Matrix/operator inequalities and applications to quantum information and free probability

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400103
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: Hiai Fumio 東北大学, 情報科学研究科, 名誉教授 (30092571)
• Research Collaborator: UEDA YOSHIMICHI 九州大学, 大学院数理学研究院, 准教授 ; Bourin Jean-Christophe Universite de Franche Comte ; Ruskai M. B. University of Massachusetts Lowell, Emeritus Professor ; Audenaert K. M. R. Royal Holloway, University of London
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 関数解析学 / 作用素 / 行列解析 / 作用素環 / 量子情報 / 自由確率論

Outline of Final Research Achievements

In the course of this research project we obtained various results in the three subjects of matrix/operator analysis, quantum information, and free probability. In matrix/operator analysis, we studied the higher order extension, called k-tone functions, of operator momotone and concave functions in connection with the kth derivative of functional calculus. We introduced the notion of symmetric anti-norms for matrices/operators and discuseed the Jensen-Minkowski type cancavity and superadditivity inequalities of matrix trace/norm functions involving operator means and symmetric anti-norms. We also studied extensions of Araki's and Ando-Hiai's log-majorizations from different aspects.

In quantum information, we studied contraction coefficients for quantum channels with respect to quantum f-divergences and quantum monotone metrics. In free probability, we studied the orbital free entropy and free Fisher information from the viewpoint of variational principle based on Legendre transform.

Free Research Field

関数解析学