{"id":10176,"date":"2021-04-20T15:38:09","date_gmt":"2021-04-20T07:38:09","guid":{"rendered":"https:\/\/virtual.math.ntnu.edu.tw\/?p=10176"},"modified":"2021-04-20T15:38:09","modified_gmt":"2021-04-20T07:38:09","slug":"001-21","status":"publish","type":"post","link":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/2021\/04\/20\/001-21\/","title":{"rendered":"<span style=\"color:#3566BD\">[\u5c08\u984c\u6f14\u8b1b] <\/span>\u30104\u670828\u65e5\u3011Explicit constructions of exact unitary designs"},"content":{"rendered":"<p>Speaker: Eiichi Bannai \u6559\u6388<br \/>\nJob title: \u7406\u8ad6\u4e2d\u5fc3\u8a2a\u554f\u7814\u7a76\u54e1\u4ee5\u53ca\u4e5d\u5dde\u5927\u5b78\u69ae\u8b7d\u6559\u6388<br \/>\nTitle : Explicit constructions of exact unitary designs<br \/>\nAbstract\uff1aThe 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.<\/p>\n<p>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.)<\/p>\n<p>In the latter part of my talk, I will present some of our recent results on unitary $t$-designs, including:\\\\<br \/>\n(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).\\\\<br \/>\n(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).\\\\<br \/>\n(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).\\\\<br \/>\nOur 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.<br \/>\nTime: April 28 (Wed.), 2:00 p.m., 2021<br \/>\nPlace: Room 212, Department of Mathematics, NTNU<br \/>\nTea Time: April 28 (Wed.), 1:30 p.m., 2021<br \/>\nTea Place: Room 107, Department of Mathematics, NTNU<\/p>\n<p class=\"wpf_wrapper\"><a class=\"print_link\" href=\"\" target=\"_blank\">\u53cb\u5584\u5217\u5370<\/a><\/p><!-- .wpf_wrapper -->","protected":false},"excerpt":{"rendered":"<p>Speaker: Eiichi Bannai \u6559\u6388 Job title: \u7406\u8ad6\u4e2d\u5fc3\u8a2a\u554f\u7814\u7a76\u54e1\u4ee5\u53ca\u4e5d\u5dde\u5927\u5b78\u69ae\u8b7d\u6559 [&hellip;]<\/p>\n","protected":false},"author":18,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1,18,122,124],"tags":[],"_links":{"self":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/10176"}],"collection":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/users\/18"}],"replies":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/comments?post=10176"}],"version-history":[{"count":1,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/10176\/revisions"}],"predecessor-version":[{"id":10177,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/10176\/revisions\/10177"}],"wp:attachment":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/media?parent=10176"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/categories?post=10176"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/tags?post=10176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}