Codes and Expansions (CodEx) Seminar

Marcus Appleby (University of Sydney)
SICs and Number Theory

SICs (i.e. what are known in the mathematics literature as maximal equiangular tight frames, and in the physics literature as symmetric informationally complete positive operator valued measures) are most obviously geometric structures living in a finite dimensional complex vector space. As such they have a rich set of symmetries described by unitary operators. However, although it has yet to be proved, there is much evidence to support the conjecture that they can also be viewed as number-theoretic structures. This number-theoretic aspect is less obvious than the geometric one, and for that reason was only recently discovered. However it may in the end prove to be no less important. A SIC is specified by a set of algebraic numbers. There is a remarkable interplay between the unitary symmetries of the SIC and the Galois symmetries of the algebraic numbers specifying it. Furthermore it turns out that the numbers specifying the SIC are Galois conjugated to the numbers featuring in a major open problem in algebraic number theory (as described in Gene Kopp's talk). The purpose of the talk is to review the number theoretic aspects of SICs. No prior knowledge of Galois theory or number theory will be assumed.