Title: On discrete structures in finite Hilbert spaces,
Abstract: We present a brief review of discrete structures in a finite Hilbert space,
relevant for the theory of quantum information. Unitary operator bases,
mutually unbiased bases, Clifford group and stabilizer states, discrete Wigner
function, symmetric informationally complete measurements, projective and
unitary t--designs are discussed. Some recent results in the field are covered
and several important open questions are formulated. We advocate a geometric
approach to the subject and emphasize numerous links to various mathematical
problems.