Research on operator inequalities via operator means and applications to quantum information theory

2023 Fiscal Year Final Research Report

• Project/Area Number: 16K05181
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Maebashi Institute of Technology
• Principal Investigator: Masatoshi ITO 前橋工科大学, 工学部, 教授 (00510702)
• Project Period (FY): 2016-04-01 – 2024-03-31
• Keywords: 作用素不等式 / 作用素平均 / 作用素単調関数 / 作用素エントロピー

Outline of Final Research Achievements

For two positive numbers or operators, the arithmetic, geometric and harmonic means are well known, and also many researchers investigate their extensions and related means. In this research, we investigate inequalities based on consideration of relations among several means and quantum information theory as their application. As a result, we get clear relations among several means, for example, the power difference mean, the Lehmer mean, the Heron mean, the weighted logarithmic mean and so on. Moreover, we have properties of quantities related to the operator entropy and the operator divergence.

Free Research Field

作用素論

Academic Significance and Societal Importance of the Research Achievements

正の実数あるいは作用素の平均については，多くの研究者により様々な角度から研究が行われている．本研究では，従来には無かった視点から多くの平均の間の関係について明らかにし，平均と関連した多くの不等式を開発した．このことは，数や作用素の平均の研究において大きな進歩をもたらした．また，作用素エントロピーや作用素ダイバージェンスに関連した量について研究成果が得られた．これらの成果により，本研究は作用素平均を用いた量子情報理論の作用素論的基礎付けに寄与することができた．