| 研究課題/領域番号 |
23K03230
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分12040:応用数学および統計数学関連
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| 研究機関 | 名古屋大学 |
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研究代表者 |
Buscemi F. 名古屋大学, 情報学研究科, 教授 (80570548)
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| 研究期間 (年度) |
2023-04-01 – 2026-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,680千円 (直接経費: 3,600千円、間接経費: 1,080千円)
2025年度: 1,560千円 (直接経費: 1,200千円、間接経費: 360千円)
2024年度: 1,560千円 (直接経費: 1,200千円、間接経費: 360千円)
2023年度: 1,560千円 (直接経費: 1,200千円、間接経費: 360千円)
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| キーワード | macroscopic entropy / observational entropy / retrodiction / coarse-graining / Umegaki relative entropy / Petz recovery theorem / Bayes theorem / statistical inference |
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| 研究開始時の研究の概要 |
The goal of this research proposal is to establish the mathematical foundations of observational entropy, which is a quantity that plays a central role in the unification of thermodynamic and information-theoretic entropy, and in discussing the physical properties of systems out of equilibrium.
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| 研究実績の概要 |
The properties of observational entropy as a functional of quantum states and POVMs have been studied with the tools of operator algebras and mathematical statistics. It has been found that observational entropy has many connections with Petz's theory of statistical sufficiency. Such connections allowed us to clarify the meaning of "coarse-graining" as a form of statistical retrodiction. This interpretation greatly simplifies the conceptual foundations of statistical mechanics. A paper about the theory of statistical retrodiction and its formulation in quantum theory has been published. Moreover, the idea of connecting statistical mechanics and retrodiction was at the basis of another paper, published on Physical Review Letters, in which we reformulate the second law of thermodynamics as an equilibrium condition in expected utility theory. Again in connection with observational entropy, two works of the resource theory of quantum measurements have been published: one about quantum measurements incompatibility, another one about quantum measurement sharpness. Finally, in exploring the process of retrodiction in quantum theory, we published a paper about the new idea of "virtual quantum broadcasting", which shows how time-like correlations can be treated within the formalism of quantum theory.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
The research plan is progressing exactly as anticipated, without any delay. The first paper discussing the most important mathematical properties of observational entropy has been published as expected. Moreover, there is a series of follow-up papers that are already under preparation.
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| 今後の研究の推進方策 |
In the next stage, we plan to generalize the observational entropy to priors different from the uniform distribution. We also plan to study the evolution of observational entropy in isolated systems. Finally, we plan to study the structure of macroscopic states from an algebraic viewpoint. All these projects are very important for our understanding of the foundations of statistical mechanics.
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