Mathematical aspects of quantum statistical physics and quantum information theory

量子统计物理和量子信息论的数学方面

• 批准号: 17K05267
• 负责人: HANSEN FRANK
• 金额: $ 2.83万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2017
• 资助国家: 日本
• 起止时间: 2017-04-01 至 2019-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17K05267/
• 关键词: geodesic convexity; quantum information; statistical mechanics; trace function; convexity; non-commutativity

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.

今年的研究活动符合申请中所述的计划的主要目的。1.今年研究的主要结果是对测地线凸迹函数集的完整而简单的表征。该结果表明a的测地线凸性。由实函数的函数微积分给出的迹函数与底层希尔伯特空间的维数无关，该结果在机器学习、金融数学和多元算子映射等不同领域（以及量子应用）中具有广泛的凸优化应用。我们专注于最后一个应用领域，并发展了一种相当普遍的凸多元算子均值理论。2.在多元算子均值的背景下，我们引入了所谓的熵超均值及其在量子中的应用。信息论.3.我们开发了变形对数函数和指数函数的变分表示，并将其应用于量子 Tsallis 熵。我们最终将 Golden-Thompson 的迹不等式扩展到具有变形参数的变形指数。 0<q<1，从而补充了变形参数 1<q<2 和 2<q<3 的早期结果。

项目成果

Quantum entropy derived from first principles

源自第一原理的量子熵

• 发表时间: 2018
• 作者: Frank Hansen

Peierls-Bogolyubov's inequality for deformed exponentials

Peierls-Bogolyubov 变形指数不等式

• 发表时间: 2017
• 作者: Frank Hansen

Peierls-Bogolyubov's inequality for deformed exponentials

Peierls-Bogolyubov 变形指数不等式

• 发表时间: 2017
• 作者: Frank Hansen

Convex multivariate operator means

凸多元算子均值

• DOI: 10.1016/j.laa.2018.11.032
• 发表时间: 2019
• 影响因子: 1.1
• 作者: Hansen Frank

Variational representations related to Tsallis relative entropy

与 Tsallis 相对熵相关的变分表示

• 发表时间: 2019
• 作者: Hansen Frank;Guanghua Shi
• 通讯作者: Guanghua Shi