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

## Tuan-Yow Chien and Shayne Waldron

## Abstract:

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

## Availability: