Although symmetric informationally complete positive operator valued measures
(SIC POVMs, or SICs for short) have been constructed in every dimension up to
67, a general existence proof remains elusive. The purpose of this paper is to
show that the SIC existence problem is equivalent to three other, on the face
of it quite different problems. Although it is still not clear whether these
reformulations of the problem will make it more tractable, we believe that the
fact that SICs have these connections to other areas of mathematics is of some
intrinsic interest. Specifically, we reformulate the SIC problem in terms of
(1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result
being a greatly strengthened version of one previously obtained by Appleby,
Flammia and Fuchs). The connection between these three reformulations is
non-trivial: It is not easy to demonstrate their equivalence directly, without
appealing to their common equivalence to SIC existence. In the course of our
analysis we obtain a number of other results which may be of some independent
interest.Comment: 36 pages, to appear in Quantum Inf. Compu