Studies of symmetries that give special quantum states the "right to exist"

Abstract: In this thesis we study symmetric structures in Hilbert spaces known as symmetric informationally complete positive operator-valued measures (SIC-POVMs), mutually unbiased bases (MUBs), and MUB-balanced states. Our tools include symmetries such as the Weyl-Heisenberg (WH) group symmetry, Clifford unitaries, Zauner symmetry, and Galois-unitaries (g-unitaries), which are non-linear operators defined to generalize the notion of anti-unitaries.