Equiangular lines, mutually unbiased bases, and difference sets - Aidan Roy

In 1975, Delsarte, Goethals and Seidel found an upper bound for the size of a "system of complex lines": a set of vectors whose scalar products have only a small number of distinct absolute values. Recently, applications in quantum tomography have renewed interest in constructing systems of maximal size. We consider two particular problems: equiangular lines, which are sets in which only one absolute value occurs; and mutually unbiased bases, in which two values occur and the lines are partitioned into orthonormal bases. For real vector spaces, these systems are closely related to regular two-graphs and Hadamard matrices. However, the complex versions are not nearly as well understood. In this talk, we will see that combinatorial structures such as difference sets and Cayley graphs can also be used for constructions in complex case. In particular, we use difference sets in abelian groups to produce equiangular lines, and we use relative difference sets to produce maximal sets of mutually unbiased bases. This talk is based on joint work with Chris Godsil.