Speaker: Bannai Eiichi (Kyushu University)

Title: On unitary t-designs


Abstract: 

The purpose of design theory is for a given space $M$ to find good finite subsets $X$ of $M$ that approximate the whole space $M$ well. There are many design theories for various spaces $M$. If $M$ is the sphere $S^{n-1}$ then such $X$ are called spherical designs. If $M$ is the unitary group $U(d)$, then such $X$ are called unitary designs. 

In this seminar talk, we start with a brief survey on the theory of spherical $t$-designs, on what kind of problems we are interested in. Then we define unitary $t$-designs, and discuss what are the current status of the study of unitary $t$-designs. (Unitary $t$-designs are very much interested in physics, in particularly in quantum information theory.)  

In the latter part of my talk, I will present some of our recent results on unitary $t$-designs, including:
(i) The classification of unitary $t$-groups (unitary $t$-designs that are groups) (Bannai-Navarro-Rizo-Tiep, On unitary $t$-groups, to appear in J. Math. Soc. of Japan).
(ii) Explicit constructions of certain unitary $4$-designs from certain unitary $3$-groups (Bannai-Nakahara-Zhao-Zhu, Explicit constructions of certain unitary $t$-designs, J. Phys. A, 2019).
(iii) Explicit constructions of exact unitary $t$-designs in $U(n)$ for all $t$ and $n$ (Bannai-Nakata-Okuda-Zhao, in preparation).