Applications of Quantum Groups and Representation Theory to Quantum Information Theory

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K18734
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Kyoto University
• Principal Investigator: COLLINS BENOIT 京都大学, 理学研究科, 教授 (20721418)
• Project Period (FY): 2017-06-30 – 2022-03-31
• Keywords: quantum information

Outline of Final Research Achievements

This research project was related to the study of properties of quantum channels obtained through group and quantum group symmetries. Together with a group of collaborators including Brannan and Youn, we used properties of rapid decay of compact quantum groups to obtain highly entangled quantum channels and we studied their properties. With Cadilhac we also obtained a characterisation of freeness that was partly inspired from problems of quantum expanders. On the other hand, together with my former PhD students Sapra and Al Nuwairan, we studied equivariant quantum groups. Part of this study also involved Bardet. Among others, we proved that all positivity can be detected through U(n) equivariant maps. With Cleve, Paulsen and Li, we proved that there is a macroscopic between spatial self embezzlement and with its version in the commuting framework, irrespective of the dimension.

Free Research Field

quantum information

Academic Significance and Societal Importance of the Research Achievements

本研究課題は既存研究がランダム行列などを用いた手法である意味、「偶発的に」量子情報理論で必要なものを構成していたのに対して、量子群を用いて、明示的な構成を提案し、今までに得られなかった深い結果を与えた。量子情報理論は古くから研究があるが、量子コンピュータなどに関連し、注目が高まり、理論の側面だけでなく応用の側面からも、より詳細な量子の世界の描像が求められている。今後の量子情報理論の応用のため、より深い世界を明示的に見せたことは量子の可能性を広げ、より汎用的かつ意義深く、社会に大きく貢献する量子的な手法を作り出す土台の一部になることが期待され、大きな意義があると思われる。