| Project/Area Number |
15K04981
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Chubu University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 典型性原理 / 量子力学 / アルゴリズム的ランダムネス / 確率解釈 / 多世界解釈 / 確率概念の操作的特徴付け / Martin-Loefランダム性 / アルゴリズム的情報理論 |
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| Outline of Final Research Achievements |
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In this research, based on the toolkit of algorithmic randomness, we presented a refinement of the Born rule, called the principle of typicality, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. We did this from the point of view of the many-worlds interpretation of quantum mechanics. We then made applications of our framework to the BB84 quantum key distribution protocol, quantum teleportation, superdense coding, Bell's inequality vs quantum mechanics argument in order to demonstrate how properly our framework works in practical problems in quantum mechanics.
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| Academic Significance and Societal Importance of the Research Achievements |
量子力学の中核にあるボルン則（波動関数の確率解釈）とアルゴリズム的ランダムネスを直接結び付ける試みは、これまでほとんど全く存在しなかった。特に、量子力学の公理系としての形式に深く立ち入り、アルゴリズム的ランダムネスと結び付ける研究は、私の知る限り、本研究が世界的に初めてのものである。本研究はアルゴリズム的ランダムネスに基づいて量子力学の物理理論としての完全化を達成しようとするものであり、量子力学に対する人類の理解を格段に深めるものであると私は考える。本研究の基礎科学に対する貢献、そして本研究の人類の世界認識・宇宙認識に対する貢献は計り知れないものであると私は考える。
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