2014 Fiscal Year Research-status 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 | trace inequality / regular operator mapping / geometric mean / concave trace function |
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| Outline of Annual Research Achievements |
1. We extended Golden-Thompson’s trace inequality in two separate directions. We found a multivariate extension that may be considered an interpolation inequality between Golden-Thompson’s inequality and Jensen’s inequality. We also extended the two-variable Golden-Thompson inequality to deformed exponentials with parameters in the interval [1,3]. We believe the result is useful in the theory of non-additive entropies and non-extensive statistical mechanics.
2. Regular operator mappings share the same regularity properties as the operator function defined by the functional calculus of a real function of one variable. They are also known in the literature as non-commutative functions. We show that convex regular operator mappings satisfy a type of Jensen inequality. We define the perspectives of regular operator mappings and show that they preserve convexity. We finally prove that a convex positively homogeneous regular operator mapping of k+1 variables is the perspective of its restriction to k variables.
3. We propose a general procedure to construct multivariate geometric means based on the theory of perspectives of regular operator mappings. The method is general enough to encompass all known examples of multivariate geometric means and also provides new interesting examples. The means constructed in this way all generalize the well-known geometric mean of two operators. A novel feature of the theory is that we extend a mean from k to k+1 variables by an updating procedure that may be adapted to a specific problem. We believe this is useful in information processing.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
This year’s research may be divided into three distinct parts that all fall within the main purpose of the program as stated in the application.
1.Inequalities related to Golden-Thompson’s trace inequality.
2.Regular operator mappings.
3.Multivariate geometric means of operators.
We have thus advanced the research during the present program corresponding to one year of effort.
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| Strategy for Future Research Activity |
We intend to continue the research program in accordance with the objectives laid out in the research plan mentioned in the application.
In conjunction with this effort we have separate research programs with Paolo Gibilisco (Rome, Tor Vergata), Edward Effros (UCLA), Elliott Lieb (Princeton) and Zhihua Zhang, a Chinese post doc.
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| Causes of Carryover |
旅費の見込額との差異が生じたため。
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| Expenditure Plan for Carryover Budget |
次年度の科研費は、主に国内外で行われる学会への参加ならびに国外の科学者との研究に関する協議等を行うための旅費ならびに学会参加費として執行する予定である.
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