SICs and the elements of order three in the Clifford group

Len Bos and Shayne Waldron


For over a decade, there has been intensive work on the numerical and analytic construction of SICs ($d^2$ equiangular lines in $\Cd$) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vectors for such SICs are eigenvectors of symplectic operations in the Clifford group with canonical order $3$. Here we describe the Clifford group and the subgroup of symplectic operations in terms of a natural set of generators. From this, we classify all its elements of canonical order three. In particular, we show (contrary to prior claims) that there are symplectic operations of canonical order $3$ for $d\equiv 6\bmod 9$. It is as yet unknown whether these give rise to SICs.

Keywords: finite tight frames, SIC (symmetric informationally complete positive operator valued measure), ghost SIC, Heisenberg group, Clifford group, symplectic operation, complex equiangular lines, quadratic Gauss sum.

Math Review Classification: Primary 05B30, 94A12, 81P15, 81R05; Secondary 42C15, 51F25, 65D30.

Length: 33 pages

Last Updated: 20 September 2018