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

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