Speaker: Eiichi Bannai 教授
Job title: 理論中心訪問研究員以及九州大學榮譽教授
Title : Explicit constructions of exact unitary 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 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) We give the classification of unitary $t$-groups (unitary $t$-designs that are groups) (Bannai-Navarro-Rizo-Tiep, On unitary $t$-groups, J. Math. Soc. of Japan, 2020).\\
(ii) We give the 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) We give the explicit constructions of exact unitary $t$-designs in $U(n)$ for all $t$ and $n$ (Bannai-Nakata-Okuda-Zhao, Explicit constructions of exact unitary designs, arXiv: 2009.11170).\\
Our method also gives the explicit constructions of spherical $t$-designs on the sphere $S^{n-1}$ for all $t$ and $n$ by induction on $n$. Also, we mention that our explicit constructions of unitary designs have practical applications to physics. See our physics paper arXiv: 2102.12617.
Time: April 28 (Wed.), 2:00 p.m., 2021
Place: Room 212, Department of Mathematics, NTNU
Tea Time: April 28 (Wed.), 1:30 p.m., 2021
Tea Place: Room 107, Department of Mathematics, NTNU