Trace functionals and operator inequalities with applications in quantum information

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400104
• 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: Frank Hansen 東北大学, 高度教養教育・学生支援機構, 教授 (00600678)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 量子情報 / Golden Thompson's ineq. / Quantum information / Statistical mechanics / Quantum entropy / geometric means / Rao's inequality

Outline of Final Research Achievements

We found a multivariate extension of Golden-Thompson’s trace inequality, which may be considered as an interpolation inequality between Golden-Thompson’s inequality and Jensen’s inequality, and we extended the two variable Golden-Thompson inequality to deformed exponentials. We proposed a general procedure to construct multivariate geometric means based on the theory of perspectives of regular operator maps. The method is general enough to encompass all known examples of multivariate geometric means, and it also provides new interesting examples. A novel feature is that we may impose an updating procedure adapted to data acquisition. We characterised the von Neumann entropy as the only possible entropic measure satisfying two fundamental properties coming from thermodynamics. We obtained an inequality for expectations of means of positive random variables.

Free Research Field

数理物理学