Codes and designs on the complex unit sphere

A "spherical code" is a collection of points on a sphere such that the distance between any two points is large. Recent work in quantum information theory has led to renewed interest in codes on the complex unit sphere, which are not nearly as well understood as codes on the real sphere. In this talk, I will present Delsarte's linear programming bounds for complex codes and explain the duality between codes and designs, and then I will describe two open problems of great interest in quantum state tomography: mutually unbiased bases and equiangular lines.