Nice error frames, canonical abstract error groups and the construction of SICs

Abstract

Nice error bases are generalisations of the Pauli matrices which have applications in quantum information theory. These orthonormal bases for the d×d matrices Md(C) also generalise the projective action of the Heisenberg group on Cd. Here we extend nice error bases to nice error frames. These are equal-norm tight frames for Md(C) consisting of d×d unitary matrices with a group indexing structure. We show that each nice error frame (irreducible faithful projective representation) is associated with a canonical abstract error group. This is calculated in number of examples, e.g., for all nice error bases for d<14, which then allows us to investigate which nice error bases might give rise to SICs (symmetric informationally complete positive operator valued measures). These results show that the current catalogue of nice error bases over counts. In particular, we give an explicit example of a SIC for d=6 with a nonabelian index group, and show that the Hoggar lines appear for various nice error bases, some of which are subgroups of the Clifford group. Thus all known SICs appear as orbits of subgroups of the Clifford group.