC-POVMs. These are highly symmetric sets of vectors in Hilbert space, interesting because of their applications in quantum tomography, quantum cryptography and other areas. It is known that these objects form complex projective 2-designs, that is, they satisfy Welch bounds for k = 2 with equality. Using this fact, we derive a necessary and sufficient condition for a set of vectors to be a complete system of MUBs or a SIC-POVM. This condition uses the orthonormality of a specific set of vectors. Then we define homogeneous systems, as a special case of systems of vectors for which the condition takes an especially elegant form. We show how known results and some new results naturally follow from this construction.

In this thesis we investigate complete systems of MUBs and SIC-POVMs. These are highly symmetric sets of vectors in Hilbert space, interesting because of their applications in quantum tomography, quantum cryptography and other areas. It is known that these objects form complex projective 2-designs, that is, they satisfy Welch bounds for k = 2 with equality. Using this fact, we derive a necessary and sufficient condition for a set of vectors to be a complete system of MUBs or a SIC-POVM. This condition uses the orthonormality of a specific set of vectors. Then we define homogeneous systems, as a special case of systems of vectors for which the condition takes an especially elegant form. We show how known results and some new results naturally follow from this construction.

In this thesis for the degree of Master of Science from Stockholm University we explore the ideas of Symmetric Informationally Complete Positive Valued Measures (SIC-POVMs; commonly just SICs). This is an emerging concept in quantum information theory with ambitious claims, such as being a candidate for standard measurements [23] and perhaps being of importance to error correcting universal quantum computing [32]. While the definition of a SIC is exceedingly simple they have proven notoriously hard to find. This thesis explores new approaches to finding SICs. It is our ambition that this thesis shall provide the reader unfamiliar with SICs with a thorough introduction to the subject along with both the necessary quantum theory and group theory. We also hope to intrigue the reader already attuned to SICs by establishing a link between how close to a SIC a state is and how close to a MUS (Minimum Uncertainty State) it is. This is the main result of this thesis and we leave the reader with several open questions relating to this discovery to provoke further scrutiny of the matter. The thesis is divided into two parts: the first part provides the necessary background and theory; while the second part presents our results. There are also three appendices attached to this thesis where we delve into a discussion about computing power and also present some of the code used. Being appendices these are not essential to the thesis per se – they are rather supplied as a reference for the curious reader who might be interested in recreating some of our results.

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.