Codes and Expansions (CodEx) Seminar

Tim Kunisky (Yale University)
Dual bounds for the positive definite functions approach to mutually unbiased bases

The maximal number of complex mutually unbiased bases—collections of bases such that lines along vectors from different bases meet at a fixed angle—in dimensions other than prime powers is a long-standing open problem. Even this number in 6 dimensions has been unknown for decades: the best known upper bound is 7 bases, while the best known constructions (widely believed to be optimal) produce 3 bases. Recent work of Kolountzakis, Matolcsi, and Weiner (2016) suggested a new convex optimization technique, based on the "method of positive definite functions," for proving better bounds. They gave a new proof of the upper bound of 7 bases, and proposed constructions that might improve on this bound.

In this talk, I will give evidence that this method cannot improve on the best known upper bound. In particular, using a convex duality argument, I will show that positive definite functions that are low-degree polynomials do not give the improvement we might hope for. The proof is a computer-assisted one involving the representation theory of the unitary group; time permitting, I will also discuss the obstacles to proving stronger dual bounds and what kinds of combinatorial and representation-theoretic results would be required for such improvements.

The talk is based on joint work with Nikolaus Doppelbauer and Afonso Bandeira.