{"id":10176,"date":"2021-04-20T15:38:09","date_gmt":"2021-04-20T07:38:09","guid":{"rendered":"https:\/\/cantor.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:\/\/cantor.math.ntnu.edu.tw\/index.php\/2021\/04\/20\/001-21\/","title":{"rendered":"[\u5c08\u984c\u6f14\u8b1b] <\/span>\u30104\u670828\u65e5\u3011Explicit constructions of exact unitary designs"},"content":{"rendered":"

Speaker: Eiichi Bannai \u6559\u6388
\nJob title: \u7406\u8ad6\u4e2d\u5fc3\u8a2a\u554f\u7814\u7a76\u54e1\u4ee5\u53ca\u4e5d\u5dde\u5927\u5b78\u69ae\u8b7d\u6559\u6388
\nTitle : Explicit constructions of exact unitary designs
\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

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

In the latter part of my talk, I will present some of our recent results on unitary $t$-designs, including:\\\\
\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).\\\\
\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).\\\\
\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).\\\\
\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.
\nTime: April 28 (Wed.), 2:00 p.m., 2021
\nPlace: Room 212, Department of Mathematics, NTNU
\nTea Time: April 28 (Wed.), 1:30 p.m., 2021
\nTea Place: Room 107, Department of Mathematics, NTNU<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":18,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"off","ocean_gallery_id":[],"footnotes":""},"categories":[1,18,122,124],"tags":[],"class_list":["post-10176","post","type-post","status-publish","format-standard","hentry","category-news","category-events","category-gallery","category-speeches","entry"],"_links":{"self":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/10176","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/users\/18"}],"replies":[{"embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/comments?post=10176"}],"version-history":[{"count":1,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/10176\/revisions"}],"predecessor-version":[{"id":10177,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/10176\/revisions\/10177"}],"wp:attachment":[{"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/media?parent=10176"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/categories?post=10176"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cantor.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/tags?post=10176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}