Markus Grassl: SIC-POVMs and MUBs: Sets of Quantum States with Special Properties

In the talk, I will consider the problem of finding particular sets of quantum states which play a role, for example, in quantum state tomography as well as quantum cryptography. While the problems have a simple mathematical description, they appear to be rather hard to be solved.

So-called SIC-POVMs correspond to maximal sets of equiangular lines in complex space. While we have numerical solutions in dimensions up to 67, algebraic solutions are only known for certain dimensions, and a proof of their (non)existence in all dimensions is elusive. The second class of states are complete sets of Mutually Unbiased Bases which are known how to construct in prime power dimensions, but in general we only have a lower bound of three bases for arbitrary oddly even dimensions.

More recently, the concept of unextendible MUBs has been introduced, which comes in different flavours. Constructions of such unextndible sets of MUBs with connections to finite geometry will be presented.