Welch Bounds and Quantum State Tomography

Abstract

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.