2016 Fiscal Year Annual Research Report
Trace functionals and operator inequalities with applications in quantum information
| Project/Area Number |
26400104
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| Research Institution | Tohoku University |
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Principal Investigator |
HANSEN FRANK 東北大学, 高度教養教育・学生支援機構, 教授 (00600678)
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | quantum entropy / GT-inequality / regular operator map / geometric mean / perspective / operator inequality |
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| Outline of Annual Research Achievements |
We extended Golden-Thompson’s trace inequality in two separate directions. We first found a multivariate extension that may be considered as an interpolation inequality between Golden-Thompson’s inequality and Jensen’s inequality. We then extended the two-variable Golden-Thompson inequality to deformed exponentials with parameters in the interval [1,3].
We defined the perspective of a regular operator map and found that it preserves convexity. By considering the filtering of a perspective through a completely positive map we discovered important multivariate operator inequalities. 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 of the theory is that we may impose an updating procedure adapted to data acquisition.
We reconsidered the von Neumann entropy and found a surprisingly simple proof of an important classical result by connecting it with the theory of matrix entropies. We then 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.
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