Informationally Complete Characters for Quark and Lepton Mixings

A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters \(\kappa\) of a finite group \(G\). Here we introduce a \(d\)-dimensional Hilbert space with \(d=cc(G)\), the number of conjugacy classes of \(G\). Groups under consideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a \(d\)-dimensional Pauli group over the characters of these representations. Groups with small \(d\) that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices. Groups leading to physical $CP$ violation are singled out.