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Although symmetric informationally complete positive operator valued measures(SIC POVMs, or SICs for short) have been constructed in every dimension up to67, a general existence proof remains elusive. The purpose of this paper is toshow that the SIC existence problem is equivalent to three other, on the faceof it quite different problems. Although it is still not clear whether thesereformulations of the problem will make it more tractable, we believe that thefact that SICs have these connections to other areas of mathematics is of someintrinsic interest. Specifically, we reformulate the SIC problem in terms of(1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second resultbeing a greatly strengthened version of one previously obtained by Appleby,Flammia and Fuchs). The connection between these three reformulations isnon-trivial: It is not easy to demonstrate their equivalence directly, withoutappealing to their common equivalence to SIC existence. In the course of ouranalysis we obtain a number of other results which may be of some independentinterest.

Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem