2018 Fiscal Year Annual Research Report
Mathematical aspects of quantum statistical physics and quantum information theory
| Project/Area Number |
17K05267
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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) |
2017-04-01 – 2019-03-31
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| Keywords | geodesic convexity / quantum information / statistical mechanics / trace function |
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| Outline of Annual Research Achievements |
This year’s research activities fall within the main purpose of the program as stated in the application.
1. The main result in this year’s research is a complete and simple characterization of the set of geodesically convex trace functions. The result implies that geodesic convexity of a trace function given by the functional calculus of a real function is independent of the dimension of the underlying Hilbert space. This result has vast applications for convex optimization in diverse areas like machine learning, financial mathematics and multivariate operator maps with applications in quantum physics. We have focused on the last area of applications and have developed a quite general theory of convex multivariate operator means.
2. In the context of multivariate operator means we have introduced the so-called entropic hyper-mean with applications in quantum information theory.
3. We developed variational representations for the deformed logarithmic and exponential functions and applied them to the quantum Tsallis entropy. We finally extended Golden-Thompson’s trace inequality to deformed exponentials with deformation parameter 0<q<1, thus complementing earlier results for deformation parameter 1<q<2 and 2<q<3.
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