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

Tuan-Yow Chien and Shayne Waldron


Nice error bases are generalisations of the Pauli matrices which have applications in quantum information theory. These orthonormal bases for the $d\times d$ matrices $M_d(\C)$ also generalise the projective action of the Heisenberg group on $\C^d$. Here we extend nice error bases to nice error frames. These are equal-norm tight frames for $M_d(\C)$ consisting of $d\times d$ 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). In particular, we give 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.

Keywords: Nice error frame, nice error basis, SIC-POVM (symmetric informationally complex positive operator valued measure), equiangular tight frame, error operator basis, Zauner's conjecture, projective representation, Heisenberg group

Math Review Classification: Primary 20C25, 42C15, 81P15, 94A15; Secondary 11L03, 14N20, 20C15, 52B11

Length: 24 pages

Last Updated: 13 February 2015